Symetry methods in geometrics mechanics

28/10/2015
Auteurs : Tudor Ratiu
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14276
DOI :

Résumé

The goal of these lectures is to show the influence of symmetry in various aspects of theoretical mechanics. Canonical actions of Lie groups on Poisson manifolds often give rise to conservation laws, encoded in modern language by the concept of momentum maps. Reduction methods lead to a deeper understanding of the dynamics of mechanical systems. Basic results in singular Hamiltonian reduction will be presented. The Lagrangian version of reduction and its associated variational principles will also be discussed. The understanding of symmetric bifurcation phenomena in for Hamiltonian systems are based on these reduction techniques. Time permitting, discrete versions of these geometric methods will also be discussed in the context of examples from elasticity. 

Symetry methods in geometrics mechanics

Média

Voir la vidéo

Métriques

118
8
228.68 Ko
 application/pdf
bitcache://7c1f78288f2990be7045ec2cde24341d6e28dddf

Licence

Creative Commons Attribution-ShareAlike 4.0 International

Sponsors

Organisateurs

logo_see.gif
logocampusparissaclay.png

Sponsors

entropy1-01.png
springer-logo.png
lncs_logo.png
Séminaire Léon Brillouin Logo
logothales.jpg
smai.png
logo_cnrs_2.jpg
gdr-isis.png
logo_gdr-mia.png
logo_x.jpeg
logo-lix.png
logorioniledefrance.jpg
isc-pif_logo.png
logo_telecom_paristech.png
csdcunitwinlogo.jpg
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/11784/14276</identifier><creators><creator><creatorName>Tudor Ratiu</creatorName></creator></creators><titles>
            <title>Symetry methods in geometrics mechanics</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Sat 7 Nov 2015</date>
	    <date dateType="Updated">Wed 31 Aug 2016</date>
            <date dateType="Submitted">Fri 20 Jul 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">7c1f78288f2990be7045ec2cde24341d6e28dddf</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>24707</version>
        <descriptions>
            <description descriptionType="Abstract">
The goal of these lectures is to show the influence of symmetry in various aspects of theoretical mechanics. Canonical actions of Lie groups on Poisson manifolds often give rise to conservation laws, encoded in modern language by the concept of momentum maps. Reduction methods lead to a deeper understanding of the dynamics of mechanical systems. Basic results in singular Hamiltonian reduction will be presented. The Lagrangian version of reduction and its associated variational principles will also be discussed. The understanding of symmetric bifurcation phenomena in for Hamiltonian systems are based on these reduction techniques. Time permitting, discrete versions of these geometric methods will also be discussed in the context of examples from elasticity. 

</description>
        </descriptions>
    </resource>
.

SYMMETRY METHODS IN GEOMETRIC MECHANICS Tudor S. Ratiu Section de Math´ematiques Ecole Polytechnique F´ed´erale de Lausanne, Switzerland tudor.ratiu@epfl.ch Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 1 PLAN OF THE PRESENTATION • Lie group actions and reduction of dynamics • The above in the Hamiltonian case • Properties of the momentum map • Regular reduction • Singular reduction • Regular cotangent bundle reduction Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 2 M, N manifolds, N ⊂ M as subsets. N is an initial submanifold of M if the inclusion map i : N → M is an immersion satisfying the following condition: for any smooth manifold P and any map g : P → N, g is smooth if and only if i ◦ g : P → M is smooth. The smooth manifold structure that makes N into an initial submanifold of M is unique. P g◦i // // g M N . i == == The integral manifolds of an integrable generalized distribution (thus forming a generalized foliation) are initial. Infinitesimal generator ξM ∈ X(M) associated to ξ ∈ g : Lie(G) ξM(m) := d dt t=0 Φexp tξ(m) = TeΦm · ξ. ξM is a complete vector field with flow (t, m) → exp tξ · m. ξ ∈ g → ξM ∈ X(M) is a Lie algebra antihomomorphism Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 3 Isotropy, stabilizer, symmetry subgroup of m ∈ M Gm := {g ∈ G | Φg(m) = m} ⊂ G, Gg·m = gGmg−1, ∀g ∈ G closed subgroup of G whose Lie algebra gm equals gm = {ξ ∈ g | ξM(m) = 0}. Om ≡ G · m := {Φg(m) | g ∈ G} G-orbit of m Om g · m ∼ ←→ gGm ∈ G/Gm diffeomorphism Om initial submanifold of M • Transitive action: only one orbit, that is, Om = M • Free action: Gm = {e} for all m ∈ M • Proper action: if Φ : G × M (g, m) −→ (m, g · m) ∈ M × M is proper. Equivalent to: for any two convergent sequences {mn} and {gn · mn} in M, there exists a convergent subsequence {gnk} in G. Examples of proper actions: compact group actions, SE(n) acting on Rn, Lie groups acting on themselves by translation. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 4 Fundamental facts about proper Lie group actions (i) The isotropy subgroups Gm are compact. (ii) The orbit space M/G is a Hausdorff topological space. (iii) If the action is free, M/G is a smooth manifold, and the canoni- cal projection π : M → M/G defines on M the structure of a smooth left principal G–bundle. (iv) If all the isotropy subgroups of the elements of M under the G– action are conjugate to a given subgroup H, then M/G is a smooth manifold and π : M → M/G defines the structure of a smooth locally trivial fiber bundle with structure group N(H)/H and fiber G/H. Normalizer of H is N(H) := {g ∈ G | gH = Hg}. (v) If the manifold M is paracompact then there exists a G-invariant Riemannian metric on it. (Palais) (vi) If the manifold M is paracompact then smooth G-invariant functions separate the G-orbits. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 5 Twisted product H ⊂ G Lie subgroup acting (left) on the manifold A. Right twisted action of H on G × A, defined by (g, a) · h = (gh, h−1 · a), g, h ∈ G, a ∈ A, is free and proper. Twisted product G ×H A := (G × A)/H. Tube G acts properly on M. For m ∈ M, let H := Gm. A tube around the orbit G · m is a G-equivariant diffeomorphism ϕ : G ×H A −→ U, where U is a G-invariant neighborhood of G · m and A is some manifold on which H acts. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 6 Slice Theorem Let G be a Lie group acting properly on M at the point m ∈ M, H := Gm. Then there exists a tube ϕ : G ×H B −→ U about G·m. B is an open H-invariant neighborhood of 0 in a vector space which is H-equivariantly isomorphic to TmM/Tm(G·m), where the H-representation is given by h · (v + Tm(G · m)) := TmΦh · v + Tm(G · m). Slice S := ϕ([e, B]) so that U = G · S. From now on, we assume that G acts on M properly. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 7 Dynamical consequences Let X ∈ X(U)G, U ⊂ M open G-invariant, S slice at m ∈ U. Then: • ∃ XT ∈ X(G·S)G, XT (z) = ξ(z)M(z) for z ∈ G·S, where ξ : G·S → g is smooth G-equivariant and ξ(z) ∈ Lie(N(Gz)) for all z ∈ G·S. The flow Tt of XT is given by Tt(z) = exp tξ(z) · z, so XT is complete. • ∃ XN ∈ X(S)Gm. • If z = g · s, for g ∈ G and s ∈ S, then X(z) = XT (z) + TsΦg (XN(s)) = TsΦg (XT (s) + XN(s)) • If Nt is the flow of XN (on S) then the integral curve of X ∈ X(U)G through g · s ∈ G · S is Ft(g · s) = g(t) · Nt(s), where g(t) ∈ G is the solution of ˙g(t) = TeLg(t) ξ(Nt(s)) , g(0) = g. This is the tangential-normal decomposition of a G-invariant vec- tor field (or Krupa decomposition in bifurcation theory). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 8 Geometric consequences M(H) = {z ∈ M | Gz ∈ (H)}, orbit type set MH = {z ∈ M | H ⊂ Gz}, fixed point set MH = {z ∈ M | H = Gz}, isotropy type set are (embedded) submanifolds of M, MH open in MH, but, in gen- eral, MH is not closed in M. Let N(H) := {g ∈ G | gH = Hg} be the normalizer of H in G. N(H)/H acts freely and properly on MH. m ∈ M is regular if ∃U m such that dim Oz = dim Om, ∀z ∈ U. Principal Orbit Theorem: M connected. Mreg := {m ∈ M | m regular} is connected, open, and dense in M. M/G contains only one principal orbit type, which is connected, open, dense in M/G. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 9 The Stratification Theorem: The connected components of all orbit type manifolds M(H) and their projections onto M(H)/G con- stitute a Whitney stratification of M and M/G, respectively. This stratification of M/G is minimal among all Whitney stratifications of M/G. G-Codostribution Theorem: Let G be a Lie group acting prop- erly on the smooth manifold M and m ∈ M a point with isotropy subgroup H := Gm. Then Tm(G · m) ◦ H = df(m) | f ∈ C∞(M)G . This is due to Ortega [1998]. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 10 Reduction of general vector fields G × M → M proper, X ∈ X(M)G (G-equivariant) with flow Ft Law of conservation of isotropy: Every isotropy type submanifold MH := {m ∈ M | Gm = H} is preserved by Ft. πH : MH → MH/(N(H)/H) projection , iH : MH → M inclusion X induces a unique H-isotropy type reduced vector field XH on MH/(N(H)/H) by XH ◦ πH = TπH ◦ X ◦ iH, whose flow FH t is given by FH t ◦ πH = πH ◦ Ft ◦ iH. G compact linear action, then the construction of MH/(N(H)/H) can be implemented by using the invariant polynomials of the ac- tion and the theorems of Hilbert and Schwarz-Mather. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 11 The Hamiltonian case (M, ω) symplectic manifold, G connected Lie group with Lie algebra g, G × M → M left free proper symplectic action: Φ∗ gω = ω, ∀g ∈ G. J : M → g∗ momentum map: XJξ = ξM, where Jξ := J, ξ . Non-equivariance (Souriau) group g∗-valued 1-cocycle: c(g) := J(g · m) − Ad∗ g−1 J(m), independent of m ∈ M if M connected. (M, ω) connected. G × g∗ (g, µ) Θ −→ Ad∗ g−1 µ + c(g) ∈ g∗ affine action. J : M → g∗ is Θ-equivariant. Noether’s Theorem: J is conserved along the flow of any G- invariant Hamiltonian. g∗ ± is an affine Lie-Poisson space {f, h}(µ) := ± µ, δf δµ , δh δµ Σ δf δµ , δh δµ , f, h ∈ C∞(g∗) Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 12 The infinitesimal non-equivariance two-cocycle Σ ∈ Z2(g, R) is Σ : g × g (ξ, η) −→ dση(e) · ξ ∈ R, where ση : G → R defined by ση(g) = σ(g), η . Its symplectic leaves (reachable sets) are the Θ-orbits Oµ: ω± Oµ (ν)(ξg∗(ν), ηg∗(ν)) = ± ν, [ξ, η] Σ(ξ, η). J : M → g∗ + is a Poisson map. Example: lifted actions on cotangent bundles. G acts on the manifold Q and then by lift on its cotangent bundle T∗Q. J(αq), ξ = αq, ξQ(q) , ∀ αq ∈ T∗Q, ∀ ξ ∈ g. This is an Ad∗-equivariant momentum map. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 13 Special case 1: linear momentum. Configuration space of N particles in space is R3N. R3 acts on R3N by v · (qi, ) = (qi + v). Then J : T∗R3N → R3 is the linear momentum J(qi, pi) = N i=1 pi. Special case 2: angular momentum. SO(3) acts naturally on R3. Then J : T∗R3N → R3 is the angular momentum J(q, p) = q×p. Example: symplectic linear actions. (V, ω) symplectic vector space, G ⊆ Sp(V, ω), acting naturally on V . Ad∗-equivariant mo- mentum map J : V → sp(V, ω)∗ is J(v), ξ = 1 2 ω(ξV (v), v). Special case: Cayley-Klein parameters and the Hopf fibration. SU(2) acts on C2, J : C2 → su(2)∗ given, as above, by J(z, w), ξ = 1 2 ω(ξ(z, w)T, (z, w)), z, w ∈ C, ξ ∈ su(2). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 14 Lie algebra isomorphism (su(2), [ , ]) → (R3, ×) given by R3 x = (x1, x2, x3) ∼ ←→ x := 1 2 −ix3 −ix1 − x2 −ix1 + x2 ix3 ∈ su(2). Identify su(2)∗ with R3 by the map µ ∈ su(2)∗ → ˇµ ∈ R3 defined by ˇµ · x := −2 µ, x , ∀ x ∈ R3. Then ˇJ : C2 → R3 has the expression ˇJ(z, w) = − 1 2 (2wz, |z|2 − |w|2) ∈ R3. (z, w) are the Cayley-Klein parameters or the Kustaanheimo- Stiefel coordinates. ˇJ|S3 : S3 → S2 1/2 is the Hopf fibration. Similar construction in fluid dynamics: Clebsch variables. The momentum map of the SU(2)-action on C2, the Cayley-Klein parameters, the Kustaanheimo-Stiefel coordinates, and the family of Hopf fibrations on concentric three-spheres in C2 are the same map. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 15 Properties of the momentum map • range TmJ = (gm)◦. Points with symmetry are points of bifurca- tion. Freeness of the action is equivalent to the regularity of J. • ker TmJ = (g · m)ω. • The obstruction to the existence of J is the vanishing of the map H1(g, R) := g/[g, g] [ξ] −→ iξM ω ∈ H1(M, R). • J[ξ, η] = {Jξ, Jη} ⇐⇒ TmJ (ξM(m)) = − ad∗ ξ J(m) ∀m ∈ M, ξ, η ∈ g Among all possible choices of momentum maps for a given action, there is at most one infinitesimally Ad∗-equivariant one. G connected, then infinitesimal Ad∗-equivariance ⇔ Ad∗-equivariance. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 16 • H1(g; R) = 0 or H1(M, R) = 0 ⇒ J exists. H2(g; R) = 0 ⇒ J equiv. Whitehead lemmas: g is semisimple =⇒ H1(g; R) = H2(g; R) = 0. • If G is compact J can always be chosen to be Ad∗-equivariant • Reduction Lemma: gJ(m) · m = g · m ∩ ker TmJ = g · m ∩ (g · m)ω. Gµ • z J–1(µ) G • z • z symplectically orthogonal spaces Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 17 Momentum maps and isotropy type manifolds • MGm is a symplectic submanifold of M for any m ∈ M. This is based on: H compact Lie group and (V, ω) symplectic rep- resentation space. Then V H is a symplectic subspace of V . • Let Mm Gm be the connected component of MGm containing m and N(Gm)m := {n ∈ N(Gm) | n · z ∈ Mm Gm for all z ∈ Mm Gm }. N(Gm)m is a closed subgroup of N(Gm) that contains the con- nected component of the identity. So it is also open and hence Lie(N(Gm)m) = Lie(N(Gm)). In addition, (N(Gm)/Gm)m = N(Gm)m/Gm so that Lie (N(Gm)m/Gm) = Lie (N(Gm)/Gm) . • Lm := N(Gm)m/Gm acts freely properly and canonically on Mm Gm by Ψ(nGm, z) := n · z. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 18 • The free proper canonical action of Lm := N(Gm)m/Gm on Mm Gm has a momentum map JLm : Mm Gm → (Lie(Lm))∗ given by JLm(z) := Λ(J|Mm Gm (z) − J(m)), z ∈ Mm Gm . In this expression Λ : (g◦ m)Gm → (Lie(Lm))∗ denotes the natural Lm-equivariant isomorphism given by Λ(β), d dt t=0 (exp tξ) Gm = β, ξ , for any β ∈ (g◦ m)Gm, ξ ∈ Lie(N(Gm)m) = Lie(N(Gm)). • The non-equivariance one-cocycle τ : Mm Gm → (Lie(Lm))∗ of the momentum map JLm is given by the map τ(l) = Λ(c(n) + n · J(m) − J(m)), l = nGm ∈ Lm, n ∈ N(Gm)m. So, even if J is equivariant, the induced momentum map JLm is not, in general! Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 19 Convexity J : M → g∗ equivariant, G, M compact connected. The intersection of range J with a Weyl chamber is a compact and convex polytope, the momentum polytope (Atiyah, Guillemin, Kirwan, Sternberg). Delzant polytope in Rn is a convex polytope that is also: (i) Simple: there are n edges meeting at each vertex. (ii) Rational: the edges meeting at a vertex p are of the form p + tui, 0 ≤ t < ∞, ui ∈ Zn, i ∈ {1, . . . , n}. (iii) Smooth: the vectors {u1, . . . , un} can be chosen to be an integral basis of Zn. Delzant’s Theorem: There is a biection {symplectic toric manifolds} ∼ ←→ {Delzant polytopes} (M, ω, Tn, J : M → Rn) ∼ ←→ J(M) Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 20 Marsden-Weinstein Reduction Theorem • If µ ∈ J(M) ⊂ g∗ regular value of J and • Gµ-action on J−1(µ) is free and proper; Gµ := {g ∈ G | Θgµ = µ}, then (Mµ := J−1(µ)/Gµ, ωµ) is symplectic: π∗ µωµ = i∗ µω, iµ : J−1(µ) → M inclusion πµ : J−1(µ) → J−1(µ)/Gµ projection. The flow Ft of Xh, h ∈ C∞(M)G, leaves the connected components of J−1(µ) invariant and commutes with the G-action, so it induces a flow F µ t on Mµ by πµ ◦ Ft ◦ iµ = F µ t ◦ πµ. F µ t is Hamiltonian on (Mµ, ωµ) for the reduced Hamiltonian hµ ∈ C∞(Mµ) given by hµ ◦ πµ = h ◦ iµ. Moreover, if h, k ∈ C∞(M)G, then {h, k}µ = {hµ, kµ}Mµ. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 21 Orbit symplectic form from reduction G a Lie group, Lg(h) = gh, Rg(h) = hg, left and right translations Oµ := {Ad∗ g µ | g ∈ G} coadjoint G-orbit through µ ∈ g∗ Take the special case M = G and the left action g · h := gh, for all g, h ∈ G. The momentum map JL : T∗G → g∗ has the expression JL(αg) = T∗ e Rg(αg) ∈ g∗, ∀αg ∈ T∗G. Then, (J−1 L (µ)/Gµ, Ωµ) ∼= (Oµ, ω− Oµ ); orbit symplectic form is ω± Oµ (ν)(ad∗ ξ ν, ad∗ η ν) = ± ν, [ξ, η] , ∀ ξ, η ∈ g, ν ∈ Oµ g∗ is a Lie-Poisson space for the bracket ((T∗G)/G JR ←→ g∗ −) {f, h}(µ) := ± µ, δf δµ , δh δµ , f, h ∈ C∞(g∗) and its symplectic leaves (reachable sets) are Oµ. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 22 Reconstruction of dynamics Given is an integral curve cµ(t) of Xhµ ∈ X(Mµ). Let m0 ∈ J−1(µ). Find integral curve c(t) of Xh ∈ X(M) with initial condition m0. Pick a smooth curve d(t) ⊂ J−1(µ) such that d(0) = m0 and πµ(d(t)) = cµ(t). If c(t) is the integral curve of Xh with initial condition c(0) = m0, then there is a curve g(t) ⊂ Gµ such that c(t) = g(t) · d(t). 1.) Find smooth curve ξ(t) ⊂ gµ s.t. ξ(t)M(d(t)) = Xh(d(t)) − ˙d(t). 2.) With this ξ(t), solve ˙g(t) = TeLg(t)ξ(t), g(0) = e. Let A ∈ Ω1 J−1(µ); gµ be a connection on the Gµ-principal bundle J−1(µ) → Mµ. Choose d(t) to be the horizontal lift of cµ(t) through m0, i.e., A(d(t))( ˙d(t)) = 0, πµ(d(t)) = cµ(t), d(0) = m0. Then the solution of 1.) is ξ(t) = A(d(t)) Xh(d(t)) . Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 23 Orbit reduction • Φ : G × M → M free proper symplectic action. • The action admits a momentum map J : M → g∗. • M is connected; if J equivariant, this is not needed. • The affine coadjoint orbit Oµ := {Ad∗ g−1 µ + c(g) | g ∈ G} is an initial submanifold of g∗. • Bifurcation Lemma (range (TmJ) = (gm)◦) + the freeness of the action (hence gm = {0}) =⇒ J is a submersion onto some open subset of g∗. So J is transversal to Oµ, i.e., for any z ∈ J−1(Oµ), we have (TzJ)(TzM) + TJ(z)Oµ = g∗. So J−1(Oµ) is an initial sub- manifold of M of dimension dim(J−1(Oµ)) = dim M − dim Gµ whose tangent space at z ∈ J−1(Oµ) equals Tz(J−1(Oµ)) = (TzJ)−1(TJ(z)Oµ) = g · z + ker(TzJ). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 24 • G-action restricts to a free and proper G-action on the G-invariant initial submanifold J−1(Oµ). Why is this restricted action smooth? Action Φ : G × M → M is smooth, so ΦOµ : G × J−1(Oµ) → M is smooth (restriction: composition of smooth maps, J−1(Oµ) → M). But ΦOµ(G×J−1(Oµ)) ⊂ J−1(Oµ). Since J−1(Oµ) is initial, it follows that ΦOµ : G × J−1(Oµ) → J−1(Oµ) is smooth. • Hence MOµ := J−1(Oµ)/G is a manifold and the projection πOµ : J−1(Oµ) → MOµ is a surjective submersion. (i) On MOµ := J−1(Oµ)/G there is a unique symplectic form ωOµ characterized by ι∗ Oµ ω = π∗ Oµ ωOµ + J∗ Oµ ω+ Oµ . ιOµ : J−1(Oµ) → M, JOµ := J|J−1(Oµ), and ω+ Oµ is the +-symplectic structure on the affine orbit Oµ. (MOµ, ωOµ) is the symplectic orbit reduced space. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 25 (ii) h ∈ C∞(M)G. The flow Ft of Xh leaves the connected compo- nents of J−1(Oµ) invariant and commutes with the G-action, so it induces a flow F Oµ t on MOµ, uniquely determined by πOµ ◦ Ft ◦ iOµ = F Oµ t ◦ πOµ. (iii) The vector field generated by the flow F Oµ t on (MOµ, ωOµ) is Hamiltonian with associated reduced Hamiltonian hOµ ∈ C∞(MOµ) defined by hOµ ◦ πOµ = h ◦ iOµ. The vector fields Xh and XhOµ are πOµ-related. (iv) h, k ∈ C∞(M)G ⇒ {h, k} ∈ C∞(M)G and {h, k}Oµ = {hOµ, kOµ}MOµ , where {·, ·}MOµ denotes the Poisson bracket associated to the sym- plectic form ωOµ on MOµ. This is a theorem in the Poisson category whereas the point reduc- tion theorem is in the symplectic category. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 26 Problems with the hypotheses of the Reduction Theorem The hypotheses are too restrictive, even in classical examples, such as Jacobi’s elimination of the nodes. Properness of the action cannot be eliminated because one needs the theory of G-manifolds. 1.) How does one recover the conservation of isotropy? The momentum map seems incapable to get this. J−1(µ) are not the smallest invariant sets. Reduction completely ignores this point. 2.) If the G-action is not free, Mµ is not a smooth manifold. Then what is the structure of the reduced topological space? What is left that remains symplectic? 3.) If G is discrete, the momentum map is zero. What is reduction in that case? These are questions in bifurcation theory with symmetry. For generic vector fields, a lot is known. For Hamiltonian vector fields, almost nothing (a few papers). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 27 Singular point reduction Given: (M, ω) connected, m ∈ M G acting symplectically on M J : (M, ω) → g∗ momentum map c : G → g∗ group 1-cocycle defined by c(g) := J(g · z) − Ad∗ g−1 J(z) affine G-action Θ(g, ν) := Ad∗ g−1 ν + c(g) on g∗ Gµ the Θ-isotropy at µ Notation: Mm H connected component of MH containing m, H := Gm ⊆ G µ := J(m) ∈ g∗ Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 28 Singular symplectic point strata (i) J−1(µ) ∩ (Gµ · Mm H ) is embedded in M. (ii) M (H) µ := [J−1(µ)∩(Gµ·Mm H )]/Gµ has a unique quotient manifold structure such that π (H) µ : J−1(µ) ∩ (Gµ · Mm H ) −→ M (H) µ is a surjective submersion. (iii) There is a unique symplectic form ω (H) µ on M (H) µ characterized by ι (H) ∗ µ ω = π (H) ∗ µ ω (H) µ , ι (H) µ : J−1(µ) ∩ (Gµ · Mm H ) → M inclusion. (M (H) µ , ω (H) µ ) are the singular symplectic point strata. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 29 (iv) h ∈ C∞(M)G. Flow Ft of Xh leaves the connected components of J−1(µ) ∩ (Gµ · Mm H ) invariant and commutes with the Gµ-action, so it induces flow F µ t on M (H) µ : π (H) µ ◦ Ft ◦ i (H) µ = F µ t ◦ π (H) µ . (v) F µ t is Hamiltonian on M (H) µ for the reduced Hamiltonian h (H) µ : M (H) µ → R, h (H) µ ◦ π (H) µ = h ◦ i (H) µ . Xh and X h (H) µ are π (H) µ -related. (vi) h, k ∈ C∞(M)G ⇒ {h, k} ∈ C∞(M)G and {h, k} (H) µ = {h (H) µ , k (H) µ } M (H) µ where {·, ·} M (H) µ is the Poisson bracket induced by the symplectic structure on M (H) µ . Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 30 Sjamaar point reduction principle GOAL: Realize the strata as usual reduced spaces Recall: we start with a proper symplectic G-action on (M, ω) • m ∈ M is fixed, H := Gm, µ := J(m). • N(H)m := {n ∈ N(H) | n · Mm H ⊂ Mm H }. N(H)m is open, hence closed, in N(H). Also H ⊂ N(H)m. Thus Lie(N(H)m/H) = Lie(N(H)/H) =: l • Lm := N(H)m/H acts freely, properly, and symplectically on Mm H with momentum map JLm : Mm H z −→ Λ(J|Mm H (z) − µ) ∈ (Lie(Lm))∗ • Λ : (g◦ m)H → (Lie(Lm))∗, Lm-equivariant isomorphism Λ(β), d dt t=0 (exp tξ)H = β, ξ , β ∈ (g◦ m)H, ξ ∈ Lie(N(H)m) = Lie(N(H)) Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 31 • g◦ m ⊆ g∗ denotes the annihilator of gm in g∗ • (g◦ m)H are the H-fixed points in g◦ m • Non-equivariance one-cocycle of JLm τ : Lm l −→ Λ(c(n) + n · µ − µ) ∈ (Lie(Lm))∗ for l = nH ∈ Lm and n ∈ N(H)m. (i) π (H) µ : J−1(µ) ∩ (Gµ · Mm H ) → M (H) µ := [J−1(µ) ∩ (Gµ · Mm H )]/Gµ is a smooth fiber bundle with fiber Gµ/H and structure group NGµ(H)m/H. (ii) (Mm H )0 := J−1 Lm(0)/Lm 0 = [J−1(µ) ∩ Mm H ]/(NGµ(H)m/H) Lm 0 = Lm, in general (recall, the Lm-action is affine). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 32 (iii) π0 : J−1 Lm(0) → (Mm H )0 is a principal Lm 0 -bundle. Gµ/H is a right (NGµ(H)m/H)-space and J−1(µ)∩Mm H is a left (NGµ(H)m/H)-space. The associated bundle with fiber Gµ/H Gµ/H ×NGµ(H)m/H J−1(µ) ∩ Mm H −→ [J−1(µ) ∩ Mm H ]/(NGµ(H)m/H). is Gµ-symplectomorphic to π (H) µ : J−1(µ) ∩ (Gµ · Mm H ) −→ M (H) µ , which means that • Gµ/H ×NGµ(H)m/H J−1(µ) ∩ Mm H ∼ ←→ J−1(µ) ∩ (Gµ · Mm H ) is a Gµ-diffeomorphism • (Mm H )0 = J−1 Lm(0)/Lm 0 = (J−1(µ)∩Mm H )/(NGµ(H)m/H) is symplec- tomorphic to M (H) µ . Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 33 • {J−1(µ)∩(Gµ ·Mm H ) | J(z) = µ} forms a Whitney (B) stratification of J−1(µ). • {M (H) µ | (H)} is a symplectic Whitney (B) stratification of the cone space Mµ := J−1(µ)/Gµ. • Each connected component of Mµ contains a unique open stratum that is connected, open, and dense in the connected component of Mµ that contains it. There are similar theorems for orbit reduction. In the diagram, at every level, the corresponding spaces are isomorphic and in the respective category. In the diagram below: • Lµ is an isomorphism of cone (hence Whitney (B)) stratified spaces; in particular, Lµ it is a homeomorphism • L (H) µ is the restriction of Lµ to the stratum determined by H := Gm • f (H) µ and f (H) Oµ are the Sjamaar principle symplectomorphisms Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 34 J−1(µ) J−1(Oµ) inclusion G · (J−1(µ) ∩ Mm H )/G symplectomorphism J−1 Lm (O0)/Lm symplectomorphism J−1(N(H)m · µ) ∩ Mm H / (N(H)m/H) J−1(µ)/Gµ J−1(µ) ∩ (Gµ · Mm H )/Gµ J−1 Lm (0)/Lm 0 J−1(µ) ∩ Mm H / NGµ (H)m/H J−1(Oµ)/G stratified isomorphism c c T T T T E E E E                         d d d d d d d d d d d d πOµ projectionsπµ stratum inclusions f(H) µ f(H) Oµ symplectomorphism lµ Lµ L(H) µ L0 35 Cotangent bundle reduction – embedding Φ : G × Q → Q left free proper action =⇒ Qµ := Q/Gµ is a smooth manifold and πQ,Qµ : Q → Qµ is a principal Gµ-bundle. Lift Φ to a G-action on (T∗Q, ωQ); it is free, proper, and it admits an equivariant momentum map J : T∗Q → g∗ given by J(αq), ξ = αq(ξQ(q)), ∀αq ∈ T∗Q, ξ ∈ g. Reduce at µ ∈ g∗ to get a symplectic manifold ((T∗Q)µ, Ωµ). HYPOTHESIS: ∃ αµ ∈ Ω2(Q), Gµ-invariant, taking values in J−1(µ). ∃! βµ ∈ Ω2(Qµ) such that π∗ Q,Qµ βµ = dαµ. βµ is closed (not exact, in general). Note: αµ does not drop to Qµ whereas dαµ does. Define Bµ := π∗ Qµ βµ ∈ Ω2(T∗Q), where πQµ : T∗Qµ → Qµ projection. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 36 Cotangent Bundle Reduction – Embedding Version. There is a symplectic embedding ϕµ : ((T∗Q)µ, (ωQ)µ) −− (T∗Qµ, ωQµ − Bµ), onto the vector subbundle [TπQ,Gµ(V )]◦ ⊆ T∗Qµ, where V ⊂ TQ is the vector subbundle consisting of vectors tangent to the G-orbits, i.e., its fiber at q ∈ Q equals Vq = {ξQ(q) | ξ ∈ g}, and ◦ denotes the annihilator for the natural duality pairing between TQµ and T∗Qµ. ϕµ : ((T∗Q)µ, (ωQ)µ) ∼ −→ (T∗Qµ, ωcan − Bµ) symplectic ⇐⇒ g = gµ. Let A ∈ Ω1(Q; g) be a principal connection on the G-principal bundle πQ,Q/G : Q → Q/G and B ∈ Ω2(Q; g) its curvature. Can choose αµ(q) := A(q)∗µ =⇒ dαµ = µ, B + 1 2[A ∧, A] ∈ Ω2(Q). Recall: Bµ = π∗ Qµ βµ ∈ Ω2(T∗Qµ), βµ = π∗ Q,Qµ dαµ ∈ Ω2(Qµ). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 37 Cotangent bundle reduction – fibration Φ : G × Q → Q left free proper action Cotangent Bundle Reduction—Bundle Version Reduced space (T∗Q)µ → T∗(Q/G) is a locally trivial fiber bundle, typical fiber Oµ. This is not good enough because it does not say anything about the symplectic form on (T∗Q)µ in terms of the symplectic structure of T∗(Q/G) and the orbit symplectic structure on Oµ. Need to study first the Poisson situation to fix the setup, also easier. Let A ∈ Ω1(Q; g) be a principal connection on πQ,Q/G : Q → Q/G. Hq = {vq ∈ TqQ | A(vq) = 0} horizontal space at q ∈ Q Vq = {ξQ(q) | ξ ∈ g} vertical space at q ∈ Q Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 38 TqQ vq −→ verq(vq) := [A(q)(vq)]Q(q) ∈ Vq vertical projection TqQ vq −→ horq(vq) := vq − verq(vq) horizontal projection TqπQ,Q/G|Hq : Hq → T[q](Q/G) isomorphism with inverse Horq := TqπQ,Q/G|Hq −1 : T[q](Q/G) → Hq, horizontal lift at q ∈ Q πQ×g,Q/G : ˜g = (Q × g)/G → Q/G, the adjoint bundle; a vector bundle with fibers isomorphic to g; πQ×g,˜g : Q × g → ˜g projection Vector bundle isomorphism αA : TQ/G [vq] −→ (Tqπ(vq), [q, A(q)(vq)]) ∈ T(Q/G) ⊕ g with inverse α−1 A : T(Q/G) ⊕ g v[q], [q, ξ] −→ [Horq v[q] + ξQ(q)] ∈ TQ/G (α−1 A )∗ : T∗Q/G [αq] −→ Hor∗ q αq, [q, J(αq)] ∈ T∗(Q/G) ⊕ g∗ where Hor∗ q : T∗ q Q → T∗ [q] (Q/G) is dual to Horq : T[q](Q/G) → TqQ. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 39 A ∈ Ω1(Q; g) induces an affine connection on T∗(Q/G) ⊕ g∗ → Q/G. For f ∈ C∞(T∗(Q/G) ⊕ g∗), w = (α[q], [q, µ]) ∈ W := T∗(Q/G) ⊕ g∗, vα[q] ∈ Tα[q] (T∗(Q/G)), the exterior covariant derivative is: d ˜Af(w) ∈ Tα[q] (T∗(Q/G)), πQ/G : T∗(Q/G) → Q/G d ˜Af(w) vα[q] := df(w) vα[q] , T(q,µ)πQ×g,g Horq Tα[q] πQ/G vα[q] , 0 Push forward by (α−1 A )∗ Poisson bracket. If f, g ∈ C∞ (T∗(Q/G) ⊕ g∗) {f, g}W (w) = ωQ/G α[q] d ˜Af(w) , d ˜Ag(w) + [q, µ], B α[q] d ˜Af(w) , d ˜Ag(w) − w, δf δw , δg δw δf δw ∈ (T(Q/G) ⊕ g)α[q] is the fiber derivative w , δf δw := d dt t=0 f(w + tw ), w, w ∈ T∗(Q/G) ⊕ g∗ α[q] B := π∗ Q/GB ∈ Ω2(T∗(Q/G); g), B ∈ Ω2(T∗(Q/G); g) B([q]) TqπQ/Guq, TqπQ/Gvq := [q, CurvA(q)(uq, vq)] Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 40 Determine the symplectic leaves of the gauged Lie-Poisson bracket on T∗(Q/G) ⊕ g∗? Solved by Perlmutter in his 1999 thesis and in final form Marsden-Perlmutter [2000]. O ⊂ g∗ codajoint orbit O := (Q×O)/G → Q/G associated fiber bundle. πQ×O,O : Q×O → O T∗(Q/G) ×Q/G O := α[q], [q, ν] | q ∈ Q, ν ∈ O, α[q] ∈ Tα[q] (T∗Q) is a fiber bundle over Q/G whose fiber at [q] ∈ Q/G is T∗ [q] (Q/G) × O[q] (α−1 A )∗ J−1(O/G) = T∗(Q/G) ×Q/G O ⊂ T∗(Q/G) ⊕ g So, reduced symplectic form ωO on (T∗Q)O := J−1(O)/G pushes forward by (α−1 A )∗ to a symplectic form ωA on T∗(Q/G) ×Q/G O: Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 41 ωA = ωQ/G − β where β ∈ Ω2 O is uniquely determined by π∗ Q×O,O β = dα + π∗ Q×O,O ω+ O , α ∈ Ω1(Q × O), α(q, ν) uq, − ad∗ ξ ν := ν, A(q)(uq) , q ∈ Q, uq ∈ TqQ, ν ∈ O, ξ ∈ g. dα has the explicit expression dα(q, ν) uq, − ad∗ ξ ν , vq, − ad∗ η ν = ν, [η , ξ] + [η, ξ ] + [ξ, η] + CurvA(q)(uq, vq) q ∈ Q, ν ∈ O, ξ, ξ , η, η ∈ g, uq, vq ∈ TqQ, where uq = ξQ(q) + horq uq, vq = ηQ(q) + horq vq is the vertical-horizontal splitting on TqQ given by A. So, (T∗Q)O, (ωQ)O ∼ ←→ T∗(Q/G) ×Q/G O, ωA symplectomorphism Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 42 Reconstruction of dynamics Given: Integral curve cµ(t) of Xhµ ∈ X ((T∗Q)µ). Let αq ∈ J−1(µ). Find integral curve c(t) of Xh ∈ X(T∗Q) with initial condition αq. Solution: c(t) = g(t)·d(t). Let A ∈ Ω1(J−1(µ); gµ) be a connection and take d(t) to be the horizontal lift through αq of cµ(t). Solve ˙g(t) = TeLg(t)ξ(t), g(0) = e. So, it all comes down to: • Choice of a convenient connection A ∈ Ω1(J−1(µ); gµ). • Finding ξ(t) ⊂ gµ in terms of d(t). 1.) Gµ = S1 or R. Let ζ ∈ gµ be a basis. Identify R a ∼ ←→ aζ ∈ gµ. Connection A = 1 µ,ζ θµ ∈ Ω1(J−1(µ)), where θµ is the pull back to J−1(µ) of the canonical θQ ∈ Ω1(T∗Q); ωQ = −dθQ canonical symplectic form on T∗Q. The curvature is CurvA = − 1 µ,ζ ωµ ∈ Ω2((T∗Q)µ). Then ξ(t) = dh(Λ)(d(t)), where Λ = pi ∂ ∂pi (unique vector field on T∗Q satisfying dθQ(Λ, ·) = θQ). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 43 2.) Let A ∈ Ω1(Q; gµ) be a connection on the left Gµ-principal bundle Q → Q/Gµ. A induces a connection A ∈ Ω1(J−1(µ); gµ) by A(αq) Vαq := A(q) TαqπQ Vαq , q ∈ Q, αq ∈ T∗ q Q, Vαq ∈ Tαq(T∗Q). Then ξ(t) = A(q(t)) Fh(d(t) ⊂ gµ, Fh : T∗Q → TQ fiber derivative, q(t) := πQ(d(t)) ⊂ Q. 3.) Let (Q, ·, · ) be a Riemannian manifold and G act by isome- tries. The mechanical connection is defined by requiring that its horizontal bundle is the orthogonal to the vertical bundle. Amech(q)(uq) := Iµ(q)−1J uq , q ∈ Q, uq ∈ TqQ uq := uq, · ∈ T∗ q Q, Iµ(q) : gµ ∼ −→ g∗ µ is the µ-locked inertia tensor defined for each q ∈ Q by Iµ(q)(ζ)(η) := ζQ(q), ηQ(q) . Special situation of 2.). Then ξ(t) = Amech(q(t)) d(t) . Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 44 4.) Simple mechanical systems. The Hamiltonian is of the form h = k +v ◦πQ, where k is the kinetic energy of the cometric on T∗Q determined by a Riemannian metric ·, · on Q and v ∈ C∞(Q). G acts by isometries on Q and the potential energy v is G-invariant. The reconstruction method is quite explicit in this case. Given is αq ∈ J−1(µ) ⊂ T∗ q Q and the solution cµ(t) ⊂ (T∗Q)µ of Xhµ with initial condition [αq] ∈ (T∗Q)µ. Step 1.) ϕµ : (T∗Q)µ, (ωQ)µ → T∗(Q/Gµ), ωQ/Gµ − Bµ , sym- plectic embedding onto a vector subbundle, Bµ induced by the me- chanical connection. Then ϕµ(cµ(t)) is an integral curve of the Hamiltonian system on T∗(Q/Gµ), ωQ/Gµ − Bµ given by the kinetic energy of the quotient Riemannian metric on Q/Gµ and the quo- tient of the amended potential vµ := h ◦ αµ ∈ C∞(Q). Compute the curves ϕµ(cµ(t)) ⊂ T∗(Q/Gµ) and qµ(t) := πQ/Gµ (ϕµ(cµ(t))) ⊂ Q/Gµ. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 45 Step 2.) Using the mechanical connection Amech ∈ Ω1(Q; gµ), horizontally lift qµ(t) ∈ Q/Gµ to a curve qh(t) ⊂ Q with qh(0) = q. Step 3.) Determine ξ(t) ⊂ gµ from the algebraic equation ξ(t)Q(qh(t)), ηQ(qh(t)) = µ, η , ∀η ∈ gµ. So, ˙qh(0) and ξ(0)Q(q) are the horizontal and vertical components of the vector αq ∈ TqQ. Step 4.) Solve ˙g(t) = TeLg(t)ξ(t) in Gµ with g(0) = e. Step 5.) With qh(t) from Step 2.) and g(t) from Step 4.), define q(t) := g(t) · qh(t). This is the base integral curve of the simple mechanical system with Hamiltonian h = k+v ◦πQ satisfying q(0) = q. The curve ˙q(t) ⊂ T∗Q is the integral curve of Xh with ˙q(t) (0) = αq. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 46 Interesting special cases (a) If Gµ is Abelian, equation in Step 4.) has the solution g(t) = t 0 ξ(s)ds. (b) Gµ = S1, ζ basis of gµ. Can solve for ξ(t) in Step 3.), namely ξ(t) = µ, ζ ζQ(qh(t)) 2 ζ and hence q(t) = exp µ, ζ t 0 ds ζQ(qh(s)) 2 · qh(t) (c) If G is compact and (·, ·) is a positive definite metric, invariant under the adoint G-action on g, and satisfying (ζ, η) = ζQ(q), ηQ(q) , ∀q ∈ Q, ζ, η ∈ g, then ξ ∈ gµ is uniquely determined by (ξ, ·) = µ|gµ and g(t) = exp(tξ). Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 47 (d) If G is solvable, let {ξ1, . . . , ξn} ⊂ g be a basis. Write g(t) = exp(f1(t)ξ1) · · · exp(fn(t)ξn). Wei and Norman [1964] have shown that ˙g(t) = TeLg(t)ξ(t) can be solved by quadratures for the all the functions f1(t), . . . , fn(t). (e) If ˙ξ(t) = α(t)ξ(t) for a known function α(t), then g(t) = exp(f(t)ξ(t)) solves ˙g(t) = TeLg(t)ξ(t), where f(t) = t 0 exp s t α(r)dr ds. The conditions in (c) are very strong, but they hold for the Kaluza- Klein construction. Many of these formulas are very useful when one wants to compute geometric phases. What happens if the action of G on Q is not free? Only partial results of Perlmutter and Rodr´ıguez-Olmos. General case is open. Geometric Science of Information, Ecole Polytechnique, Paris-Saclay, October 28-30, 2015 48