Geometric Degree of Non Conservativeness

07/11/2017
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Symplectic structure is powerful especially when it is applied to Hamiltonian systems.We show here how this symplectic structure may de ne and evaluate an integer index that measures the defect for the system to be Hamiltonian. This defect is called the Geometric Degree of Non Conservativeness of the system. Darboux theorem on di erential forms is the key result. Linear and non linear frameworks are investigated.

Geometric Degree of Non Conservativeness

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application/pdf Geometric Degree of Non Conservativeness Jean Lerbet, Noël Challamel, François Nicot, Félix Darve
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Geometric Degree of Non Conservativeness

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            <description descriptionType="Abstract">Symplectic structure is powerful especially when it is applied to Hamiltonian systems.We show here how this symplectic structure may de ne and evaluate an integer index that measures the defect for the system to be Hamiltonian. This defect is called the Geometric Degree of Non Conservativeness of the system. Darboux theorem on di erential forms is the key result. Linear and non linear frameworks are investigated.
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Geometric Degree of Non Conservativeness Jean Lerbet1 , Noël Challamel2 , François Nicot3 , and Félix Darve4 1 Evry Val d’Essonne University, IBISC Laboratory, jlerbet@gmail.com 2 South Brittain University, LIMATB 3 IRSTEA 4 Grenoble Alpes University, 3SR Abstract. Symplectic structure is powerful especially when it is applied to Hamiltonian systems. We show here how this symplectic structure may define and evaluate an integer index that measures the defect for the sys- tem to be Hamiltonian. This defect is called the Geometric Degree of Non Conservativeness of the system. Darboux theorem on differential forms is the key result. Linear and non linear frameworks are investigated. Keywords: Hamiltonian system, symplectic geometry, Geometric De- gree of Non Conservativeness, kinematic constraints 1 Position of the problem Beyond the hamiltonian framework where external actions (like gravity) and internal actions (like in elasticity) may be described by a potential function, we are concerned here by mechanical systems whose actions are positional but without potential. For external actions, this is the case for example of the so- called follower forces ([2] for example). For internal actions, this is the case of the so-called hypoelasticity ([10] for example). One main characteristic of these questions is the loss of symmetry of the stiffness matrix K(p) in the investigated equilibrium configuration and for the load parameter p. For such systems, the stability issue presents some interesting paradoxical prop- erties . For example, a divergence stable equilibrium configuration can become unstable as the system is subjected to appropriate additional kinematic con- straints (see [9],[6] for example). This problem and the associate Kinematic Structural Stability concept have been deeply investigated for some years mainly in the linear framework ([5],[6] for example). In the present work, we are con- cerned by the dual question: for such a non conservative system Σ, what is the minimal number of additional kinematic constraints that transform the non conservative system into a conservative one? This minimal number d is called the geometric degree of nonconservativeness (GDNC). The second issue consists in finding the set of appropriate constraints. This issue will be tackled in the framework of discrete mechanics. More precisely, the set of configurations is a n-dimensional manifold M and the non hamiltonian actions are described by a section ω of the cotangent bundle T∗ M. This one form ω is supposed to be a non closed one form: dω 6= 0 where d is the usual exterior derivative of differential forms. With the differential geometry concepts, the geometric meaning of the GDNC issue is: What is the highest dimension n − d of embedded submanifolds N of M such that the ”restriction” (in a well defined meaning) ωN to N is closed. We do not tackled in this work the very difficult global issues on N and, by Poincaré’s theorem, the closed form ωN will be locally exact. 2 Solution 2.1 Linear framework In this subsection, we are concerned by the linearized version of the general prob- lem. A configuration me ∈ M (we can think to me as an equilibrium position) is fixed and a coordinate system q = (q1, . . . , qn) is given. We are looking for solution of the linear GDNC issue at me. There is here a real geometric issue to build the linearized counterpart of ω at me because it should be obtained by derivative of ω. But there is no connection on M to make the derivative of ω. We will come back to this problem in the last part. However, as usual, in a coordinate system q, the linearized counterpart of ω is the so-called stiffness matrix K = K(qe) of the system at me whose coordinate system is qe. In this framework, the issue is pulled back on the tangent space Tme M which will be identified with Rn thanks to the natural basis of Tme M associated with the coordinate system q on M. We indifferently note E = Rn and E∗ its dual space, the vector space of the linear forms on E. Thus, let φ the exterior 2-form defined on E = Rn by: φ(u, v) = uT Kav (1) where Ka is the skew-symmetric part of K. Usual linear algebra says that there is a basis B = (e1, . . . , en) of Rn and a number r = 2s ≤ n such that φ(e2i−1, e2i) = −φ(e2i, e2i−1) = 1 for i ≤ s and φ(ei, ej) = 0 for the other values of i and j. In the dual basis (e∗ 1, . . . , e∗ n) of (e1, . . . , en), the form φ then reads: φ = e∗ 1 ∧ e∗ 2 + . . . + e∗ 2s−1 ∧ e∗ 2s (2) The solution of the linear GDNC issue at me is then given by the following: Proposition 1. d = s is the GDNC of the mechanical system Σ and a possible set C = {C1, . . . , Cs} of linear kinematic constraints making the constrained system ΣC conservative is such that Ci is any in < e∗ 2i−1, e∗ 2i > for i = 1, . . . , s. In this framework, it is possible to find the set of all such possible constraints. Let then F be the kernel of φ. Then (Rn /F, φ̃) is a 2s-dimensional symplectic vector space where φ̃ is canonically defined by φ̃(ū, v̄) = φ(x, y) with x (resp, y) any vector of the class ū (resp. v̄). Proposition 2. The set of solutions of the GDNC is (isomorphic with) the set of Lagrangian subspaces of (Rn /F, φ̃). One can find in [8] a concrete construction of this set and in [7] the proof of these results. 2.2 Non linear framework The key of the solution in the nonlinear framework is related to Darboux theorem about the class of 1-form and 2-forms ([3] for example). We suppose now that the 2-form dω is regular on M meaning that its class r is constant on M. Then here, since the form dω is itself a closed form (d2 = 0), its class is also equal to its rank and is even: r = 2s. s is the unique number such that (dω)s 6= 0 and (dω)s+1 = 0. We then deduce that 2s ≤ n. Darboux’s theorem gives the local modeling of dω on an open set U of M and reads: dω = s X k=1 dyk ∧ dyk+s (3) where y1 , . . . , y2s are 2s independent functions on U. We then deduce the fol- lowing Proposition 3. Suppose that the class of dω is constant at m ∈ M (namely maximal). The (non linear) GDNC of Σ (in a neighborhood of m ∈ M is then the half s of the class 2s of dω. The local definition of a submanifold N solution of the problem is given by the family f1 = 0, . . . , fs = 0 of equations on M where fi is any linear combination (in the vector space on R and not in the modulus on the ring on the functions on R) of the above yi and yi+s for all i = 1, . . . , s. 3 Open issues Two open issues are related to this GDNC issue. The first one concerns the derivative of sections in T∗ M. The dual issue is the KISS issue that involves, in a linearized version at me, the symmetric part Ks(qe) of the stiffness matrix K(qe). It is worth noting that the skew-symmetric aspect Ka(qe) may be extended to the nonlinear framework through the exterior derivative dω whereas no similar extension is possible for the symmetric part without specify a connection on M. This issue is today partially solved and will be the subject of a forthcoming paper. The second one concerns the extension to continuum mechanics and infinite dimension spaces. Regarding the dual KISS issue, it is has been performed and will be soon published in an already accepted paper. Regarding the GDNC issue, it remains an interesting challenge because the tools, involved for the finite dimensional solution, are not naturally extendable to the case of infinite dimensional (Hilbert) vector spaces. References 1. Arnold V.: Mathematical Methods of Classical Mechanics, Graduate texts in Math- ematics, Springer, 1989 2. Bolotin V.V.: Nonconservative problems of the theory of elastic stability, Pergamon Press, 1963 3. Godbillon C.: Géométrie différentielle et mécanique analytique Herman, Paris, 1969 4. Lerbet J., Challamel N., Kirillov O. , Nicot F., Darve F.: Geometric degree of Non- conservativity, Math. and Mech. of Complex Systems Vol.2, N. 2, 2014 5. Lerbet J., Challamel N., Nicot F., Darve F.: Variational Formulation of Di- vergence Stability for constrained systems, Applied Mathematical Modelling http://dx.doi.org/10.1016/j.apm.2015.02.052 6. Lerbet J., Challamel N., Nicot F., Darve F.: Kinematical Structural Stability, Dis- crete and Continuous Dynamical Systems - Series S (DCDS-S) of American Institute of Mathematical Sciences (AIMS), DCDS-S 9-2 June 2016 special issue 7. Lerbet J., Challamel N., Nicot F., Darve F.: Geometric Degree of Nonconservativity: set of solutions for the linear case and extension to the differentiable non linear case, Appl. Math. Modell., doi: 10.1016/j.apm.2016.01.030, 2016 8. Souriau J.M.: Construction explicite de l’indice de Maslov. Applications. Lecture Notes in Phys., Vol. 50, pp 117-148, Springer, Berlin, 1976 9. Thompson J. M. T.: Paradoxical mechanics under fluid flow, Nature, 5853, vol. 296, pp 135-137 (1982) 10. Truesdell C.: Hypoelasticity, J. Rationa. Mech. Anal. 4, 83-133, 1019-1020,1955