Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems

28/10/2015
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I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.

Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems

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I will present some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Lie group and Lie algebra actions on symplectic and Poisson manifolds, momentum maps and their equivariance properties, first integrals associated to symmetries of Hamiltonian systems will be discussed. Reduction methods taking advantage of symmetries will be discussed.

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Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems Charles-Michel Marle Universit´e Pierre et Marie Curie Geometric Science of Information ´Ecole Polytechnique, 28-th–30-th October 2015 Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 1/84 Summary I 1. Introduction 2. The Lagrangian formalism 2.1. The Euler-Lagrange equations 2.2. Hamilton’s principle of stationary action 2.3. The Euler-Cartan theorem 3. Lagrangian symmetries 3.1. Infinitesimal symmetries of the Poincar´e-Cartan form 3.2. The Noether theorem in Lagrangian formalism 3.3. The Lagrangian momentum map 4. The Hamiltonian formalism 4.1. Hyper-regular Lagrangians 4.2. Presymplectic manifolds 4.3. The Hamiton equation 4.4. The Tulczyjew isomorphisms 4.5. The Hamiltonian formalism on symplectic manifolds 4.6. The Hamiltonian formalism on Poisson manifolds Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 2/84 Summary II 5. Hamiltonian symmetries 5.1. Presymplectic, symplectic and Poisson diffeomorphisms 5.2. Presymplectic, symplectic and Poisson vector fields 5.3. Lie algebras and Lie groups actions 5.4. Hamiltonian actions 5.5. Momentum maps of a Hamiltonian action 5.6. The Noether theorem in Hamiltonian formalism 5.7. Symplectic cocycles 5.8. First application: symmetries of the phase space 5.9. Second application: symmetries of the space of motions 6. Souriau’s thermodynamics of Lie groups 6.1. Statistical states 6.2. Action of the group of time translations 6.3. Thermodynamic equilibrium state 6.4. Generalization for a Hamiltonian Lie group action Thanks Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 3/84 Summary III The Euler-Poincar´e equation References Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 4/84 1. Introduction I present in this talk some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 5/84 1. Introduction I present in this talk some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. In parts 2 and 3 I discuss the Lagrangian formalism and Lagrangian symmetries, and in parts 4 and 5 the Hamiltonian formalism and Hamiltonian symmetries. The Tulczyjew isomorphisms, which explain some aspects of the relations between the Lagrangian and Hamiltonian formalisms, are presented at the end of part 4. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 5/84 1. Introduction I present in this talk some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. In parts 2 and 3 I discuss the Lagrangian formalism and Lagrangian symmetries, and in parts 4 and 5 the Hamiltonian formalism and Hamiltonian symmetries. The Tulczyjew isomorphisms, which explain some aspects of the relations between the Lagrangian and Hamiltonian formalisms, are presented at the end of part 4. Part 6 discusses Jean-Marie Souriau’s theory of Thermodynamics on Lie groups. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 5/84 1. Introduction I present in this talk some tools in Symplectic and Poisson Geometry in view of their applications in Geometric Mechanics and Mathematical Physics. In parts 2 and 3 I discuss the Lagrangian formalism and Lagrangian symmetries, and in parts 4 and 5 the Hamiltonian formalism and Hamiltonian symmetries. The Tulczyjew isomorphisms, which explain some aspects of the relations between the Lagrangian and Hamiltonian formalisms, are presented at the end of part 4. Part 6 discusses Jean-Marie Souriau’s theory of Thermodynamics on Lie groups. Finally, the Euler-Poincar´e equation is presented in an Appendix. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 5/84 2. The Lagrangian formalism The principles of Mechanics were stated by the great English mathematician Isaac Newton (1642–1727) in his book Philosophia Naturalis Principia Mathematica published in 1687 [27]. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 6/84 2. The Lagrangian formalism The principles of Mechanics were stated by the great English mathematician Isaac Newton (1642–1727) in his book Philosophia Naturalis Principia Mathematica published in 1687 [27]. On this basis, a little more than a century later, Joseph Louis Lagrange (1736–1813) in his book M´ecanique analytique [16] derived the equations (today known as the Euler-Lagrange equations) which govern the motion of a mechanical system made of any number of material points or rigid material bodies, eventually submitted to external forces, interacting between themselves by very general forces. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 6/84 2. The Lagrangian formalism The principles of Mechanics were stated by the great English mathematician Isaac Newton (1642–1727) in his book Philosophia Naturalis Principia Mathematica published in 1687 [27]. On this basis, a little more than a century later, Joseph Louis Lagrange (1736–1813) in his book M´ecanique analytique [16] derived the equations (today known as the Euler-Lagrange equations) which govern the motion of a mechanical system made of any number of material points or rigid material bodies, eventually submitted to external forces, interacting between themselves by very general forces. The configuration space and the space of kinematic states of the system are, respectively, a smooth n-dimensional manifold N and its tangent bundle TN, which is 2n-dimensional. In local coordinates a configuration of the system is determined by the n coordinates x1, . . . , xn of a point in N, and a kinematic state by the 2n coordinates x1, . . . , xn, v1, . . . vn of a vector tangent to N at some point in N. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 6/84 2. The Lagrangian formalism 2.1. The Euler-Lagrange equations When the mechanical system is conservative, the Euler-Lagrange equations involve a single real valued function L called the Lagrangian of the system, defined on the product of the real line R (spanned by the variable t representing the time) with the manifold TN of kinematic states of the system. In local coordinates, the Lagrangian L is expressed as a function of the 2n + 1 variables, t, x1, . . . , xn, v1, . . . , vn and the Euler-Lagrange equations have the remarkably simple form d dt ∂L ∂vi t, x(t), v(t) − ∂L ∂xi t, x(t), v(t) = 0 , 1 i n , where x(t) stands for x1(t), . . . , xn(t) and v(t) for v1(t), . . . , vn(t) with, of course, vi (t) = dxi (t) dt , 1 i n . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 7/84 2. The Lagrangian formalism 2.2. Hamilton’s principle of stationary action The great Irish mathematician William Rowan Hamilton (1805–1865) observed [8, 9] that the Euler-Lagrange equations can be obtained by applying the standard techniques of Calculus of Variations, due to Leonhard Euler (1707–1783) and Joseph Louis Lagrange, to the action integral IL(γ) = t1 t0 L t, x(t), v(t) = dx(t) dt dt , where γ : [t0, t1] → N is a smooth curve in N parametrized by the time t. These equations express the fact that the action integral IL(γ) is stationary with respect to any smooth infinitesimal variation of γ with fixed end-points t0, γ(t0) and t1, γ(t1) . This fact is today called Hamilton’s principle of stationary action. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 8/84 2. The Lagrangian formalism 2.2. Hamilton’s principle of stationary action The great Irish mathematician William Rowan Hamilton (1805–1865) observed [8, 9] that the Euler-Lagrange equations can be obtained by applying the standard techniques of Calculus of Variations, due to Leonhard Euler (1707–1783) and Joseph Louis Lagrange, to the action integral IL(γ) = t1 t0 L t, x(t), v(t) = dx(t) dt dt , where γ : [t0, t1] → N is a smooth curve in N parametrized by the time t. These equations express the fact that the action integral IL(γ) is stationary with respect to any smooth infinitesimal variation of γ with fixed end-points t0, γ(t0) and t1, γ(t1) . This fact is today called Hamilton’s principle of stationary action. This principle does not appear explicitly in Lagrange’s book in which the Euler-Lagrange equations are obtained by a very clever evaluation of the virtual work of inertial forces for a smooth infinitesimal variation of the motion. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 8/84 2. The Lagrangian formalism 2.3. The Euler-Cartan theorem The Lagrangian formalism is the use of Hamilton’s principle of stationary action for the derivation of the equations of motion of a system. It is widely used in Mathematical Physics, often with more general Lagrangians involving more than one independent variable and higher order partial derivatives of dependent variables. For simplicity I will consider here only the Lagrangians of (maybe time dependent) conservative mechanical systems. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 9/84 2. The Lagrangian formalism 2.3. The Euler-Cartan theorem The Lagrangian formalism is the use of Hamilton’s principle of stationary action for the derivation of the equations of motion of a system. It is widely used in Mathematical Physics, often with more general Lagrangians involving more than one independent variable and higher order partial derivatives of dependent variables. For simplicity I will consider here only the Lagrangians of (maybe time dependent) conservative mechanical systems. An intrinsic geometric expression of the Euler-Lagrange equations, wich does not use local coordinates, was obtained by the great French mathematician ´Elie Cartan (1869–1951). Let T∗N be the cotangent space to the configuration manifold N (often called the phase space of the mechanical system), θN be its Liouville 1-form, LL = dvertL : R × TN → T∗N be the Legendre map and E : R × TN → R be the energy function EL(t, v) = dvertL(t, v), v − L(t, v) , v ∈ TN . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 9/84 The Lagrangian formalism (5) 2.3. The Euler-Cartan theorem (2) The 1-form on R × TN L = L∗ LθN − EL(t, v)dt is called the Euler-Poincar´e 1-form. The Euler-Cartan theorem, due to ´Elie Cartan, asserts that the action integral IL(γ) is stationary at a smooth parametrized curve γ : [t0, t1] → N, with respect to smooth infinitesimal variations of γ with fixed end-points, if and only if i d dt t, dγ(t) dt d L t, dγ(t) dt = 0 . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 10/84 The Lagrangian formalism (5) 2.3. The Euler-Cartan theorem (2) The 1-form on R × TN L = L∗ LθN − EL(t, v)dt is called the Euler-Poincar´e 1-form. The Euler-Cartan theorem, due to ´Elie Cartan, asserts that the action integral IL(γ) is stationary at a smooth parametrized curve γ : [t0, t1] → N, with respect to smooth infinitesimal variations of γ with fixed end-points, if and only if i d dt t, dγ(t) dt d L t, dγ(t) dt = 0 . In his beautiful book [?], Jean-Marie Souriau uses a slightly different terminology: for him the odd-dimensional space R × TN is the evolution space of the system, and the exact 2-form d L on that space is the Lagrange form. He defines that 2-form in a setting more general than that of the Lagrangian formalism. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 10/84 3. Lagrangian symmetries Let N be the configuration space of a conservative Lagrangian mechanical system with a smooth Lagrangian, maybe time dependent, L : R × TN → R. Let L be the Poincar´e-Cartan 1-form on the evolution space R × TN. Several kinds of symmetries can be defined, which very often are special cases of infinitesimal symmetries of the Poincar´e-Cartan form, which play an important part in the famous Noether theorem. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 11/84 3. Lagrangian symmetries Let N be the configuration space of a conservative Lagrangian mechanical system with a smooth Lagrangian, maybe time dependent, L : R × TN → R. Let L be the Poincar´e-Cartan 1-form on the evolution space R × TN. Several kinds of symmetries can be defined, which very often are special cases of infinitesimal symmetries of the Poincar´e-Cartan form, which play an important part in the famous Noether theorem. 3.1. Infinitesimal symmetries of the Poincar´e-Cartan form Definition An infinitesimal symmetry of the Poincar´e-Cartan form L is a vector field Z on R × TN such that L(Z) L = 0 , L(Z) denoting the Lie derivative of differential forms with respect to Z. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 11/84 3. Lagrangian symmetries 3.1. Infinitesimal symmetries of the Poincar´e-Cartan form (2) Examples 1 Let us assume that the Lagrangian L does not depend on the time t ∈ R, i.e. is a smooth function on TN. The vector field on R × TN denoted by ∂ ∂t , whose projection on R is equal to 1 and whose projection on TN is 0, is an infinitesimal symmetry of L. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 12/84 3. Lagrangian symmetries 3.1. Infinitesimal symmetries of the Poincar´e-Cartan form (2) Examples 1 Let us assume that the Lagrangian L does not depend on the time t ∈ R, i.e. is a smooth function on TN. The vector field on R × TN denoted by ∂ ∂t , whose projection on R is equal to 1 and whose projection on TN is 0, is an infinitesimal symmetry of L. 2 Let X be a smooth vector field on N and X be its canonical lift to the tangent bundle TN. We still assume that L does not depend on the time t. Moreover we assume that X is an infinitesimal symmetry of the Lagrangian L, i.e. that L(X)L = 0. Considered as a vector field on R × TN whose projection on the factor R is 0, X is an infinitesimal symmetry of L. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 12/84 3. Lagrangian symmetries 3.2. The Noether theorem in Lagrangian formalism Theorem (E. Noether’s theorem in Lagrangian formalism) Let Z be an infinitesimal symmetry of the Poincar´e-Cartan form L. For each possible motion γ : [t0, t1] → N of the Lagrangian system, the function, defined on R × TN, i(Z) L keeps a constant value along the parametrized curve t → t, dγ(t) dt . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 13/84 3. Lagrangian symmetries 3.2. The Noether theorem in Lagrangian formalism Theorem (E. Noether’s theorem in Lagrangian formalism) Let Z be an infinitesimal symmetry of the Poincar´e-Cartan form L. For each possible motion γ : [t0, t1] → N of the Lagrangian system, the function, defined on R × TN, i(Z) L keeps a constant value along the parametrized curve t → t, dγ(t) dt . Example When the Lagrangian L does not depend on time, application of Emmy Noether’s theorem to the vector field ∂ ∂t shows that the energy EL remains constant during any possible motion of the system, since i ∂ ∂t L = −EL. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 13/84 3. Lagrangian symmetries 3.2. The Noether theorem in Lagrangian formalism (2) Remark There exists many generalizations of the Noether theorem. For example, if instead of being an infinitesimal symmetry of L, i.e. instead of satisfying L(Z) L = 0 the vector field Z satisfies L(Z) L = df , where f : R × TM → R is a smooth function, which implies of course L(Z)(d L) = 0 , the function i(Z) L − f keeps a constant value along t → t, dγ(t) dt . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 14/84 3. Lagrangian symmetries 3.3. The Lagrangian momentum map The Lie bracket of two infinitesimal symmetries of L is also an infinitesimal symmetry of L. Let us therefore assume that there exists a finite dimensional Lie algebra of vector fields on R × TN whose elements are infinitesimal symmetries of L. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 15/84 3. Lagrangian symmetries 3.3. The Lagrangian momentum map The Lie bracket of two infinitesimal symmetries of L is also an infinitesimal symmetry of L. Let us therefore assume that there exists a finite dimensional Lie algebra of vector fields on R × TN whose elements are infinitesimal symmetries of L. Definition Let ψ : G → A1(R × TN) be a Lie algebras homomorphism of a finite-dimensional real Lie algebra G into the Lie algebra of smooth vector fields on R × TN such that, for each X ∈ G, ψ(X) is an infinitesimal symmetry of L. The Lie algebras homomorphism ψ is said to be a Lie algebra action on R × TN by infinitesimal symmetries of L. The map KL : R × TN → G∗, which takes its values in the dual G∗ of the Lie algebra G, defined by KL(t, v), X = i ψ(X) L(t, v) , (t, v) ∈ R × TN , is called the Lagrangian momentum of the Lie algebra action ψ. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 15/84 3. Lagrangian symmetries 3.3. The Lagrangian momentum map (2) Corollary (of E. Noether’s theorem) Let ψ : G → A1(R × TM) be an action of a finite-dimensional real Lie algebra G on the evolution space R × TN of a conservative Lagrangian system, by infinitesimal symmetries of the Poincar´e-Cartan form L. For each possible motion γ : [t0, t1] → N of that system, the Lagrangian momentum map KL keeps a constant value along the parametrized curve t → t, dγ(t) dt . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 16/84 3. Lagrangian symmetries 3.3. The Lagrangian momentum map (3) Example Let us assume that the Lagrangian L does not depend explicitly on the time t and is invariant by the canonical lift to the tangent bundle of the action on N of the six-dimensional group of Euclidean diplacements (rotations and translations) of the physical space. The corresponding infinitesimal action of the Lie algebra of infinitesimal Euclidean displacements (considered as an action on R × TN, the action on the factor R being trivial) is an action by infinitesimal symmetries of L. The six components of the Lagrangian momentum map are the three components of the total linear momentum and the three components of the total angular momentum. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 17/84 3. Lagrangian symmetries 3.3. The Lagrangian momentum map (3) Example Let us assume that the Lagrangian L does not depend explicitly on the time t and is invariant by the canonical lift to the tangent bundle of the action on N of the six-dimensional group of Euclidean diplacements (rotations and translations) of the physical space. The corresponding infinitesimal action of the Lie algebra of infinitesimal Euclidean displacements (considered as an action on R × TN, the action on the factor R being trivial) is an action by infinitesimal symmetries of L. The six components of the Lagrangian momentum map are the three components of the total linear momentum and the three components of the total angular momentum. Remark These results are valid without any assumption of hyper-regularity of the Lagrangian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 17/84 4. The Hamiltonian formalism The Lagrangian formalism can be applied to any smooth Lagrangian. Its application yields second order differential equations on R × TN (in local coordinates, the Euler-Lagrange equations) which in general are not solved with respect to the second order derivatives of the unknown functions with respect to time. The classical existence and unicity theorems for the solutions of differential equations (such as the Cauchy-Lipschitz theorem) therefore cannot be applied to these equations. 1 Lagrange obtained however Hamilton’s equations before Hamilton, but only in a special case, for the slow “variations of constants” such as the orbital parameters of planets in the solar system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 18/84 4. The Hamiltonian formalism The Lagrangian formalism can be applied to any smooth Lagrangian. Its application yields second order differential equations on R × TN (in local coordinates, the Euler-Lagrange equations) which in general are not solved with respect to the second order derivatives of the unknown functions with respect to time. The classical existence and unicity theorems for the solutions of differential equations (such as the Cauchy-Lipschitz theorem) therefore cannot be applied to these equations. Under the additional assumption that the Lagrangian is hyper-regular, a very clever change of variables discovered by William Rowan Hamilton 1 allows a new formulation of these equations in the framework of symplectic geometry. The Hamiltonian formalism is the use of these new equations. It was later generalized independently of the Lagrangian formalism. 1 Lagrange obtained however Hamilton’s equations before Hamilton, but only in a special case, for the slow “variations of constants” such as the orbital parameters of planets in the solar system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 18/84 4. The Hamiltonian formalism 4.1. Hyper-regular Lagrangians Assume that for each fixed value of the time t ∈ R, the map v → LL(t, v) is a smooth diffeomorphism of the tangent bundle TN onto the cotangent bundle T∗N. Equivalent assumption: the map (idR, LL) : (t, v) → t, LL(t, v) is a smooth diffeomorphism of R × TN onto R × T∗N. The Lagrangian L is then said to be hyper-regular. The equations of motion can be written on R × T∗N instead of R × TN. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 19/84 4. The Hamiltonian formalism 4.1. Hyper-regular Lagrangians Assume that for each fixed value of the time t ∈ R, the map v → LL(t, v) is a smooth diffeomorphism of the tangent bundle TN onto the cotangent bundle T∗N. Equivalent assumption: the map (idR, LL) : (t, v) → t, LL(t, v) is a smooth diffeomorphism of R × TN onto R × T∗N. The Lagrangian L is then said to be hyper-regular. The equations of motion can be written on R × T∗N instead of R × TN. Let HL : R × T∗N → R be the function, called the Hamiltonian associated to the Lagrangian L, HL(t, p) = EL ◦ (idR, LL)−1 (t, p) , t ∈ R , p ∈ T∗ N , EL : R × TN → R being the energy function. The Poincar´e-Cartan 1-form L on R × TN is the pull-back, by the diffeomorphism (idR, LL) : R × TN → R × T∗N, of the 1-form on R × T∗N H = θN − Hdt , where θN is the Liouville 1-form on T∗N. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 19/84 4. The Hamiltonian formalism 4.2. Presymplectic manifolds The 1-form HL on R × T∗N is called the Poincar´e-Cartan 1-form in Hamiltonian formalism. It is related to the Poincar´e-Cartan 1-form L on R × TN, called the Poincar´e-Cartan 1-form in Lagrangian formalism, by L = (idR, LL)∗ HL . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 20/84 4. The Hamiltonian formalism 4.2. Presymplectic manifolds The 1-form HL on R × T∗N is called the Poincar´e-Cartan 1-form in Hamiltonian formalism. It is related to the Poincar´e-Cartan 1-form L on R × TN, called the Poincar´e-Cartan 1-form in Lagrangian formalism, by L = (idR, LL)∗ HL . The exterior derivatives d L and d HL of the Poincar´e-Cartan 1-forms in the Lagrangian and Hamiltonian formalisms both are presymplectic 2-forms on the odd-dimensional manifolds R × TN and R × T∗N, respectively. At any point of these manifolds, the kernels of these closed 2 forms are 1-dimensional, therefore determine a foliation into smooth curves of these manifolds. The Euler-Cartan theorem shows that each of these curves is a possible motion of the system, described either in the Lagrangian formalism, or in the Hamiltonian formalism, respectively. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 20/84 4. The Hamiltonian formalism 4.2. Presymplectic manifolds (2) The set of all possible motions of the system, called by Jean-Marie Souriau the manifold of motions of the system, is described in the Lagrangian formalism by the quotient of the Lagrangian evolution space R × TM by its foliation into curves determined by ker d L, and in the Hamiltonian formalism by the quotient of the Hamiltonian evolution space R × T∗M by its foliation into curves determined by ker d HL . Both are (maybe non-Hausdorff) symplectic manifolds, the projections on these quotient manifolds of the presymplectic forms d L and d h both being symplectic forms. Of course the diffeomorphism (idR, LL) : R × TN → R × T∗N projects onto a symplectomorphism between the Lagrangian and Hamiltonian descriptions of the manifold of motions of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 21/84 4. The Hamiltonian formalism 4.3. The Hamiton equation Let ψ : [t0, t1] → T∗N be the map ψ(t) = LL t, dγ(t) dt . Since d H = dθN − dHL ∧ dt, the parametrized curve t → γ(t) is a motion of the system if and only if the parametrized curve t → ψ(t) satisfies both    i dψ(t) dt dθN = −dHL t , d dt HL t, ψ(t) = ∂HL ∂t t, ψ(t) , where dHL t = dHL − ∂HL ∂t dt is the differential of the function HL t : T∗N → R in which the time t is considered as a parameter with respect to which there is no differentiation. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 22/84 4. The Hamiltonian formalism 4.3. The Hamilton equation (2) The first equation i dψ(t) dt dθN = −dHL t is the Hamilton equation. In local coordinates x1, . . . , xn, p1, . . . pn on T∗N associated to the local coordinates x1, . . . , xn on N, it is expressed as    dxi (t) dt = ∂HL(t, x, p) ∂pi , dpi (t) dt = − ∂HL(t, x, p) ∂xi , 1 i n . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 23/84 4. The Hamiltonian formalism 4.3. The Hamilton equation (2) The first equation i dψ(t) dt dθN = −dHL t is the Hamilton equation. In local coordinates x1, . . . , xn, p1, . . . pn on T∗N associated to the local coordinates x1, . . . , xn on N, it is expressed as    dxi (t) dt = ∂HL(t, x, p) ∂pi , dpi (t) dt = − ∂HL(t, x, p) ∂xi , 1 i n . The second equation d dt HL t, ψ(t) = ∂HL ∂t t, ψ(t) is the energy equation. It is automatically satisfied when the Hamilton equation is satisfied. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 23/84 4. The Hamiltonian formalism 4.3. The Hamilton equation (3) The 2-form dθN is a symplectic form on the cotangent bundle T∗N, called its canonical symplectic form. We have shown that when the Lagrangian L is hyper-regular, the equations of motion can be written in three equivalent manners: 1 as the Euler-Lagrange equations on R × TM, Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 24/84 4. The Hamiltonian formalism 4.3. The Hamilton equation (3) The 2-form dθN is a symplectic form on the cotangent bundle T∗N, called its canonical symplectic form. We have shown that when the Lagrangian L is hyper-regular, the equations of motion can be written in three equivalent manners: 1 as the Euler-Lagrange equations on R × TM, 2 as the equations given by the kernels of the presymplectic forms d L or d HL which determine the foliations into curves of the evolution spaces R × TM in the Lagrangian formalism, or R × T∗M in the Hamiltonian formalism, Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 24/84 4. The Hamiltonian formalism 4.3. The Hamilton equation (3) The 2-form dθN is a symplectic form on the cotangent bundle T∗N, called its canonical symplectic form. We have shown that when the Lagrangian L is hyper-regular, the equations of motion can be written in three equivalent manners: 1 as the Euler-Lagrange equations on R × TM, 2 as the equations given by the kernels of the presymplectic forms d L or d HL which determine the foliations into curves of the evolution spaces R × TM in the Lagrangian formalism, or R × T∗M in the Hamiltonian formalism, 3 as the Hamilton equation associated to the Hamiltonian HL on the symplectic manifold (T∗N, dθN), often called the phase space of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 24/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms Around 1974, W.M. Tulczyjew [35, 36] discovered 2 two remarkable vector bundles isomorphisms αN : TT∗N → T∗TN and βN : TT∗N → T∗T∗N. 2 βN was probably known long before 1974, but I believe that αN , much more hidden, was noticed by Tulczyjew for the first time. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 25/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms Around 1974, W.M. Tulczyjew [35, 36] discovered 2 two remarkable vector bundles isomorphisms αN : TT∗N → T∗TN and βN : TT∗N → T∗T∗N. The first one αN is an isomorphism of the bundle (TT∗N, TπN, TN) onto the bundle (T∗TN, πTN, TN), while the second βN is an isomorphism of the bundle (TT∗N, τT∗N, T∗N) onto the bundle (T∗T∗N, πT∗N, T∗N). T∗T∗N πT∗N  TT∗N βN oo τT∗Nyy TπN %% αN // T∗TN πTN  T∗N πN %% TN τN yy N 2 βN was probably known long before 1974, but I believe that αN , much more hidden, was noticed by Tulczyjew for the first time. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 25/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (2) Since they are the total spaces of cotangent bundles, the manifolds T∗TN and T∗T∗N are endowed with the Liouville 1-forms θTN and θT∗N, and with the canonical symplectic forms dθTN and dθT∗N, respectively. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 26/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (2) Since they are the total spaces of cotangent bundles, the manifolds T∗TN and T∗T∗N are endowed with the Liouville 1-forms θTN and θT∗N, and with the canonical symplectic forms dθTN and dθT∗N, respectively. Using the isomorphisms αN and βN, we can therefore define on TT∗N two 1-forms α∗ NθTN and β∗ NθT∗N, and two symplectic 2-forms α∗ N(dθTN) and β∗ N(dθT∗N). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 26/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (2) Since they are the total spaces of cotangent bundles, the manifolds T∗TN and T∗T∗N are endowed with the Liouville 1-forms θTN and θT∗N, and with the canonical symplectic forms dθTN and dθT∗N, respectively. Using the isomorphisms αN and βN, we can therefore define on TT∗N two 1-forms α∗ NθTN and β∗ NθT∗N, and two symplectic 2-forms α∗ N(dθTN) and β∗ N(dθT∗N). The very remarkable property of the isomorphisms αN and βN is that the two symplectic forms so obtained on TT∗N are equal! α∗ N(dθTN) = β∗ N(dθT∗N) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 26/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (2) Since they are the total spaces of cotangent bundles, the manifolds T∗TN and T∗T∗N are endowed with the Liouville 1-forms θTN and θT∗N, and with the canonical symplectic forms dθTN and dθT∗N, respectively. Using the isomorphisms αN and βN, we can therefore define on TT∗N two 1-forms α∗ NθTN and β∗ NθT∗N, and two symplectic 2-forms α∗ N(dθTN) and β∗ N(dθT∗N). The very remarkable property of the isomorphisms αN and βN is that the two symplectic forms so obtained on TT∗N are equal! α∗ N(dθTN) = β∗ N(dθT∗N) . The 1-forms α∗ NθTN and β∗ NθT∗N are not equal, their difference is the differential of a smooth function. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 26/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (3) Let L : TN → R and H : T∗ → R be two smooth real valued functions, defined on TN and on T∗N, respectively. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 27/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (3) Let L : TN → R and H : T∗ → R be two smooth real valued functions, defined on TN and on T∗N, respectively. The graphs dL(TN) and dH(T∗N) of their differentials are Lagrangian submanifolds of the symplectic manifolds (T∗TN, dθTN) and (T∗T∗N, dθT∗N). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 27/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (3) Let L : TN → R and H : T∗ → R be two smooth real valued functions, defined on TN and on T∗N, respectively. The graphs dL(TN) and dH(T∗N) of their differentials are Lagrangian submanifolds of the symplectic manifolds (T∗TN, dθTN) and (T∗T∗N, dθT∗N). Their pull-backs α−1 N dL(TN) and β−1 N dH(T∗N) by the symplectomorphisms αN and βN are therefore two Lagrangian submanifolds of the manifold TT∗N endowed with the symplectic form α∗ N(dθTN), which is equal to the symplectic form β∗ N(dθT∗N). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 27/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (3) Let L : TN → R and H : T∗ → R be two smooth real valued functions, defined on TN and on T∗N, respectively. The graphs dL(TN) and dH(T∗N) of their differentials are Lagrangian submanifolds of the symplectic manifolds (T∗TN, dθTN) and (T∗T∗N, dθT∗N). Their pull-backs α−1 N dL(TN) and β−1 N dH(T∗N) by the symplectomorphisms αN and βN are therefore two Lagrangian submanifolds of the manifold TT∗N endowed with the symplectic form α∗ N(dθTN), which is equal to the symplectic form β∗ N(dθT∗N). The following theorem enlightens some aspects of the relationships between the Hamiltonian and the Lagrangian formalisms. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 27/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (4) Theorem (W.M. Tulczyjew) Let XH : T∗N → TT∗N the Hamiltonian vector field on the symplectic manifold (T∗N, dθN) associated to the Hamiltonian H : T∗N → R, defined by i(XH)dθN = −dH. Then XH(T∗ N) = β−1 N dH(T∗ N) . Moreover, the equality α−1 N dL(TN) = β−1 N dH(T∗ N) if and only if the Lagrangian L is hyper-regular and such that dH = d EL ◦ L−1 L , where LL : TN → T∗N is the Legendre map and EL : TN → R the energy associated to the Lagrangian L. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 28/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (5) When L is not hyper-regular, α−1 N dL(TN) still is a Lagrangian submanifold of the symplectic manifold TT∗N, α∗ N(dθTN) , but it is no more the graph of a smooth vector field XH defined on T∗N. Tulczyjew proposes to consider this Lagrangian submanifold as an implicit Hamilton equation on T∗N. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 29/84 4. The Hamiltonian formalism 4.4. The Tulczyjew isomorphisms (5) When L is not hyper-regular, α−1 N dL(TN) still is a Lagrangian submanifold of the symplectic manifold TT∗N, α∗ N(dθTN) , but it is no more the graph of a smooth vector field XH defined on T∗N. Tulczyjew proposes to consider this Lagrangian submanifold as an implicit Hamilton equation on T∗N. These results can be extended to Lagrangians and Hamiltonians which may depend on time. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 29/84 4. The Hamiltonian formalism 4.5. The Hamiltonian formalism on symplectic manifolds In pure mathematics as well as in applications of mathematics to Mechanics and Physics, symplectic manifolds other than cotangent bundles are encountered. A theorem due to the french mathematician Gaston Darboux (1842–1917) asserts that any symplectic manifold (M, ω) is of even dimension 2n and is locally isomorphic to the cotangent bundle to a n-dimensional manifold: in a neighbourhood of each of its point there exist local coordinates (x1, . . . , xn, p1, . . . , pn) with which the symplectic form ω is expressed exactly as the canonical symplectic form of a cotangent bundle: ω = n i=1 dpi ∧ dxi . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 30/84 4. The Hamiltonian formalism 4.5. The Hamiltonian formalism on symplectic manifolds (2) Let (M, ω) be a symplectic manifold and H : R × M → R a smooth function, said to be a time-dependent Hamiltonian. It determines a time-dependent Hamiltonian vector field XH on M, such that i(XH)ω = −dHt , Ht : M → R being the function H in which the variable t is considered as a parameter with respect to which no differentiation is made. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 31/84 4. The Hamiltonian formalism 4.5. The Hamiltonian formalism on symplectic manifolds (2) Let (M, ω) be a symplectic manifold and H : R × M → R a smooth function, said to be a time-dependent Hamiltonian. It determines a time-dependent Hamiltonian vector field XH on M, such that i(XH)ω = −dHt , Ht : M → R being the function H in which the variable t is considered as a parameter with respect to which no differentiation is made. The Hamilton equation determined by H is the differential equation dψ(t) dt = XH t, ψ(t) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 31/84 4. The Hamiltonian formalism 4.5. The Hamiltonian formalism on symplectic manifolds (2) Let (M, ω) be a symplectic manifold and H : R × M → R a smooth function, said to be a time-dependent Hamiltonian. It determines a time-dependent Hamiltonian vector field XH on M, such that i(XH)ω = −dHt , Ht : M → R being the function H in which the variable t is considered as a parameter with respect to which no differentiation is made. The Hamilton equation determined by H is the differential equation dψ(t) dt = XH t, ψ(t) . The Hamiltonian formalism can therefore be applied to any smooth, maybe time dependent Hamiltonian on M, even when there is no associated Lagrangian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 31/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds The Hamiltonian formalism is not limited to symplectic manifolds: it can be applied, for example, to Poisson manifolds [20]. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 32/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds The Hamiltonian formalism is not limited to symplectic manifolds: it can be applied, for example, to Poisson manifolds [20]. Definition A Poisson manifold is a smooth manifold P whose algebra of smooth functions C∞(P, R) is endowed with a bilinear composition law, called the Poisson bracket, which associates to any pair (f , g) of smooth functions on P another smooth function denoted by {f , g}, that composition satisfying the three properties 1 it is skew-symmetric, {g, f } = −{f , g}, Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 32/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds The Hamiltonian formalism is not limited to symplectic manifolds: it can be applied, for example, to Poisson manifolds [20]. Definition A Poisson manifold is a smooth manifold P whose algebra of smooth functions C∞(P, R) is endowed with a bilinear composition law, called the Poisson bracket, which associates to any pair (f , g) of smooth functions on P another smooth function denoted by {f , g}, that composition satisfying the three properties 1 it is skew-symmetric, {g, f } = −{f , g}, 2 it satisfies the Jacobi identity f , {g, h} + g, {h, f } + h, {f , g} = 0, Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 32/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds The Hamiltonian formalism is not limited to symplectic manifolds: it can be applied, for example, to Poisson manifolds [20]. Definition A Poisson manifold is a smooth manifold P whose algebra of smooth functions C∞(P, R) is endowed with a bilinear composition law, called the Poisson bracket, which associates to any pair (f , g) of smooth functions on P another smooth function denoted by {f , g}, that composition satisfying the three properties 1 it is skew-symmetric, {g, f } = −{f , g}, 2 it satisfies the Jacobi identity f , {g, h} + g, {h, f } + h, {f , g} = 0, 3 it satisfies the Leibniz identity {f , gh} = {f , g}h + g{f , h}. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 32/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds (2) On a Poisson manifold P, the Poisson bracket {f , g} of two smooth functions f and g can be expressed by means of a smooth field of bivectors Λ: {f , g} = Λ(df , dg) , f and g ∈ C∞ (P, R) , called the Poisson bivector field of P. The considered Poisson manifold is denoted by (P, Λ). The Poisson bivector field Λ satisfies [Λ, Λ] = 0 , where the bracket [ , ] in the left hand side is the Schouten-Nijenhuis bracket. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 33/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds (2) On a Poisson manifold P, the Poisson bracket {f , g} of two smooth functions f and g can be expressed by means of a smooth field of bivectors Λ: {f , g} = Λ(df , dg) , f and g ∈ C∞ (P, R) , called the Poisson bivector field of P. The considered Poisson manifold is denoted by (P, Λ). The Poisson bivector field Λ satisfies [Λ, Λ] = 0 , where the bracket [ , ] in the left hand side is the Schouten-Nijenhuis bracket. It determines a vector bundle morphism Λ : T∗P → TP, defined by Λ(η, ζ) = ζ, Λ (η) , where η and ζ ∈ T∗P are two covectors attached to the same point in P. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 33/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds (3) Let (P, Λ) be a Poisson manifold. A (maybe time-dependent) vector field on P can be associated to each (maybe time-dependent) smooth function H : R × P → R. It is called the Hamiltonian vector field associated to the Hamiltonian H, and denoted by XH. Its expression is XH(t, x) = Λ (x) dHt(x) , where dHt(x) = dH(t, x) − ∂H(t, x) ∂t dt is the differential of the function deduced from H by considering t as a parameter with respect to which no differentiation is made. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 34/84 4. The Hamiltonian formalism 4.6. The Hamiltonian formalism on Poisson manifolds (3) Let (P, Λ) be a Poisson manifold. A (maybe time-dependent) vector field on P can be associated to each (maybe time-dependent) smooth function H : R × P → R. It is called the Hamiltonian vector field associated to the Hamiltonian H, and denoted by XH. Its expression is XH(t, x) = Λ (x) dHt(x) , where dHt(x) = dH(t, x) − ∂H(t, x) ∂t dt is the differential of the function deduced from H by considering t as a parameter with respect to which no differentiation is made. The Hamilton equation determined by the (maybe time-dependent) Hamiltonian H is dϕ(t) dt = XH( t, ϕ(t) = Λ (dHt) ϕ(t) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 34/84 5. Hamiltonian symmetries 5.1. Presymplectic, symplectic and Poisson diffeomorphisms Let M be a manifold endowed with some structure, which can be either a presymplectic structure, determined by a presymplectic form, i.e., a 2-form ω which is closed (dω = 0), a symplectic structure, determined by a symplectic form ω, i.e., a 2-form ω which is both closed (dω = 0) and nondegenerate (ker ω = {0}), a Poisson structure, determined by a smooth Poisson bivector field Λ satisfying [Λ, Λ] = 0. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 35/84 5. Hamiltonian symmetries 5.1. Presymplectic, symplectic and Poisson diffeomorphisms Let M be a manifold endowed with some structure, which can be either a presymplectic structure, determined by a presymplectic form, i.e., a 2-form ω which is closed (dω = 0), a symplectic structure, determined by a symplectic form ω, i.e., a 2-form ω which is both closed (dω = 0) and nondegenerate (ker ω = {0}), a Poisson structure, determined by a smooth Poisson bivector field Λ satisfying [Λ, Λ] = 0. Definition A presymplectic (resp. symplectic, resp. Poisson) diffeomorphism of a presymplectic (resp., symplectic, resp. Poisson) manifold (M, ω) (resp. (M, Λ)) is a smooth diffeomorphism f : M → M such that f ∗ω = ω (resp. f ∗Λ = Λ). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 35/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields Definition A smooth vector field X on a presymplectic (resp. symplectic, resp. Poisson) manifold (M, ω) (resp. (M, Λ)) is said to be a presysmplectic (resp. symplectic, resp. Poisson) vector field if L(X)ω = 0 (resp. if L(X)Λ = 0), where L(X) denotes the Lie derivative of forms or mutivector fields with respect to X. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 36/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields Definition A smooth vector field X on a presymplectic (resp. symplectic, resp. Poisson) manifold (M, ω) (resp. (M, Λ)) is said to be a presysmplectic (resp. symplectic, resp. Poisson) vector field if L(X)ω = 0 (resp. if L(X)Λ = 0), where L(X) denotes the Lie derivative of forms or mutivector fields with respect to X. Definition Let (M, ω) be a presymplectic or symplectic manifold. A smooth vector field X on M is said to be Hamiltonian if there exists a smooth function H : M → R, called a Hamiltonian for X, such that i(X)ω = −dH . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 36/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields Definition A smooth vector field X on a presymplectic (resp. symplectic, resp. Poisson) manifold (M, ω) (resp. (M, Λ)) is said to be a presysmplectic (resp. symplectic, resp. Poisson) vector field if L(X)ω = 0 (resp. if L(X)Λ = 0), where L(X) denotes the Lie derivative of forms or mutivector fields with respect to X. Definition Let (M, ω) be a presymplectic or symplectic manifold. A smooth vector field X on M is said to be Hamiltonian if there exists a smooth function H : M → R, called a Hamiltonian for X, such that i(X)ω = −dH . Not any smooth function on a presymplectic manifold can be a Hamiltonian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 36/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields (2) Definition Let (M, Λ) be a Poisson manifold. A smooth vector field X on M is said to be Hamiltonian if there exists a smooth function H ∈ C∞(M, R), called a Hamiltonian for X, such that X = Λ (dH). An equivalent definition is that i(X)dg = {H, g} for anyg ∈ C∞ (M, R) , where {H, g} = Λ(dH, dg) denotes the Poisson bracket of the functions H and g. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 37/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields (2) Definition Let (M, Λ) be a Poisson manifold. A smooth vector field X on M is said to be Hamiltonian if there exists a smooth function H ∈ C∞(M, R), called a Hamiltonian for X, such that X = Λ (dH). An equivalent definition is that i(X)dg = {H, g} for anyg ∈ C∞ (M, R) , where {H, g} = Λ(dH, dg) denotes the Poisson bracket of the functions H and g. On a symplectic or a Poisson manifold, any smooth function can be a Hamiltonian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 37/84 5. Hamiltonian symmetries 5.2. Presymplectic, symplectic and Poisson vector fields (2) Definition Let (M, Λ) be a Poisson manifold. A smooth vector field X on M is said to be Hamiltonian if there exists a smooth function H ∈ C∞(M, R), called a Hamiltonian for X, such that X = Λ (dH). An equivalent definition is that i(X)dg = {H, g} for anyg ∈ C∞ (M, R) , where {H, g} = Λ(dH, dg) denotes the Poisson bracket of the functions H and g. On a symplectic or a Poisson manifold, any smooth function can be a Hamiltonian. Proposition A Hamiltonian vector field on a presymplectic (resp. symplectic, resp. Poisson) manifold automatically is a presymplectic (resp. symplectic, resp. Poisson) vector field. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 37/84 5. Hamiltonian symmetries 5.3. Lie algebras and Lie groups actions An action on the left (resp. an action on the right) of a Lie group G on a smooth manifold M is a smooth map Φ : G × M → M (resp. a smooth map Ψ : M × G → M) such that for each fixed g ∈ G, the map Φg : M → M defined by Φg (x) = Φ(g, x) (resp. the map Ψg : M → M defined by Ψg (x) = Ψ(x, g)) is a smooth diffeomorphism of M, Φe = idM (resp. Ψe = idM), e being the neutral element of G, for each pair (g1, g2) ∈ G × G, Φg1 ◦ Φg2 = Φg1g2 (resp. Ψg1 ◦ Ψg2 = Ψg2g1 ). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 38/84 5. Hamiltonian symmetries 5.3. Lie algebras and Lie groups actions An action on the left (resp. an action on the right) of a Lie group G on a smooth manifold M is a smooth map Φ : G × M → M (resp. a smooth map Ψ : M × G → M) such that for each fixed g ∈ G, the map Φg : M → M defined by Φg (x) = Φ(g, x) (resp. the map Ψg : M → M defined by Ψg (x) = Ψ(x, g)) is a smooth diffeomorphism of M, Φe = idM (resp. Ψe = idM), e being the neutral element of G, for each pair (g1, g2) ∈ G × G, Φg1 ◦ Φg2 = Φg1g2 (resp. Ψg1 ◦ Ψg2 = Ψg2g1 ). An action of a Lie algebra G on a smooth manifold M is a Lie algebras morphism of G into the Lie algebra A1(M) of smooth vector fields on M, i.e. a map ψ : G → A1(M) which associates to each X ∈ G a smooth vector field ψ(X) on M such that for each pair (X, Y ) ∈ G × G, ψ [X, Y ] = ψ(X), ψ(Y ) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 38/84 5. Hamiltonian symmetries 5.3. Lie algebras and Lie groups actions (2) An action Ψ, either on the left or on the right, of a Lie group G on a smooth manifold M automatically determines an action of its Lie algebra G on that manifold, which associates to each X ∈ G the vector field ψ(X) on M defined by ψ(X)(x) = d ds (Ψexp(sX)(x) s=0 , x ∈ M , with the following convention: ψ a Lie algebras homomorphism when we take for Lie algebra G of the Lie group G the Lie algebra or right invariant vector fields on G if Ψ is an action on the left, and the Lie algebra of left invariant vector fields on G if Ψ is an action on the right. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 39/84 5. Hamiltonian symmetries 5.3. Lie algebras and Lie groups actions (2) An action Ψ, either on the left or on the right, of a Lie group G on a smooth manifold M automatically determines an action of its Lie algebra G on that manifold, which associates to each X ∈ G the vector field ψ(X) on M defined by ψ(X)(x) = d ds (Ψexp(sX)(x) s=0 , x ∈ M , with the following convention: ψ a Lie algebras homomorphism when we take for Lie algebra G of the Lie group G the Lie algebra or right invariant vector fields on G if Ψ is an action on the left, and the Lie algebra of left invariant vector fields on G if Ψ is an action on the right. When M is a presymplectic (resp. symplectic, resp. Poisson) manifold, an action Ψ of a Lie group on M is called a presymplectic (resp. symplectic, resp. Poisson) action if for each g ∈ G, Ψg is a presymplectic (resp. symplectic, resp. Poisson) diffeomorphism of M. Similar definitions hold for Lie algebras actions. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 39/84 5. Hamiltonian symmetries 5.4. Hamiltonian actions Definitions An action ψ of a Lie algeba G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), is said to be Hamiltonian if for each X ∈ G, the vector field ψ(X) on M is Hamiltonian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 40/84 5. Hamiltonian symmetries 5.4. Hamiltonian actions Definitions An action ψ of a Lie algeba G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), is said to be Hamiltonian if for each X ∈ G, the vector field ψ(X) on M is Hamiltonian. An action Ψ (either on the left or on the right) of a Lie group G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), is said to be Hamiltonian if that action is presymplectic, or symplectic, or Poisson (according to the structure of M), and if in addition the associated action of the Lie algebra G of G is Hamiltonian. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 40/84 5. Hamiltonian symmetries 5.5. Momentum maps of a Hamiltonian action Proposition Let ψ be a Hamiltonian action of a finite-dimensional Lie algebra G on a presymplectic, symplectic or Poisson manifold (M, ω) or (M, Λ). There exists a smooth map J : M → G∗, taking its values in the dual space G∗ of the Lie algebra G, such that for each X ∈ G the Hamiltonian vector field ψ(X) on M admits as Hamiltonian the function JX : M → R, defined by JX (x) = J(x), X , x ∈ M . The map J is called a momentum map for the Lie algebra action ψ. When ψ is the action of the Lie algebra G of a Lie group G associated to a Hamiltonian action Ψ of a Lie group G, J is called a momentum map for the Hamiltonian Lie group action Ψ. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 41/84 5. Hamiltonian symmetries 5.5. Momentum maps of a Hamiltonian action (2) The momentum map J is not unique: Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 42/84 5. Hamiltonian symmetries 5.5. Momentum maps of a Hamiltonian action (2) The momentum map J is not unique: when (M, ω) is a connected presymplectic or symplectic manifold, J is determined up to addition of an arbitrary constant element in G∗; Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 42/84 5. Hamiltonian symmetries 5.5. Momentum maps of a Hamiltonian action (2) The momentum map J is not unique: when (M, ω) is a connected presymplectic or symplectic manifold, J is determined up to addition of an arbitrary constant element in G∗; when (M, Λ) is a connected Poisson manifold, the momentum map J is determined up to addition of an arbitrary G∗-valued smooth map which, coupled with any X ∈ G, yields a Casimir of the Poisson algebra of (M, Λ), i.e. a smooth function on M whose Poisson bracket with any other smooth function on that manifold is the function identically equal to 0. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 42/84 5. Hamiltonian symmetries 5.6. Noether’s theorem in Hamiltonian formalism Theorem (Noether’s theorem in Hamiltonian formalism) Let XH and Z be two Hamiltonian vector fields on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), which admit as Hamiltonians the smooth functions H and g on the manifold M. The function H remains constant on each integral curve of Z if and only if g remains constant on each integral curve of XH. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 43/84 5. Hamiltonian symmetries 5.6. Noether’s theorem in Hamiltonian formalism Theorem (Noether’s theorem in Hamiltonian formalism) Let XH and Z be two Hamiltonian vector fields on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), which admit as Hamiltonians the smooth functions H and g on the manifold M. The function H remains constant on each integral curve of Z if and only if g remains constant on each integral curve of XH. Corollary (of Noether’s theorem in Hamiltonian formalism) Let ψ : G → A1(M) be a Hamiltonian action of a finite-dimensional Lie algebra G on a presymplectic or symplectic manifold (M, ω), or on a Poisson manifold (M, Λ), and let J : M → G∗ be a momentum map of this action. Let XH be a Hamiltonian vector field on M admitting as Hamiltonian a smooth function H. If for each X ∈ G we have i ψ(X) (dH) = 0, the momentum map J remains constant on each integral curve of XH. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 43/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles Theorem (J.M. Souriau) Let Φ be a Hamiltonian action (either on the left or on the right) of a Lie group G on a symplectic manifold (M, ω) and J : M → G∗ be a moment map of this action. There exists an affine action A (either on the left or on the right) of the Lie group G on the dual G∗ of its LIe algebra G such that the momentum map J is equivariant with respect to the actions of G Φ on M and A on G∗: J ◦ Φg (x) = Ag ◦ J(x) for all g ∈ G , x ∈ M . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 44/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles Theorem (J.M. Souriau) Let Φ be a Hamiltonian action (either on the left or on the right) of a Lie group G on a symplectic manifold (M, ω) and J : M → G∗ be a moment map of this action. There exists an affine action A (either on the left or on the right) of the Lie group G on the dual G∗ of its LIe algebra G such that the momentum map J is equivariant with respect to the actions of G Φ on M and A on G∗: J ◦ Φg (x) = Ag ◦ J(x) for all g ∈ G , x ∈ M . The action A can be written, with g ∈ G and ξ ∈ G∗, A(g, ξ) = Ad∗ g−1 (ξ) + θ(g) if Φ is an action on the left, A(ξ, g) = Ad∗ g (ξ) − θ(g−1) if Φ is an action on the right. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 44/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (2) Proposition Under the assumptions and with the notations of the previous theorem, the map θ : G → G∗ is a cocycle of the Lie group G with values in G∗, for the coadjoint representation. It means that is satisfies, for all g and h ∈ G, θ(gh) = θ(g) + Ad∗ g−1 θ(h) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 45/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (2) Proposition Under the assumptions and with the notations of the previous theorem, the map θ : G → G∗ is a cocycle of the Lie group G with values in G∗, for the coadjoint representation. It means that is satisfies, for all g and h ∈ G, θ(gh) = θ(g) + Ad∗ g−1 θ(h) . More precisely θ is a symplectic cocycle. It means that its differential Teθ : TeG ≡ G → G∗ at the neutral element e ∈ G can be considered as a skew-symmetric bilinear form on G: Θ(X, Y ) = Teθ(X), Y = − Teθ(Y ), X . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 45/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (3) The bilinear form Θ on the Lie algebra G is a symplectic cocycle of that Lie algebra. It means that it is skew-symmetric and satisfies, for all X, Y and Z ∈ G, Θ [X, Y ], Z + Θ [Y , Z], X + Θ [Z, X], Y = 0 . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 46/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (3) The bilinear form Θ on the Lie algebra G is a symplectic cocycle of that Lie algebra. It means that it is skew-symmetric and satisfies, for all X, Y and Z ∈ G, Θ [X, Y ], Z + Θ [Y , Z], X + Θ [Z, X], Y = 0 . Proposition The composition law which associates to each pair (f , g) of smooth real-valued functions on G∗ the function {f , g}Θ given by {f , g}Θ(x) = x, [df (x), dg(x)] − Θ df (x), dg(x) , x ∈ G∗ , (G being identified with its bidual G∗∗), determines a Poisson structure on G∗, and the momentum map J : M → G∗ is a Poisson map, M being endowed with the Poisson structure associated to its symplectic structure. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 46/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (4) When the momentum map J is replaced by another momentum map J = J + µ, where µ ∈ G∗ is a constant, the symplectic Lie group cocycle θ and the symplectic Lie algebra cocycle Θ are replaced by θ and Θ , respectively, given by θ (g) = θ(g) + µ − Ad∗ g−1 (µ) , g ∈ G , Θ (X, Y ) = Θ(X, Y ) + µ, [X, Y ] , X and Y ∈ G . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 47/84 5. Hamiltonian symmetries 5.7. Symplectic cocycles (4) When the momentum map J is replaced by another momentum map J = J + µ, where µ ∈ G∗ is a constant, the symplectic Lie group cocycle θ and the symplectic Lie algebra cocycle Θ are replaced by θ and Θ , respectively, given by θ (g) = θ(g) + µ − Ad∗ g−1 (µ) , g ∈ G , Θ (X, Y ) = Θ(X, Y ) + µ, [X, Y ] , X and Y ∈ G . These formulae show that θ − θ and Θ − Θ are symplectic coboudaries of the Lie group G and the Lie algebra G. In other words, the cohomology classes of the cocycles θ and Θ only depend on the Hamiltonian action Φ of G on the symplectic manifold (M, ω). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 47/84 5. Hamiltonian symmetries 5.8. Fist application: symmetries of the phase space Hamiltonian Symmetries are often used for the search of solutions of the equations of motion of mechanical systems. The symmetries considered are those of the phase space of the mechanical system. This space is very often a symplectic manifold, either the cotangent bundle to the configuration space with its canonical symplectic structure, or a more general symplectic manifold. Sometimes, after some simplifications, the phase space is a Poisson manifold. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 48/84 5. Hamiltonian symmetries 5.8. Fist application: symmetries of the phase space Hamiltonian Symmetries are often used for the search of solutions of the equations of motion of mechanical systems. The symmetries considered are those of the phase space of the mechanical system. This space is very often a symplectic manifold, either the cotangent bundle to the configuration space with its canonical symplectic structure, or a more general symplectic manifold. Sometimes, after some simplifications, the phase space is a Poisson manifold. The Marsden-Weinstein reduction procedure [25, 26] or one of its generalizations [28] is the most often method used to facilitate the determination of solutions of the equations of motion. In a first step, a possible value of the momentum map is chosen and the subset of the phas space on which the momentum map takes this value is determined. In a second step, that subset (when it is a smooth manifold) is quotiented by its isotropic foliation. The quotient manifold is a symplectic manifold of a dimension smaller than that of the original phase space, and one has an easier to solve Hamiltonian system on that reduced phase space.Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 48/84 5. Hamiltonian symmetries 5.8. First application: symmetries of the phase space (2) When Hamiltonian symmetries are used for the reduction of the dimension of the phase space of a mechanical system, the symplectic cocycle of the Lie group of symmetries action, or of the Lie algebra of symmetries action, is almost always the zero cocycle. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 49/84 5. Hamiltonian symmetries 5.8. First application: symmetries of the phase space (2) When Hamiltonian symmetries are used for the reduction of the dimension of the phase space of a mechanical system, the symplectic cocycle of the Lie group of symmetries action, or of the Lie algebra of symmetries action, is almost always the zero cocycle. For example, if the goup of symmetries is the canonical lift to the cotangent bundle of a group of symmetries of the configuration space, not only the canonical symplectic form, but the Liouville 1-form of the cotangent bundle itself remains invariant under the action of the symmetry group, and this fact implies that the symplectic cohomology class of the action is zero. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 49/84 5. Hamiltonian symmetries 5.8. First application: symmetries of the phase space (2) When Hamiltonian symmetries are used for the reduction of the dimension of the phase space of a mechanical system, the symplectic cocycle of the Lie group of symmetries action, or of the Lie algebra of symmetries action, is almost always the zero cocycle. For example, if the goup of symmetries is the canonical lift to the cotangent bundle of a group of symmetries of the configuration space, not only the canonical symplectic form, but the Liouville 1-form of the cotangent bundle itself remains invariant under the action of the symmetry group, and this fact implies that the symplectic cohomology class of the action is zero. A completely different way of using symmetries was initiated by Jean-Marie Souriau, who proposed to consider the symmetries of the manifold of motions of the mechanical system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 49/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions Jean-Marie Souriau observed that the Lagrangian and Hamiltonian formalisms, in their usual formulations, involve the choice of a particular reference frame, in which the motion is described. This choice destroys a part of the natural symmetries of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 50/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions Jean-Marie Souriau observed that the Lagrangian and Hamiltonian formalisms, in their usual formulations, involve the choice of a particular reference frame, in which the motion is described. This choice destroys a part of the natural symmetries of the system. For example, in classical (non-relativistic) Mechanics, the natural symmetry group of an isolated mechanical system must contain the symmetry group of the Galilean space-time, called the Galilean group. This group is of dimension 10. It contains not only the group of Euclidean displacements of space which is of dimension 6 and the group of time translations which is of dimension 1, but the group of linear changes of Galilean refernce frames which is of dimension 3. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 50/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions Jean-Marie Souriau observed that the Lagrangian and Hamiltonian formalisms, in their usual formulations, involve the choice of a particular reference frame, in which the motion is described. This choice destroys a part of the natural symmetries of the system. For example, in classical (non-relativistic) Mechanics, the natural symmetry group of an isolated mechanical system must contain the symmetry group of the Galilean space-time, called the Galilean group. This group is of dimension 10. It contains not only the group of Euclidean displacements of space which is of dimension 6 and the group of time translations which is of dimension 1, but the group of linear changes of Galilean refernce frames which is of dimension 3. If we use the Lagrangian formalism or the Hamiltonian formalism, the Lagrangian or the Hamiltonian of the system depends on the reference frame: it is not invariant with respect to linear changes of Galilean reference frames. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 50/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (2) It may seem strange to consider the set of all possible motions of a system, which is unknown as long as we have not determined all these possible motions. One may ask if it is really useful when we want to determine not all possible motions, but only one motion with prescribed initial data, since that motion is just one point of the (unknown) manifold of motion! Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 51/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (2) It may seem strange to consider the set of all possible motions of a system, which is unknown as long as we have not determined all these possible motions. One may ask if it is really useful when we want to determine not all possible motions, but only one motion with prescribed initial data, since that motion is just one point of the (unknown) manifold of motion! Souriau’s answers to this objection are the following. 1. We know that the manifold of motions has a symplectic structure, and very often many things are known about its symmetry properties. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 51/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (2) It may seem strange to consider the set of all possible motions of a system, which is unknown as long as we have not determined all these possible motions. One may ask if it is really useful when we want to determine not all possible motions, but only one motion with prescribed initial data, since that motion is just one point of the (unknown) manifold of motion! Souriau’s answers to this objection are the following. 1. We know that the manifold of motions has a symplectic structure, and very often many things are known about its symmetry properties. 2. In classical (non-relativistic) mechanics, there exists a natural mathematical object which does not depend on the choice of a particular reference frame (even if the decriptions given to that object by different observers depend on the reference frame used by these observers): it is the evolution space of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 51/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (3) The knowledge of the equations which govern the system’s evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 52/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (3) The knowledge of the equations which govern the system’s evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved. Moreover, the symmetry properties of the evolution space are the same as those of the manifold of motions. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 52/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (3) The knowledge of the equations which govern the system’s evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved. Moreover, the symmetry properties of the evolution space are the same as those of the manifold of motions. For example, the evolution space of a classical mechanical system with configuration manifold N is 1 in the Lagrangian formalism, the space R × TN endowed with the presymplectic form d L, whose kernel is of dimension 1 when the Lagrangian L is hyper-regular, Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 52/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (3) The knowledge of the equations which govern the system’s evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved. Moreover, the symmetry properties of the evolution space are the same as those of the manifold of motions. For example, the evolution space of a classical mechanical system with configuration manifold N is 1 in the Lagrangian formalism, the space R × TN endowed with the presymplectic form d L, whose kernel is of dimension 1 when the Lagrangian L is hyper-regular, 2 in the Hamiltonian formalism, the space R × T∗N with the presymplectic form d H, whose kernel is also of dimension 1. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 52/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (3) The knowledge of the equations which govern the system’s evolution allows the full mathematical description of the evolution space, even when these equations are not yet solved. Moreover, the symmetry properties of the evolution space are the same as those of the manifold of motions. For example, the evolution space of a classical mechanical system with configuration manifold N is 1 in the Lagrangian formalism, the space R × TN endowed with the presymplectic form d L, whose kernel is of dimension 1 when the Lagrangian L is hyper-regular, 2 in the Hamiltonian formalism, the space R × T∗N with the presymplectic form d H, whose kernel is also of dimension 1. The Poincar´e-Cartan 1-form L in the Lagrangian formalism or H in the Hamiltonian formalism depend on the choice of a particular reference frame, made for using the Lagrangian or the Hamiltonian formalism. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 52/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (4) But their exterior differentials, the presymplectic forms d L or d H, do not depend on that choice, modulo a simple change of variables in the evolution space. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 53/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (4) But their exterior differentials, the presymplectic forms d L or d H, do not depend on that choice, modulo a simple change of variables in the evolution space. Souriau defined this presymplectic form in a framework more general than those of Lagrangian or Hamiltonian formalisms, and called it the Lagrange form. In this more general setting, it may not be an exact 2-form. Souriau proposed as a new Principle, the assumption that it always projects on the space of motions of the systems as a symplectic form, even in Relativistic Mechanics in which the definition of an evolution space is not clear. He called this new principle the Maxwell Principle. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 53/84 5. Hamiltonian symmetries 5.9. Second application: symmetries of the space of motions (4) But their exterior differentials, the presymplectic forms d L or d H, do not depend on that choice, modulo a simple change of variables in the evolution space. Souriau defined this presymplectic form in a framework more general than those of Lagrangian or Hamiltonian formalisms, and called it the Lagrange form. In this more general setting, it may not be an exact 2-form. Souriau proposed as a new Principle, the assumption that it always projects on the space of motions of the systems as a symplectic form, even in Relativistic Mechanics in which the definition of an evolution space is not clear. He called this new principle the Maxwell Principle. V. Bargmann proved that the symplectic cohomology of the Galilean group is of dimension 1, and Souriau proved that the cohomology class of its action on the manifold of motions of an isolated classical (non-relativistic) mechanical system can be identified with the total mass of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 53/84 6. Souriau thermodynamics on Lie groups 6.1. Statistical states Let N be the configuration manifold of a Lagrangian system whose Lagrangian L : TN → R is hyper-regular and does not explicitly depend on the time t. Let H : T∗N → R be the corresponding Hamiltonian and (M, ω) be the manifold of motions of the system. In the Hamiltonian formalism, a motion ϕ ∈ M is a smooth curve t → ϕ(t) defined on an open interval of R, with values in T∗N. For each t ∈ R, the map ϕ → ϕ(t) is a symplectomorphism of the open subset of (M, ω) made by all motions defined on an interval containing t, onto an open subset of the phase space (T∗N, dθN). For simplicity I will assume in the following that this symplectomorphism is a global symplectomorphism of (M, ω) onto (T∗N, dθN). In other words I assume that all the motions of the system are defined for all values of the time t ∈ R. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 54/84 6. Souriau thermodynamics on Lie groups 6.1. Statistical states Let N be the configuration manifold of a Lagrangian system whose Lagrangian L : TN → R is hyper-regular and does not explicitly depend on the time t. Let H : T∗N → R be the corresponding Hamiltonian and (M, ω) be the manifold of motions of the system. In the Hamiltonian formalism, a motion ϕ ∈ M is a smooth curve t → ϕ(t) defined on an open interval of R, with values in T∗N. For each t ∈ R, the map ϕ → ϕ(t) is a symplectomorphism of the open subset of (M, ω) made by all motions defined on an interval containing t, onto an open subset of the phase space (T∗N, dθN). For simplicity I will assume in the following that this symplectomorphism is a global symplectomorphism of (M, ω) onto (T∗N, dθN). In other words I assume that all the motions of the system are defined for all values of the time t ∈ R. Definition A statistical state of the mechanical system is a probability measure on the symplectic manifold of motions (M, ω). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 54/84 6. Souriau thermodynamics on Lie groups 6.1. Statistical states (2) For simplicity I only consider in what follows statistical states which can be represented by a smooth density of probability ρ : M → [0, +∞[ with respect to natural volume form ωn of the symplectic manifold of motions (M, ω) (with n = dim N). We must therefore have M ρ(ϕ)ωn (ϕ) = 1 . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 55/84 6. Souriau thermodynamics on Lie groups 6.1. Statistical states (2) For simplicity I only consider in what follows statistical states which can be represented by a smooth density of probability ρ : M → [0, +∞[ with respect to natural volume form ωn of the symplectic manifold of motions (M, ω) (with n = dim N). We must therefore have M ρ(ϕ)ωn (ϕ) = 1 . With each statistical state with a smooth probability density ρ let us associate the number s(ρ) = − M log ρ(ϕ) ρ(ϕ)ωn (ϕ) , with the convention that if x ∈ M is such that ϕ(x) = 0, log ϕ(x) ϕ(x) = 0. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 55/84 6. Souriau thermodynamics on Lie groups 6.1. Statistical states (2) For simplicity I only consider in what follows statistical states which can be represented by a smooth density of probability ρ : M → [0, +∞[ with respect to natural volume form ωn of the symplectic manifold of motions (M, ω) (with n = dim N). We must therefore have M ρ(ϕ)ωn (ϕ) = 1 . With each statistical state with a smooth probability density ρ let us associate the number s(ρ) = − M log ρ(ϕ) ρ(ϕ)ωn (ϕ) , with the convention that if x ∈ M is such that ϕ(x) = 0, log ϕ(x) ϕ(x) = 0. The Hamiltonian H : T∗N → R remains constant along each motion of the system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 55/84 6. Souriau thermodynamics on Lie groups 6.2. Action of the group of time translations Therefore we can define on the symplectic manifold of motions (M, ω) a smooth function E : M → R, called the energy function E(ϕ) = H ϕ(t) for all t ∈ R , ϕ ∈ M . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 56/84 6. Souriau thermodynamics on Lie groups 6.2. Action of the group of time translations Therefore we can define on the symplectic manifold of motions (M, ω) a smooth function E : M → R, called the energy function E(ϕ) = H ϕ(t) for all t ∈ R , ϕ ∈ M . The Hamiltonian vector field XE on M is the infinitesimal generator of the 1-dimensional group of time translations. A time translation ∆t : R → R is a map ∆t : R → R, ∆t(t) = t + ∆t. The group of time translations can be identified with R. It acts on the manifold of motions M by the action ΦE , such that for each time translation ∆t and each motion ϕ, ΦE ∆t(ϕ) is the motion t → ΦE ∆t(ϕ)(t) = ϕ(t + ∆t) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 56/84 6. Souriau thermodynamics on Lie groups 6.2. Action of the group of time translations Therefore we can define on the symplectic manifold of motions (M, ω) a smooth function E : M → R, called the energy function E(ϕ) = H ϕ(t) for all t ∈ R , ϕ ∈ M . The Hamiltonian vector field XE on M is the infinitesimal generator of the 1-dimensional group of time translations. A time translation ∆t : R → R is a map ∆t : R → R, ∆t(t) = t + ∆t. The group of time translations can be identified with R. It acts on the manifold of motions M by the action ΦE , such that for each time translation ∆t and each motion ϕ, ΦE ∆t(ϕ) is the motion t → ΦE ∆t(ϕ)(t) = ϕ(t + ∆t) . Following the ideas of Ludwig Boltzmann (1844–1906), more recently reformulated by E.T. Jaynes [12] and G.W. Mackey [21], J.-M. Souriau [30] proposed the following definition of a thermodynamic equilibrium state. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 56/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state Definition A thermodynamic equilibrium state of the mechanical system, for a given value mean value Q of the energy function E, is a statistical state with a smooth probability density ρ 0 satisfying the two constraints M ρ(ϕ)ωn (ϕ) = 1 , M ρ(ϕ)E(ϕ)ωn (ϕ) = Q , which, moreover, is such that the integral s(ρ) = − M log ρ(ϕ) ρ(ϕ)ωn (ϕ) is stationary with respect to all infinitesimal smooth variations of the probability density ρ 0 submitted to these two constraints. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 57/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state (2) By using the standard techniques of calculus of variations, Souriau proves that for each mean value Q of the energy function for which the involved integrals are normally convergent, there exists a unique thermodynamic equilibrium state whose probability density ρ is given by ρ(ϕ) = exp −Ψ − Θ.E(ϕ) , where Ψ and Θ are two constants which satisfy the two equalities Ψ = log M exp −Θ.E(ϕ) ωn (ϕ) , Q = M E(ϕ) exp −Θ.E(ϕ) ωn (ϕ) M exp −Θ.E(ϕ) ωn (ϕ) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 58/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state (3) Souriau proves that these two equalities imply that Ψ and Q are smooth functions of the variable Θ, and that Q(Θ) = − dΨ(Θ) dΘ . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 59/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state (3) Souriau proves that these two equalities imply that Ψ and Q are smooth functions of the variable Θ, and that Q(Θ) = − dΨ(Θ) dΘ . Moreover, by using convexity arguments, he proves that when Q is given, there is at most one corresponding value of Θ, so that Ψ(Θ) and the probability density ρ are uniquely determined. Moreover, he proves that the value of s(ρ) is a strict maximum, with respect to smooth variations of ρ satisfying the two above stated constraints. That maximum is a function S of the variable Θ given by S(Θ) = Ψ(Θ) + Θ.Q(Θ) , therefore dS(Θ) dΘ = −Θ d2Ψ(Θ) dΘ2 . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 59/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state (4) Souriau proves that d2Ψ(Θ) dΘ2 > 0, therefore Ψ is a convex function. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 60/84 6. Souriau thermodynamics on Lie groups 6.3. Thermodynamic equilibrium state (4) Souriau proves that d2Ψ(Θ) dΘ2 > 0, therefore Ψ is a convex function. Physical intepretation of these results: Θ is related to the absolute temperature T by Θ = 1 kT , where k is the Boltzmann constant, S is the entropy and Q the internal energy of the system. By this means Souriau recovers the Maxwell distribution of velocities of particles in a perfect gas. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 60/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action The energy function E on the symplectic manifold of motions (M, ω) can be seen as the momentum map of the Hamiltonian action ΦE on that manifold of the one-dimensional Lie group of time translations. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 61/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action The energy function E on the symplectic manifold of motions (M, ω) can be seen as the momentum map of the Hamiltonian action ΦE on that manifold of the one-dimensional Lie group of time translations. Souriau proposes a natural generalization of the definition of a thermodynamic equilibrium state in which a (maybe multi-dimensional and maybe non-Abelian) Lie group G acts, by a Hamiltonian action Φ, on that symplectic manifold. Let G be the Lie algebra of G, G∗ be its dual space and J : M → G∗ be a momentum map of the action Φ. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 61/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action The energy function E on the symplectic manifold of motions (M, ω) can be seen as the momentum map of the Hamiltonian action ΦE on that manifold of the one-dimensional Lie group of time translations. Souriau proposes a natural generalization of the definition of a thermodynamic equilibrium state in which a (maybe multi-dimensional and maybe non-Abelian) Lie group G acts, by a Hamiltonian action Φ, on that symplectic manifold. Let G be the Lie algebra of G, G∗ be its dual space and J : M → G∗ be a momentum map of the action Φ. In [30], he calls it an equilibrium state allowed by the group G and in his later papers and book [31, 32] a Gibbs state of the Lie group G, probably because it is not so clear whether physically such a state really is an equilibrium. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 61/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (2) Definition A Gibbs state of a connected Lie group G acting on a connected symplectic manifold (M, ω) by a Hamiltonian action Φ, with a momentum map J : M → G∗, for a given value mean value Q ∈ G∗ of that momentum map, is a statistical state with a smooth probability density ρ 0 satisfying the two constraints M ρ(ϕ)ωn (ϕ) = 1 , M ρ(ϕ)J(ϕ)ωn (ϕ) = Q , which, moreover, is such that the integral s(ρ) = − M log ρ(ϕ) ρ(ϕ)ωn (ϕ) is stationary with respect to all infinitesimal smooth variations of the probability density ρ 0 submitted to these two constraints. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 62/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (3) By the same calculations as those made for a thermodynamic equilibrium, Souriau obtains the following results. For each value Q of the momentum map J for which the involved integrals are normally convergent, there exists a unique Gibbs state whose probability density ρ is given by ρ(ϕ) = exp −Ψ − Θ, J(ϕ) , where Ψ is a real constant and Θ a constant which takes its value in the Lie algebra G, considered as the dual of G∗, which satisfy the two equalities Ψ = log M exp − Θ, J(ϕ) ωn (ϕ) , Q = M J(ϕ) exp − Θ, E(ϕ) ωn (ϕ) M exp − Θ, J(ϕ) ωn (ϕ) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 63/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (4) Souriau proves that these two equalities imply that Ψ and Q are smooth functions of the variable Θ ∈ G, which take their value, respectively, in R and in G∗, and that Q(Θ) = −DΨ(Θ) , where DΨ is the first differential of Ψ : G → R. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 64/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (4) Souriau proves that these two equalities imply that Ψ and Q are smooth functions of the variable Θ ∈ G, which take their value, respectively, in R and in G∗, and that Q(Θ) = −DΨ(Θ) , where DΨ is the first differential of Ψ : G → R. Exactly as for an equilibrium state, when Q is given, there is at most one corresponding value of Θ, so that Ψ(Θ) and the probability density ρ are uniquely determined. Moreover, he proves that the value of s(ρ) is a strict maximum, with respect to smooth variations of ρ satisfying the two above stated constraints. That maximum is a function S of the variable Θ given by S(Θ) = Ψ(Θ) + Θ, Q(Θ) . The second differential D2Ψ of the function Ψ : G → R is a positive symmetric bilinear form, which moreover is definite except when J takes its value in an affine subspace of G∗. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 64/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (5) When (M, ω) is the manifold of motions of a mechanical system, Θ is interpreted as a G-valued generalized temperature and S(ρ) as the entropy function of the Gibbs state ρ. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 65/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (5) When (M, ω) is the manifold of motions of a mechanical system, Θ is interpreted as a G-valued generalized temperature and S(ρ) as the entropy function of the Gibbs state ρ. There is however an important difference between a thermodynamic equilibrium state and a Gibbs state of a Lie group G: a Gibbs state may not be invariant with respect to the action of the Lie group G on the symplectic manifold of motions (M, ω), since the expression of its probability density ρ involves the value of the momentum map J, which is equivariant with respect to the action Φ of G on (M, ω) and an affine action of G on the dual of its Lie algebra G∗, whose linear part is the coadjoint action, eventually with a symplectic cocycle of G. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 65/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (5) When (M, ω) is the manifold of motions of a mechanical system, Θ is interpreted as a G-valued generalized temperature and S(ρ) as the entropy function of the Gibbs state ρ. There is however an important difference between a thermodynamic equilibrium state and a Gibbs state of a Lie group G: a Gibbs state may not be invariant with respect to the action of the Lie group G on the symplectic manifold of motions (M, ω), since the expression of its probability density ρ involves the value of the momentum map J, which is equivariant with respect to the action Φ of G on (M, ω) and an affine action of G on the dual of its Lie algebra G∗, whose linear part is the coadjoint action, eventually with a symplectic cocycle of G. Moreover, there are Hamiltonian actions for which the set of Gibbs states is empty because the involved integrals never converge: this happens, for example, for the action of the Galilean group on the manifold of motions of an isolated classical mechanical system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 65/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (6) In [32], Souriau presents several example, both for classical and for relativistic systems, which have clear physical interpretations. For example he discusses both non-relativistic and relativistic centrifuges for isotopic separation, and recovers the velocity distribution of particles in a relativistic perfect gas which can be found in the book by J.L. Synge [34]. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 66/84 6. Souriau thermodynamics on Lie groups 6.4. Generalization for a Hamiltonian Lie group action (6) In [32], Souriau presents several example, both for classical and for relativistic systems, which have clear physical interpretations. For example he discusses both non-relativistic and relativistic centrifuges for isotopic separation, and recovers the velocity distribution of particles in a relativistic perfect gas which can be found in the book by J.L. Synge [34]. In the second part of that paper, he presents a very nice cosmological model of the Universe, founded on his ideas of thermodynamics of Lie groups, compatible with the observed isotropy of the 2.7 Kelvin degrees microwave background radiation. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 66/84 Thanks Many thanks to Fr´ed´eric Barbaresco, Frank Nielsen and all the members of the scientific and organizing committees of this international conference for inviting me to present a talk. Thanks Many thanks to Fr´ed´eric Barbaresco, Frank Nielsen and all the members of the scientific and organizing committees of this international conference for inviting me to present a talk. And my warmest thanks to all the persons who patiently listened to my talk! Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 67/84 Appendix. The Euler-Poincar´e equation In a short Note [29] published in 1901, the great french mathematician Henri Poincar´e (1854–1912) proposed a new formulation of the equations of Mechanics. Assumptions Let N be the configuration manifold of a conservative Lagrangian system, with a smooth Lagrangian L : TN → R which does not depend explicitly on time. Poincar´e assumes that there exists an homomorphism ψ of a finite-dimensional real Lie algebra G into the Lie algebra A1(N) of smooth vector fields on N, such that for each x ∈ N, the values at x of the vetor fields ψ(X), when X varies in G, completely fill the tangent space Tx N. The action ψ is then said to be locally transitive. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 68/84 Appendix. The Euler-Poincar´e equation (2) Of course these assumptions imply dim G dim N Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 69/84 Appendix. The Euler-Poincar´e equation (2) Of course these assumptions imply dim G dim N Under these assumptions, Henri Poincar´e proved that the equations of motion of the Lagrangian system could be written on N × G or on N × G∗, where G∗ is the dual of the Lie algebra G, instead of on the tangent bundle TN. When dim G = dim N (which can occur only when the tangent bundle TN is trivial) the obtained equation, called the Euler-Poincar´e equation, is perfectly equivalent to the Euler-Lagrange equations and may, in certain cases, be easier to use. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 69/84 Appendix. The Euler-Poincar´e equation (2) Of course these assumptions imply dim G dim N Under these assumptions, Henri Poincar´e proved that the equations of motion of the Lagrangian system could be written on N × G or on N × G∗, where G∗ is the dual of the Lie algebra G, instead of on the tangent bundle TN. When dim G = dim N (which can occur only when the tangent bundle TN is trivial) the obtained equation, called the Euler-Poincar´e equation, is perfectly equivalent to the Euler-Lagrange equations and may, in certain cases, be easier to use. But when dim G > dim N, the system made by the Euler-Poincar´e equation is underdetermined. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 69/84 Appendix. The Euler-Poincar´e equation (3) Let γ : [t0, t1] → N be a smooth parametrized curve in N. Poincar´e proves that there exists a smooth curve V : [t0, t1] → G in the Lie algebra G such that, for each t ∈ [t0, t1], ψ V (t) γ(t) = dγ(t) dt . (∗) Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 70/84 Appendix. The Euler-Poincar´e equation (3) Let γ : [t0, t1] → N be a smooth parametrized curve in N. Poincar´e proves that there exists a smooth curve V : [t0, t1] → G in the Lie algebra G such that, for each t ∈ [t0, t1], ψ V (t) γ(t) = dγ(t) dt . (∗) When dim G > dim N the smooth curve V in G is not uniquely determined by the smooth curve γ in N. However, instead of writing the second-order Euler-Lagrange differential equations on TN satisfied by γ when this curve is a possible motion of the Lagrangian system, Poincar´e derives a first order differential equation for the curve V and proves that it is satisfied, together with Equation (∗), if and only if γ is a possible motion of the Lagrangian system. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 70/84 Appendix. The Euler-Poincar´e equation (3) Let γ : [t0, t1] → N be a smooth parametrized curve in N. Poincar´e proves that there exists a smooth curve V : [t0, t1] → G in the Lie algebra G such that, for each t ∈ [t0, t1], ψ V (t) γ(t) = dγ(t) dt . (∗) When dim G > dim N the smooth curve V in G is not uniquely determined by the smooth curve γ in N. However, instead of writing the second-order Euler-Lagrange differential equations on TN satisfied by γ when this curve is a possible motion of the Lagrangian system, Poincar´e derives a first order differential equation for the curve V and proves that it is satisfied, together with Equation (∗), if and only if γ is a possible motion of the Lagrangian system. Let ϕ : N × G → TN and L : N × G → R be the maps ϕ(x, X) = ψ(X)(x) , L(x, X) = L ◦ ϕ(x, X) . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 70/84 Appendix. The Euler-Poincar´e equation (4) We denote by d1L : N × G → T∗N and by d2L : N × G → G∗ the partial differentials of L : N × G → R with respect to its first variable x ∈ N and with respect to its second variable X ∈ G. The map ϕ : N × G → TN is a surjective vector bundles morphism of the trivial vector bundle N × G into the tangent bundle TN. Its transpose ϕT : T∗N → N × G∗ is therefore an injective vector bundles morphism, which can be written ϕT (ξ) = πN(ξ), J(ξ) , where πN : T∗N → N is the canonical projection of the cotangent bundle and J : T∗N → G∗ is a smooth map whose restriction to each fibre T∗ x N of the cotangent bundle is linear, and is the transpose of the map X → ϕ(x, X) = ψ(X)(x). It can be seen that J is in fact a Hamiltonian momentum map. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 71/84 Appendix. The Euler-Poincar´e equation (5) Let LL = dvertL : TN → T∗N be the Legendre map. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 72/84 Appendix. The Euler-Poincar´e equation (5) Let LL = dvertL : TN → T∗N be the Legendre map. Theorem (Euler-Poincar´e equation) With the above defined notations, let γ : [t0, t1] → N be a smooth parametrized curve in N and V : [t0, t1] → G be a smooth parametrized curve such that, for each t ∈ [t0, t1], ψ V (t) γ(t) = dγ(t) dt . (∗) The curve γ is a possible motion of the Lagrangian system if and only if V satisfies the equation d dt − ad∗ V (t) J ◦LL ◦ϕ γ(t), V (t) −J ◦d1L γ(t), V (t) = 0 . (∗∗) Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 72/84 Appendix. The Euler-Poincar´e equation (6) Remark Equation (∗) is called the compatibility condition and Equation (∗∗) is the Euler-Poincar´e equation. It can be written also as d dt − ad∗ V (t) d2L γ(t), V (t) − J ◦ d1L γ(t), V (t) = 0 . (∗∗∗) Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 73/84 Appendix. The Euler-Poincar´e equation (6) Remark Equation (∗) is called the compatibility condition and Equation (∗∗) is the Euler-Poincar´e equation. It can be written also as d dt − ad∗ V (t) d2L γ(t), V (t) − J ◦ d1L γ(t), V (t) = 0 . (∗∗∗) Several examples of applications of the Euler-Poincar´e equation can be found in [23, 24]. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 73/84 Appendix. The Euler-Poincar´e equation (7) When the function L : N × G → R does not depend on its first variable x ∈ N, we have d1L = 0, and the Euler-Poincar´e equation becomes simpler: it an be written either as d dt − ad∗ V (t) J ◦ LL ◦ ϕ γ(t), V (t) = 0 , Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 74/84 Appendix. The Euler-Poincar´e equation (7) When the function L : N × G → R does not depend on its first variable x ∈ N, we have d1L = 0, and the Euler-Poincar´e equation becomes simpler: it an be written either as d dt − ad∗ V (t) J ◦ LL ◦ ϕ γ(t), V (t) = 0 , or as d dt − ad∗ V (t) d2L γ(t), V (t) = 0 . Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 74/84 Appendix. The Euler-Poincar´e equation (7) When the function L : N × G → R does not depend on its first variable x ∈ N, we have d1L = 0, and the Euler-Poincar´e equation becomes simpler: it an be written either as d dt − ad∗ V (t) J ◦ LL ◦ ϕ γ(t), V (t) = 0 , or as d dt − ad∗ V (t) d2L γ(t), V (t) = 0 . The condition that L : N × G → R does not depend on its first variable x ∈ N does not mean that the Lagrangian L : TN → R is invariant by the canonical lift to TN of the action on N of the Lie algebra G. When the Lagrangian L is hyper-regular, it does not mean that the Hamiltonian HL associated to L is invariant par the canonical lift to T∗N of that action. On the contrary, when in addition dim G = dim N, it means that the Hamiltonian HL can be written as HL = HG∗ ◦ J, where HG∗ is a smooth function defined on G∗, and the Euler-Poincar´e equation can be identified whith a Hamilton equation on G∗.Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 74/84 References Several books [1, 2, 5, 7, 10, 11, 19, 28, 31] present the mathematical tools used in Geometric Mechanics. The calculus of variations and its applications in Mechanics are presented in [33, 2, 3, 4, 17, 22]. Poisson manifolds were defined by A. Lichnerowicz [20], considered in the more general setting of local Lie algebras by A. Kirillov [13]. Their local structure was studied by A. Weinstein [38]. Their geometric properties are extensively described in the more recent books [37, 18]. The best text about the Schouten-Nijenhuis bracket, in which the sign conventions used are the most natural and the easiest to use, is [15]. The Bargmann group and its applications in Thermodynamics are discussed in the recent paper by G. de Saxc´e and C. Vall´ee [6]. The very nice recent book [14] by Y. Kosmann-Schwarzbach gives an excellent historical and mathematical presentation of the Noether theorems. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 75/84 Bibliography I [1] Abraham, R., and Marsden, J.E., Foundations of Mechanics, 2nd edn., Addison-Wesley, Reading (1978). [2] Arnold, V.I., Mathematical methods of Classical Mechanics, 2nd edn., Springer, New York (1978). [3] P. B´erest, Calcul des variations. Les cours de l’´Ecole Polytechnique, Ellipses /´editions marketing, Paris 1997. [4] J.-P. Bourguignon, Calcul variationnel. ´Ecole Polytechnique, 1991. [5] A. Cannas da Silva, lectures on symplectic geometry, Lecture Notes in Mathematics, Springer. [6] de Saxc´e, G., and Vall´ee, C., Bargmann group, momentum tensor and Galilean invariance of Clausius-Duhem inequality. International Journal of Engineering Science, Vol. 50, 1, January 2012, p. 216–232. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 76/84 Bibliography II [7] Guillemin, V. and Sternberg, S., Symplectic Techniques in Physics, Cambridge University Press, Cambridge (1984). [8] W. R. Hamilton, On a general method in Dynamics. Read April 10, 1834, Philosophical Transactions of the Royal Society, part II for 1834, pp. 247–308. In Sir William Rowan Hamilton mathematical Works, vol. II, Cambridge University Press, London, 1940. [9] W. R. Hamilton, Second essay on a general method in Dynamics. Read January 15, 1835, Philosophical Transactions of the Royal Society, part I for 1835, pp. 95–144. In Sir William Rowan Hamilton mathematical Works, vol. II, Cambridge University Press, London, 1940. [10] Darryl Holm, Geometric Mechanics, Part I: Dynamics and Symmetry (354 pages), Part II: Rotating, Translating and Rolling (294 pages). World Scientific, London, 2008. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 77/84 Bibliography III [11] Iglesias, P., Sym´etries et moment, Hermann, Paris, 2000. [12] E.T. Jaynes, Phys. Rev. 106 (1957), p. 620. [13] A. Kirillov, Local Lie algebras, Russian Math. Surveys 31 (1976), 55–75. [14] Kosmann-Schwarzbach, Y., The Noether theorems, Springer, 2011. [15] J.L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in ´E. Cartan et les math´ematiques d’aujourd’hui Ast´erisque, num´ero hors s´erie, 1985, 257–271. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 78/84 Bibliography IV [16] J.-L. Lagrange, M´ecanique analytique. Premi`ere ´edition chez la veuve Desaint, Paris 1808. R´eimprim´e par Jacques Gabay, Paris, 1989. Deuxi`eme ´edition par Mme veuve Courcier, Paris, 1811. R´eimprim´e par Albert Blanchard, Paris. Quatri`eme ´edition (la plus compl`ete) en deux volumes, avec des notes par M. Poinsot, M. Lejeune-Dirichlet, J. Bertrand, G. Darboux, M. Puiseux, J. A. Serret, O. Bonnet, A. Bravais, dans Œuvres de Lagrange, volumes XI et XII, Gauthier-Villars, Paris, 1888. [17] C. S. Lanczos, The variational principles of Mechanics. Dover, New York 1970. [18] Laurent-Gengoux, C., Pichereau, A., and Vanhaecke, P., Poisson structures, Springer, Berlin (2013). [19] Libermann, P., and Marle, C.-M., Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Company, Dordrecht (1987). Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 79/84 Bibliography V [20] Lichnerowicz, A., Les vari´et´es de Poisson et leurs alg`ebres de Lie associ´ees, Journal of Differential Geometry 12 (1977), p. 253–300. [21] G. W. Mackey, The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, Inc., New York, 1963. [22] P. Malliavin, G´eom´etrie diff´erentielle intrins`eque, Hermann, Paris 1972. [23] Marle, C.-M., On Henri Poincar´e’s note “Sur une forme nouvelle des ´equations de la M´ecanique”, Journal of Geometry and Symmetry in Physics, vol. 29, 2013, p. 1–38. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 80/84 Bibliography VI [24] Marle C.-M., Symmetries of Hamiltonian Systems on Symplectic and Poisson manifolds, in Similarity and Symmetry Methods, Applications in Elasticity and Mechanics of Materials, Lecture Notes in Applied and Computational Mechanics (J.-F. Ganghoffer and I. Mladenov, editors), Springer, 2014, pp. 183–269. [25] Marsden, J.E., and Weinstein, A., Reduction of symplectic manifolds with symmetry, Reports on Mathematical Physics 5, 1974, p. 121–130. [26] Meyer, K., Symmetries and integrals in mechanics. In Dynamical systems (M. Peixoto, ed.), Academic Press (1973) p. 259–273. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 81/84 Bibliography VII [27] I. Newton, Philosophia Naturalis Principia Mathematica, London, 1687. Translated in French by ´Emilie du Chastelet (1756). [28] J.-P. Ortega and T. S. Ratiu, Momentum maps and Hamiltonian reduction, Birkh¨auser, Boston, Basel, Berlin, 2004. [29] Poincar´e, H., Sur une forme nouvelle des ´equations de la M´eanique, C. R. Acad. Sci. Paris, T. CXXXII, n. 7 (1901), p. 369–371. [30] J.-M. Souriau, D´efinition covariante des ´equilibres thermodynamiques, Supplemento al Nuovo cimento vol. IV n.1, 1966, p. 203–216. [31] J.-M. Souriau, Structure des syst`emes dynamiques, Dunod, Paris, 1969. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 82/84 Bibliography VIII [32] J.-M. Souriau, M´ecanique statistique, groupes de Lie et cosmologie, Colloques internationaux du CNRS num´ero 237G´eom´etrie symplectique et physique math´ematique, 1974, p. 59–113. [33] S. Sternberg, Lectures on differential geometry, Prentice-Hall, Englrwood Cliffs, 1964. [34] J.L. Synge, The relativistic gas, North Holland Publishing Company, Amsterdam, 1957. [35] Tulczyjew W.M., Hamiltonian systems, Lagrangian systems and the Legendre transformation, Symposia Mathematica, 14 (1974), P. 247–258. [36] Wlodzimierz M. Tulczyjev, Geometric Formulations of Physical Theories, Monographs and Textbooks in Physical Science, Bibliopolis, Napoli, 1989. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 83/84 Bibliography IX [37] I. Vaisman, Lectures on the Geometry of Poisson manifolds, Birkh¨auser, Basel, Boston, Berlin, 1994. [38] Alan Weinstein, The local structure of Poisson manifolds, J. Differential Geometry 18 (1983), pp. 523–557 and 22 (1985), p. 255. Charles-Michel Marle, Universit´e Pierre et Marie Curie Actions of Lie groups and Lie algebras on symplectic and Poisson manifolds. Application to Lagrangian and Hamiltonian systems 84/84