Application Moment et R eduction en M ecanique Tudor S. Ratiu - PowerPoint PPT Presentation

Let ( S, {· , ·} S ) and ( M, {· , ·} M ) be two Poisson mani- folds such that S ⊂ M and the inclusion i S : S ֒ → M is an immersion. ( S, {· , ·} S ) is a Poisson submanifold of ( M, {· , ·} M ) if i S is a canonical map. An immersed submanifold Q of M is called a quasi Pois- son submanifold of ( M, {· , ·} M ) if for any q ∈ Q , any open neighborhood U of q in M , and any f ∈ C ∞ M ( U ) we have X f ( i Q ( q )) ∈ T q i Q ( T q Q ) , where i Q : Q ֒ → M is the inclusion and X f is the Hamilto- nian vector field of f on U with respect to the restricted Poisson bracket {· , ·} M U .

• On a quasi Poisson submanifold there is a unique Pois- son structure that makes it into a Poisson submanifold. • Any Poisson submanifold is quasi Poisson.

The converse is not true! Counterexample. Let ( M = R 2 , B ) where    0 y B ( x, y ) =  − y 0 and ( Q = R 2 , ω can ). The identity map id : Q → M is obviously not a Poisson diffeomorphism because one structure has leaves and the other is non-degenerate. But is is also clear that any Hamiltonian vector field relative to B is tangent to Q = R 2 and hence ( Q, ω can ) is a quasi-Poisson submanifold of ( M, B ).

Given two symplectic manifolds ( M, ω ) and ( S, ω S ) such that S ⊂ M and the inclusion i : S ֒ → M is an immersion, the manifold ( S, ω S ) is a symplectic submanifold of ( M, ω ) when i is a symplectic map. Symplectic submanifolds of a symplectic manifold ( M, ω ) are in general neither Poisson nor quasi Poisson mani- folds of M . The only quasi Poisson submanifolds of a symplectic manifold are its open sets which are, in fact, Poisson submanifolds.

Symplectic Foliation Theorem. Let ( M, {· , ·} ) be a Poisson manifold and D the associated characteristic distribution. D is a smooth and integrable generalized distribution and its maximal integral leaves form a gener- alized foliation decomposing M into initial submanifolds L , each of which is symplectic with the unique sym- plectic form that makes the inclusion i : L ֒ → M into a Poisson map, that is, L is a Poisson submanifold of ( M, {· , ·} ).

Example: Let g ∗ with the Lie-Poisson structure. The symplectic leaves of the Poisson manifolds ( g ∗ , {· , ·} ± ) coincide with the connected components of the orbits of the elements in g ∗ under the coadjoint action. In this situation, the symplectic form for the leaves is given by the Kostant–Kirillov–Souriau (KKS) or orbit sym- plectic form � � ω ± − ad ∗ ξ ν, − ad ∗ O ( ν ) = ± � ν, [ ξ, η ] � . η ν

• ( M, {· , ·} ) Poisson manifold. G acts canonically on M when Φ ∗ g { f, h } = { Φ ∗ g f, Φ ∗ g h } for all g ∈ G . • Easy Poisson reduction: ( M, {· , ·} ) Poisson manifold, G Lie group acting canonically, freely, and properly on M . The orbit space M/G is a Poisson manifold with bracket { f, g } M/G ( π ( m )) = { f ◦ π, g ◦ π } ( m )

h ∈ C ∞ ( M ) G • Reduction of Hamiltonian dynamics: reduces to h ∈ C ∞ ( M/G ) given by h ◦ π = h such that X h ◦ π = Tπ ◦ X h • What about the symplectic leaves? This is where symplectic reduction comes in. Left quotient ( T ∗ G ) /G ∼ • Lie-Poisson reduction: = g ∗ [ α g ] �→ T ∗ − . The map is: e R g ( α g ). Direct proof. Discuss later. Notice that the quotient is for a left action and the map is given by right translation. Will be proved later.

LIE GROUP ACTIONS M a manifold and G a Lie group. A left action of G on M is a smooth mapping Φ : G × M → M such that (i) Φ( e, z ) = z , for all z ∈ M and (ii) Φ( g, Φ( h, z )) = Φ( gh, z ) for all g, h ∈ G and z ∈ M . We will often write g · z := Φ( g, z ) := Φ g ( z ) := Φ z ( g ) .

The triple ( M, G, Φ) is called a G - space or a G - manifold . Examples of group actions • Translation and conjugation. The left (right) translation L g : G → G , ( R g ) h �→ gh , induces a left (right) action of G on itself. • The inner automorphism AD g : G → G , given by AD g := R g − 1 ◦ L g defines a left action of G on itself called conjugation .

• Adjoint and coadjoint action. The differential at the identity of the conjugation mapping defines a lin- ear left action of G on g called the adjoint repre- sentation of G on g Ad g := T e AD g : g − → g . g : g ∗ → g ∗ is the dual of Ad g , then the map If Ad ∗ Φ : G × g ∗ − g ∗ → → Ad ∗ ( g, ν ) �− g − 1 ν, defines also a linear left action of G on g ∗ called the of G on g ∗ . coadjoint representation

• Group representation. If the manifold M is a vector space V and G acts linearly on V , that is, Φ g ∈ GL( V ) for all g ∈ G , where GL( V ) denotes the group of all linear automorphisms of V , then the action is said to be a representation of G on V . For example, the adjoint and coadjoint actions of G defined above are representations. • Tangent lift of a group action. Φ induces a natural action on the tangent bundle TM of M by g · v m := T m Φ g ( v m ) , g ∈ G, v m ∈ T m M.

• Cotangent lift of a group action. Let Φ : G × M → M be a smooth Lie group action on the manifold M . The map Φ induces a natural action on the cotangent bundle T ∗ M of M by g · α m := T ∗ g · m Φ g − 1 ( α m ) where g ∈ G and α m ∈ T ∗ m M .

The infinitesimal generator ξ M ∈ X ( M ) associated to ξ ∈ g is the vector field on M defined by � ξ M ( m ) := d � Φ exp tξ ( m ) = T e Φ m · ξ. � � � t =0 dt The infinitesimal generators are complete vector fields. The flow of ξ M equals ( t, m ) �→ exp tξ · m . Moreover, the map ξ ∈ g �→ ξ M ∈ X ( M ) is a Lie algebra antihomo- morphism , that is, (i) ( aξ + bη ) M = aξ M + bη M , (ii) [ ξ, η ] M = − [ ξ M , η M ].

If the action is on the right, then ξ ∈ g �→ ξ M ∈ X ( M ) is a Lie algebra homomorphism . Let g be a Lie algebra and M a smooth manifold. A (left) right Lie algebra action of g on M is a Lie algebra (anti)homomorphism ξ ∈ g �− → ξ M ∈ X ( M ) such that the mapping ( m, ξ ) ∈ M × g �− → ξ M ( m ) ∈ TM is smooth. Given a Lie group action, we will refer to the Lie alge- bra action induced by its infinitesimal generators as the associated Lie algebra action .

Stabilizers and orbits. The isotropy subgroup or sta- bilizer of an element m in the manifold M acted upon by the Lie group G is the closed (hence Lie) subgroup G m := { g ∈ G | Φ g ( m ) = m } ⊂ G whose Lie algebra g m equals g m = { ξ ∈ g | ξ M ( m ) = 0 } . The orbit O m of the element m ∈ M under the group action Φ is the set O m ≡ G · m := { Φ g ( m ) | g ∈ G } .

The isotropy subgroups of the elements in a group orbit are related by the expression G g · m = gG m g − 1 for all g ∈ G. The notion of orbit allows the introduction of an equiv- alence relation in the manifold M , namely, two elements x, y ∈ M are equivalent if and only if they are in the same G –orbit, that is, if there exists an element g ∈ G such that Φ g ( x ) = y . The space of classes with respect to this equivalence relation is usually referred to as the space of orbits and, depending on the context, it is denoted by the symbol M/G .

• Transitive action : only one orbit, that is, O m = M • Free action : G m = { e } for all m ∈ M • Proper action : if Φ : G × M → M × M defined by Φ( g, z ) := ( z, Φ( g, z )) is proper. This is equivalent to: for any two conver- gent sequences { m n } and { g n · m n } in M , there exists a convergent subsequence { g n k } in G . Examples of proper actions: compact group actions, SE ( n ) acting on R n , Lie groups acting on themselves by translation.

Fundamental facts about proper Lie group actions Φ : G × M → M be a proper action of the Lie group G on the manifold M . Then: (i) The isotropy subgroups G m are compact. (ii) The orbit space M/G is a Hausdorff topological space (even when G is not Hausdorff). (iii) If the action is free, M/G is a smooth manifold, and the canonical 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 one 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 . (v) If the manifold M is paracompact then there exists a G -invariant Riemannian metric on it. (vi) If the manifold M is paracompact then smooth G - invariant functions separate the G -orbits.

Twisted product. Let G be a Lie group and H ⊂ G a subgroup. Suppose that H acts on the left on the manifold A . The right twisted action of H on the product G × A is defined by ( g, a ) · h = ( gh, h − 1 · a ) . This action is free and proper by the freeness and proper- ness of the action on the G –factor. The twisted prod- uct G × H A is defined as the orbit space ( G × A ) /H corresponding to the twisted action.

Tube. Let M be a manifold and G a Lie group acting properly on M . Let m ∈ M and denote H := G m . 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.

Slice Theorem. G a Lie group acting properly on M at the point m ∈ M , H := G m . 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 T m M/T m ( G · m ), where the H -representation is given by h · ( v + T m ( G · m )) := T m Φ h · v + T m ( G · m ) . Slice : S := ϕ ([ e, B ]) so that U = G · S .

X ∈ X ( U ) G , U ⊂ M open Dynamical consequences. G -invariant, S slice at m ∈ U . Then there exists • X T ∈ X ( G · S ) G , X T ( z ) = ξ ( z ) M ( z ) for z ∈ G · S , where ξ : G · S → g is smooth G -equivariant and ξ ( z ) ∈ Lie( N ( G z )) for all z ∈ G · S . The flow T t of X T is given by T t ( z ) = exp tξ ( z ) · z , so X T is complete. • X N ∈ X ( S ) G m • If z = g · s , for g ∈ G and s ∈ S , then X ( z ) = X T ( z ) + T s Φ g ( X N ( s )) = T s Φ g ( X T ( s ) + X N ( s ))

• If N t is the flow of X N (on S ) then the integral curve of X ∈ X ( U ) G through g · s ∈ G · S is F t ( g · s ) = g ( t ) · N t ( s ) , where g ( t ) ∈ G is the solution of � � g ( t ) = T e L g ( t ) ˙ ξ ( N t ( s )) g (0) = g. , This is the tangential-normal decomposition of a G - invariant vector field (or Krupa decomposition in bi- furcation theory).

Geometric consequences. Orbit type , fixed point , and isotropy type spaces M ( H ) = { z ∈ M | G z ∈ ( H ) } , M H = { z ∈ M | H ⊂ G z } , M H = { z ∈ M | H = G z } are submanifolds. M H . M H is open in m ∈ M is regular if ∃ U ∋ m such that dim O z = dim O m , ∀ z ∈ U .

Principal Orbit Theorem: M connected. The subset M reg is connected, open, and dense in M . M/G contains only one principal orbit type, which is a connected open and dense subset of it. The Stratification Theorem: Let M be a smooth manifold and G a Lie group acting properly on it. The connected components of the orbit type manifolds M ( H ) and their projections onto orbit space M ( H ) /G constitute a Whitney stratification of M and M/G , respectively. This stratification of M/G is minimal among all Whit- ney stratifications of M/G .

G -Codostribution Theorem: Let G be a Lie group acting properly on the smooth manifold M and m ∈ M a point with isotropy subgroup H := G m . Then � � �� � ◦ � H = d f ( m ) | f ∈ C ∞ ( M ) G T m ( G · m ) .

SIMPLE EXAMPLES • S 1 acting on R 2 Since S 1 is Abelian we do not distinguish between orbit types and isotropy types, that is, R 2 ( H ) = R 2 H for any isotropy group H of this action. x = 1 and S 1 · x is the circle centered If x � = 0 then S 1 at the origin of radius � x � . The slice is the ray through 0 and x . ( R 2 ) reg = R 2 \ { 0 } , which is open, connected, 1 = ( R 2 ) reg and ( R 2 ) reg /S 1 =]0 , ∞ [. dense. R 2

If x = 0 , then S 1 0 = S 1 . The slice is R 2 . R 2 0 = { 0 } and 0 /S 1 = { 0 } . R 2 Finally R 2 /S 1 = [0 , ∞ [. • SO(3) acting on R 3 Since SO(3) is non-Abelian, there is a distinction be- tween orbit and isotropy types. Since every rotation has an axis, if x � = 0 the isotropy subgroup SO(3) x = S 1 ( x ), the circle representing the rotations with axis x . So ( R 3 ) reg = R 3 \ { 0 } .

The orbit SO(3) · x is the sphere centered at the origin with radius � x � . The slice at x is the ray connecting the origin to x . ( R 3 ) S 1 ( x ) is the set of points in R 3 which have the same istropy group S 1 ( x ), so it is equal to the line through the origin and x with the origin eliminated. It is disconnected and not SO(3)-invariant. ( R 3 ) ( S 1 ( x )) is the set of points in R 3 which have the istropy group S 1 ( x ) conjugate to S 1 ( x ). But any two rotations are conjugate, so ( R 3 ) ( S 1 ( x )) = R 3 \ { 0 } , which

is again equal in this case to ( R 3 ) reg . This is connected, open, dense. ( R 3 ) ( S 1 ( x )) / SO(3) =]0 , ∞ [. If x = 0 , the slice is R 3 , SO(3) 0 = SO(3), ( R 3 ) SO(3) = ( R 3 ) (SO(3)) = { 0 } , and ( R 3 ) (SO(3)) = { 0 } / SO(3) = { 0 } . Finally R 3 / SO(3) = [0 , ∞ [. • Semidirect products V vector space, G Lie group σ : G → GL( V ) representation

σ ′ : g → gl( V ) induced Lie algebra representation: � ξ · v := ξ V ( v ) := σ ′ ( ξ ) v := d � � σ (exp tξ ) v � � t =0 dt S := G � V semidirect product: underlying manifold is G × V , multiplication ( g 1 , v 1 )( g 2 , v 2 ) := ( g 1 g 2 , v 1 + σ ( g 1 ) v 2 ) for g 1 , g 2 ∈ G and v 1 , v 2 ∈ V , identity element is ( e, 0) and ( g, v ) − 1 = ( g − 1 , − σ ( g − 1 ) v ). Note that V is a normal subgroup of S and that S/V = G .

Let g be the Lie algebra of G and let s := g � V be the Lie algebra of S ; it is the semidirect product of g with V using the representation σ ′ and its underlying vector space is g × V . The Lie bracket on s is given by [( ξ 1 , v 1 ) , ( ξ 2 , v 2 )] = ([ ξ 1 , ξ 2 ] , σ ′ ( ξ 1 ) v 2 − σ ′ ( ξ 2 ) v 1 ) for ξ 1 , ξ 2 ∈ g and v 1 , v 2 ∈ V . Identify s ∗ with g ∗ × V ∗ by using the duality pairing on each factor.

Adjoint action of S on s : � � Ad g ξ, σ ( g ) v − σ ′ (Ad g ξ ) u Ad ( g,u ) ( ξ, v ) = , for ( g, u ) ∈ S, ( ξ, v ) ∈ s . Coadjoint action of S on s ∗ : � � Ad ∗ Ad ∗ g − 1 ν + ( σ ′ u ) ∗ σ ∗ ( g ) a, σ ∗ ( g ) a ( g,u ) − 1 ( ν, a ) = , for ( g, u ) ∈ S , ( ν, a ) ∈ s ∗ , where σ ∗ ( g ) := σ ( g − 1 ) ∗ ∈ GL( V ∗ ) , σ ′ u : g → V is the linear map given by σ ′ u ( ξ ) := σ ′ ( ξ ) u and u ) ∗ : V ∗ → g ∗ is its dual. ( σ ′

Clasification of orbits is a major problem! Do the example of the coadjoint action of SE (3) = SO (3) � R 3 . In this case: σ : SO (3) → GL( R 3 ) is usual matrix multiplication on vectors, that is, σ ( A ) v := A v , for any A ∈ SO (3) and v ∈ R 3 . Dualizing we get σ ( A ) ∗ Γ = A ∗ Γ = A − 1 Γ , for any Γ ∈ V ∗ ∼ = R 3 .

The induced Lie algebra representation σ ′ : R 3 ∼ = so (3) → gl ( R 3 ) is given by σ ′ ( Ω ) v = σ ′ v Ω = Ω × v , for any Ω , v ∈ R 3 . � � ∗ Γ = v × Γ and σ ′ ( Ω ) ∗ Γ = Γ × Ω , for any σ ′ Therefore, v v ∈ V ∼ = R 3 , Ω ∈ R 3 ∼ = so (3), and Γ ∈ V ∗ ∼ = R 3 . We have ad ∗ Ω Π = Π × Ω So all formulas in this case become: ( A , a )( B , b ) = ( AB , Ab + a ) ( A , a ) − 1 = ( A − 1 , − A − 1 a )

[( x , y ) , ( x ′ , y ′ )] = ( x × x ′ , x × y ′ − x ′ × y ) Ad ( A , a ) ( x , y ) = ( Ax , Ay − Ax × a ) Ad ∗ ( A , a ) − 1 ( u , v ) = ( Au + a × Av , Av ) Let { e 1 , e 2 , e 3 , f 1 , f 2 , f 3 } be an orthonormal basis of se (3) = R 3 × R 3 such that e i = f i for i = 1 , 2 , 3. The dual basis of se (3) ∗ using the dot product is again { e 1 , e 2 , e 3 , f 1 , f 2 , f 3 } . Let e ∈ { e 1 , e 2 , e 3 } and f ∈ { f 1 , f 2 , f 3 } be arbitrary. What are the coadjoint orbits? SE(3) · ( 0 , 0 ) = ( 0 , 0 ). Since SE(3) ( 0 , 0 ) = SE(3) is not compact, the coadjoint action is not proper .

The orbit through ( e , 0 ), e � = 0 , is SE(3) · ( e , 0 ) = { ( Ae , 0 ) | A ∈ SO(3) } = S 2 � e � × { 0 } , the two-sphere of radius � e � . The orbit through ( 0 , f ), f � = 0 , is SE(3) · ( 0 , f ) = { ( a × Af , Af ) | A ∈ SO(3) , a ∈ R 3 } = { ( u , Af ) | A ∈ SO(3) , u ⊥ Af } = TS 2 � f � , the tangent bundle of the two-sphere of radius � f � ; note that the vector part is the first component. We can think of it also as T ∗ S 2 � f � .

The orbit through ( e , f ) , where e � = 0 , f � = 0 , equals SE(3) · ( e , f ) = { ( Ae + a × Af , Af ) | A ∈ SO(3) , a ∈ R 3 } . To get a better description of this orbit, consider the smooth map    Ae + a × Af − e · f  ∈ TS 2 ϕ : ( A , a ) ∈ SE(3) �→ � f � 2 Af , Af � f � , which is right invariant under the isotropy group SE(3) ( e , f ) = { ( B , b ) | Be + b × f = e , Bf = f } and induces hence a diffeomorphism ¯ ϕ : SE(3) / SE(3) ( e , f ) → TS 2 � f � .

The orbit through ( e , f ) is diffeomorphic to SE(3) / SE(3) ( e , f ) by the diffeomorphism ( A , a ) �→ Ad ∗ ( A , a ) − 1 ( e , f ) . Composing these two maps and identifying TS 2 and T ∗ S 2 by the natural Riemannian metric on S 2 , we get the diffeomorphism Φ : SE(3) · ( e , f ) → T ∗ S 2 � f � given by    Ae + a × Af − e · f Φ(Ad ∗  . ( A , a ) − 1 ( e , f )) = � f � 2 Af , Af Thus this orbit is also diffeomorphic to T ∗ S 2 � f � .

• SE(3) acting on R 3 This action is proper: ( A , a ) · u := Au + a . It is not The orbit through the origin is R 3 , a representation. SE(3) 0 = SO(3). This action is transitive: given u ∈ R 3 we have ( I , 0 ) · u = u . So there is only one single orbit which is R 3 .

EXAMPLE • Consider R 6 with the bracket   3  ∂f ∂g − ∂f ∂g � { f, g } =  ∂x i ∂y i ∂y i ∂x i i =1 • S 1 -action given by S 1 × R 6 R 6 Φ : − → ( e iφ , ( x , y )) �− → ( R φ x , R φ y ) • Hamiltonian of the spherical pendulum h = 1 2 � y, y � + � x, e 3 �

• Impose constraint � x, x � = 1 • Angular momentum: J ( x , y ) = x 1 y 2 − x 2 y 1 .

Hilbert-Weyl Theorem: H → Aut( V ) representation, H compact Lie group. Then the algebra P ( V ) H of H - invariant polynomials on V is finitely generated, i.e., ∀ P ∈ P ( V ) H , ∃ k ∈ N , π 1 , . . . , π k ∈ P ( V ) H , ˆ P ∈ R [ X 1 , . . . , X k ] s.t. P = ˆ P ◦ ( π 1 , . . . , π k ). Minimal set is a Hilbert basis . Hilbert basis of the algebra of S 1 -invariant polynomials on R 6 is given by σ 3 = y 2 1 + y 2 2 + y 2 σ 5 = x 2 1 + x 2 σ 1 = x 3 3 2 σ 2 = y 3 σ 4 = x 1 y 1 + x 2 y 2 σ 6 = x 1 y 2 − x 2 y 1 . Semialgebraic relations σ 2 4 + σ 2 6 = σ 5 ( σ 3 − σ 2 2 ) , σ 3 ≥ 0 , σ 5 ≥ 0 .

Hilbert map π : v ∈ V �→ ( π 1 ( v ) , . . . , π k ( v )) ∈ R k separates H -orbits. So V/H ∼ = range( π ). Schwarz Theorem: The map f ∈ C ∞ ( R k ) �→ f ◦ ( π 1 , . . . π k ) ∈ C ∞ ( V ) H is surjective. Mather Theorem: The quotient presheaf of smooth functions on V/H is isomorphic to the presheaf of Whit- ney smooth functions on π ( V ) induced by the sheaf of smooth functions on R k . Tarski-Seidenberg Theorem: Since π is a polynomial map, range( π ) ⊂ R k is semi-algebraic.

Theorem: Every semi-algebraic set admits a canoni- cal Whitney stratification into a finite number of semi- algebraic subsets. Bierstone Theorem: This canonical stratification of π ( V ) coincides with the stratification of V/H into orbit type manifolds. These theorems can be used to explicitly describe quo- tient spaces of representations as semi-algebraic subsets of a (high dimensional) Euclidean space. Return to our concrete case of the spherical pendulum.

The Hilbert map is given by T R 3 R 6 σ : − → ( x , y ) �− → ( σ 1 ( x , y ) , . . . , σ 6 ( x , y )) . The S 1 -orbit space T R 3 /S 1 can be identified with the semialgebraic variety σ ( T R 3 ) ⊂ R 6 , defined by these re- lations. TS 2 is a submanifold of R 6 given by TS 2 = { ( x , y ) ∈ R 6 | � x , x � = 1 , � x , y � = 0 } . TS 2 is S 1 -invariant.

TS 2 /S 1 can be thought of the semialgebraic variety σ ( TS 2 ) defined by the previous relations and σ 5 + σ 2 1 = 1 σ 4 + σ 1 σ 2 = 0 , which allow us to solve for σ 4 and σ 5 , yielding TS 2 /S 1 = σ ( TS 2 ) = { ( σ 1 , σ 2 , σ 3 , σ 6 ) ∈ R 4 | σ 2 1 σ 2 2 + σ 2 6 = (1 − σ 2 1 )( σ 3 − σ 2 2 ) , | σ 1 | ≤ 1 , σ 3 ≥ 0 } .

The Poisson bracket is {· , ·} TS 2 /S 1 σ 1 σ 2 σ 3 σ 6 1 − σ 2 0 2 σ 2 0 σ 1 1 − (1 − σ 2 σ 2 1 ) 0 − 2 σ 1 σ 3 0 σ 3 − 2 σ 2 2 σ 1 σ 3 0 0 σ 6 0 0 0 0 The reduced Hamiltonian is H = 1 2 σ 3 + σ 1

If µ � = 0 then ( TS 2 ) µ := J − 1 ( µ ) /S 1 appears as the graph of the smooth function σ 3 = σ 2 2 + µ 2 , | σ 1 | < 1 . 1 − σ 2 1 The case µ = 0 is singular and ( TS 2 ) 0 := J − 1 (0) /S 1 is not a smooth manifold.

ABSTRACT SYMMETRY REDUCTION The case of general vector fields M manifold G × M → M smooth proper Lie group action X ∈ X ( M ) G , G -equivariant vector field F t flow of X ∈ X ( M ) G Law of conservation of isotropy :

M H := { m ∈ M | G m = H } , the H - isotropy type sub- manifold , is preserved by F t . M H is, in general, not closed in M . Properness of the action implies: • G m is compact • the (connected components of) M H are embedded submanifolds of M

N ( H ) /H (where N ( H ) denotes the normalizer of H in G ) acts freely and properly on M H . π H : M H → M H / ( N ( H ) /H ) projection i H : M H ֒ → M inclusion X induces a unique H - isotropy type reduced vector field X H on M H / ( N ( H ) /H ) by X H ◦ π H = Tπ H ◦ X ◦ i H ,

whose flow F H is given by t F H ◦ π H = π H ◦ F t ◦ i H . t If G is compact and the action is linear, then the con- struction of M H / ( N ( H ) /H ) can be implemented in a very explicit and convenient manner by using the in- variant polynomials of the action and the theorems of Hilbert and Schwarz-Mather.

The Hamiltonian case ( M, ω ) Poisson manifold, G connected Lie group with Lie algebra g , G × M → M free proper symplectic action J : M → g ∗ momentum map if X J ξ = ξ M , where J ξ := � J , ξ � and ξ M is the infinitesimal generator given by ξ ∈ g J : M → g ∗ (infinitesimally) equivariant if J ( g · m ) = � Ad ∗ T m J ( ξ M ( m )) = − ad ∗ g − 1 J ( m ), ∀ g ∈ G ξ J ( m ) ⇐ ⇒ � J ξ , J η � � J [ ξ,η ] = .

Proof Take the derivative on M of the defining relation J ξ := � J , ξ � . Get: dJ ξ ( m )( v m ) = � T m J ( v m ) , ξ � . Hence � � � � J ξ , J η J ξ ( m ) = dJ ξ ( m ) ( X J η ( m )) ( m ) = X J η = � T m J ( X J η ( m )) , ξ � = � T m J ( η M ( m )) , ξ � . On the other hand, J [ ξ,η ] ( m ) = � J ( m ) , [ ξ, η ] � = − � J ( m ) , ad η ξ � � � ad ∗ = − η J ( m ) , ξ .

Noether’s Theorem: The fibers of J are preserved by the Hamiltonian flows associated to G -invariant Hamil- tonians. Equivalently, J is conserved along the flow of any G -invariant Hamiltonian. Proof Let h ∈ C ∞ ( M ) be G -invariant, so h ◦ Φ g = h for any g ∈ G . Take the derivative of this relation at g = e and get £ ξ M h = 0. But ξ M = X J ξ so we get { J ξ , h } = � � = £ ξ M h = 0, which shows that J ξ ∈ C ∞ ( M ) d h, X J ξ is constant on the flow of X h for any ξ ∈ g , that is J is conserved. �

Example: lifted actions on cotangent bundles. Φ : G × Q → Q Lie group action, g · q := Φ( g, q ). Its lift to the cotangent bundle T ∗ Q is g · α q := Ψ g α q := T ∗ g · q Φ g − 1 ( α q ) . Ψ admits the following equivariant momentum map: � J ( α q ) , ξ � = � α q , ξ Q ( q ) � , ∀ α q ∈ T ∗ Q, ∀ ξ ∈ g . Very important so we will give two complete proofs.

Proof 1 Recall that the cotangent lift of a diffeomor- phism preserves the canonical one-form Θ on T ∗ Q . Hence � � d Ψ ∗ exp tξ Θ = Θ. Take � � t =0 of this: dt � � 0 = £ ξ T ∗ Q Θ = i ξ T ∗ Q d Θ+ di ξ T ∗ Q Θ = − i ξ T ∗ Q Ω+ d Θ , ξ T ∗ Q which shows that a momentum map exists and is equal � � to J ξ = . However, ∀ α q ∈ T ∗ Q , we have Θ , ξ T ∗ Q � � � � �� J ξ ( α q ) = Θ( α q ) , ξ T ∗ Q ( α q ) = α q , T α q π Q ξ T ∗ Q ( α q ) . But � d � � � � � � T α q π Q ξ T ∗ Q ( α q ) = T α q π Q Ψ exp tξ ( α q ) � � t =0 dt � � = d � � � ( α q ) = d � � � � � π Q ◦ Ψ exp tξ Φ exp tξ ◦ π Q ( α q ) � � � t =0 � t =0 dt dt = ξ Q ( q ) ,

which proves the formula. We prove G -equivariance. Let g ∈ G , ξ ∈ g , α q ∈ T ∗ Q . � � � J ( g · α q ) , ξ � = g · α q , ξ Q ( g · q ) � � � � � � � � T g · q Φ − 1 = α q , ◦ ξ Q ◦ Φ g ( q ) = α q , Ad g − 1 ξ Q ( q ) g � � � � Ad ∗ = J ( α q ) , Ad g − 1 ξ = g − 1 J ( α q ) , ξ . � Proof 2 Define the momentum function of X ∈ X ( Q ) P : X ( Q ) → C ∞ ( T ∗ Q ) by P ( X )( α q ) := � α q , X ( q ) � for any α q ∈ T ∗ q Q . In coordinates P ( q i , p i ) = X j ( p i ) p j .

L ( T ∗ Q ) is the space of smooth functions linear on the fibers . In coordinates F ∈ L ( T ∗ Q ) ⇐ ⇒ F ( q i , p i ) = X j ( q i ) p j for some functions X j . If H ( q i , p i ) = Y j ( q i ) p j , { F, H } ( q i , p i ) = ∂F ∂H − ∂H ∂F ∂q j ∂q j ∂p j ∂p j = ∂X i k − ∂Y i ∂q j p i Y k δ j ∂q j p i X k δ j k    ∂X i ∂q j p i Y j − ∂Y i ∂q j p i X j  p i = so L ( T ∗ Q ) is a Lie subalgebra of C ∞ ( T ∗ Q ).

Momentum Commutator Lemma : The Lie algebras (i) ( X ( Q ) , [ · , · ]) of vector fields on Q (ii) Hamiltonian vector fields X F on T ∗ Q with F ∈ L ( T ∗ Q ) are isomorphic. Each of these Lie algebras is anti- isomorphic to ( L ( T ∗ Q ) , {· , ·} ). In particular, we have {P ( X ) , P ( Y ) } = −P ([ X, Y ]) . Proof P : X ( Q ) → L ( T ∗ Q ) is linear and satisfies the relation above because [ X, Y ] i = ∂Y i ∂q j X j − ∂X i ∂q j Y j implies

   ∂X i ∂q j p i Y j − ∂Y i ∂q j p i X j  p i = {P ( X ) , P ( Y ) } −P ([ X, Y ]) = as we saw above. So, P is a Lie algebra anti-homomorphism. ⇒ P ( X )( α q ) := � α q , X ( q ) � , ∀ α q ∈ T ∗ Q ⇐ P ( X ) = 0 ⇐ ⇒ X ( q ) = 0 , ∀ q ∈ Q , so P is injective. For each F ∈ L ( T ∗ Q ), define X ( F ) ∈ X ( Q ) by � α q , X ( F )( q ) � := F ( α q ) . Then P ( X ( F )) = F , so P is also surjective.

We know that F �→ X F is a Lie algebra anti-homomorphism (by the Jacobi identity for {· , ·} ) from ( L ( T ∗ Q ) , {· , ·} ) to ( { X F | F ∈ L ( T ∗ Q ) } , [ · , · ]). This map is surjective by def- inition. Moreover, if X F = 0 then F is constant on T ∗ Q , hence equal to zero becuase F is linear on the fibers. � If X ∈ X ( Q ) has flow ϕ t , then the flow of X P ( X ) on T ∗ Q is T ∗ ϕ − t . Call X ′ := X P ( X ) the cotangent lift of X . Proof π Q : T ∗ Q → Q cotangent bundle projection. Dif- ferentiate π Q ◦ T ∗ ϕ − t = ϕ t ◦ π Q at t = 0 and get � Y ( α q ) := d � T ∗ ϕ − t ( α q ) � Tπ Q ◦ Y = X ◦ π Q , where � � t =0 dt

So, T ∗ ϕ − t is the flow of Y , by construction. Since T ∗ ϕ − t preserves the canonical one-form Θ ∈ Ω 1 ( T ∗ Q ), it fol- lows that £ Y Θ = 0, hence i Y Ω = − i Y d Θ = di Y Θ − £ Y Θ = di Y Θ By definition of Θ, we have � � i Y Θ( α q ) = � Θ( α q ) , Y ( α q ) � = α q , T α q π Q ( Y ( α q )) = � α q , X ( q ) � = P ( X )( α q ) i Y Θ = P ( X ) , ⇐ ⇒ that is, i Y Ω = d P ( X ) ⇐ ⇒ Y = X P ( X ) . � Note: � � X P ( X ) , X P ( Y ) = − X {P ( X ) , P ( Y ) } = − X −P ([ X,Y ]) = X P ([ X,Y ])

g acts on the left on Q , so it acts on T ∗ Q by ξ T ∗ Q := This g -action on T ∗ Q is Hamiltonian with in- X P ( ξ Q ) . finitesimally equivariant momentum map J : P → g ∗ given by � � � J ( α q ) , ξ � = α q , ξ Q ( q ) = P ( ξ Q )( α q ) If G , with Lie algebra g , acts on Q and hence on T ∗ Q by cotangent lift, then J is equivariant. a ( q j ) ⇒ J a ξ a = p i ξ i In coordinates, ξ i Q ( q j ) = ξ a A i Q = p i A i a ξ a , i.e., J a ( q j , p j ) = p i A i a ( q j )

Proof For Lie group actions, the theorem follows di- rectly from the previous one, because the infinitesimal generator is given by ξ T ∗ Q := X P ( ξ Q ) , so the momentum map exists and is given by J ξ = P ( ξ Q ) for all ξ ∈ g . For Lie algebra actions we need to check first that the cotangent lift gives a canonical action. So, for ξ, η ∈ g , ξ T ∗ Q [ { F, H } ] = X P ( ξ Q ) [ { F, H } ] � � � � = X P ( ξ Q ) [ F ] , H + F, X P ( ξ Q ) [ H ] � � � � = ξ T ∗ Q [ F ] , H + F, ξ T ∗ Q [ H ] Done!

Remember that the momentum map J : T ∗ Q → g ∗ is given by J ξ = P ( ξ Q ) for any ξ ∈ g . Recall the formula [ ξ, η ] Q = − [ ξ Q , η Q ]. Then � � J [ ξ,η ] = P ([ ξ, η ] Q ) = −P ([ ξ Q , η Q ]) = P ( ξ Q ) , P ( η Q ) � � J ξ , J η = , so J is infinitesimally equivariant. Now assume that G has Lie algebra g and that G acts on Q and hence on T ∗ Q by cotangent lift. Remember: g · α q := T ∗ g · q Φ g − 1 α q .

We prove G -equivariance. Let g ∈ G , ξ ∈ g , α q ∈ T ∗ Q . � � � J ( g · α q ) , ξ � = g · α q , ξ Q ( g · q ) � � � � T g · q Φ − 1 = α q , ◦ ξ Q ◦ Φ g ( q ) g � � � � = Ad g − 1 ξ Q ( q ) α q , � � = J ( α q ) , Ad g − 1 ξ � � Ad ∗ = g − 1 J ( α q ) , ξ . � If J : M → g ∗ is an infinitesimally equivariant momentum map for a left Hamiltonian action of g on a Poisson manifold M , then J is a Poisson map: J ∗ { F 1 , F 2 } + = { J ∗ F 1 , J ∗ F 2 } , ∀ F 1 , F 2 ∈ C ∞ ( g ∗ ) .

Proof Infinitesimal equivariance ⇔ { J ξ , J η } = J [ ξ,η ] . Let m ∈ M , ξ = δF 1 /δµ , η = δF 2 /δµ , µ := J ( m ) ∈ g ∗ . Then   � �  δF 1 δµ , δF 2 J ∗ { F 1 , F 2 } + ( m ) =  = � µ, [ ξ, η ] � µ, δµ = J [ ξ,η ] ( m ) = { J ξ , J η } ( m ) . But for any m ∈ M an v m ∈ T m M , we have d ( F 1 ◦ J )( m )( v m ) = d F 1 ( µ ) ( T m J ( v m )) � � T m J ( v m ) , δF 1 = dJ ξ ( m )( v m ) = δµ i.e., F 1 ◦ J and J ξ have equal m -derivatives. The Poisson bracket depends only on the point values of the first derivatives and hence

{ F 1 ◦ J , F 2 ◦ J } ( m ) = { J ξ , J η } ( m ) . � Special case: M = T ∗ G , G -action on T ∗ G is the lift of left translation. We get: { F 1 , F 2 } + ◦ J L = { F 1 ◦ J L , F 2 ◦ J L } . Restrict this relation to g ∗ and get { F 1 , F 2 } + ( µ ) = { F 1 ◦ J L , F 2 ◦ J L } ( µ ). But ( F i ◦ J L )( α g ) = F i ( T ∗ e R g α g ) =: ( F i ) R ( α g ), where ( F i ) R : T ∗ G → g ∗ is the right invariant extension of F i to T ∗ G . So we get { F 1 , F 2 } + ( µ ) = { ( F 1 ) R , ( F 2 ) R } ( µ ) .

Identifying the set of functions on g ∗ with the set of right(left)-invariant functions on T ∗ G endows g ∗ with the ± Lie-Poisson structure. This is an a posteriori proof, i.e., one needs to already know the formula for the Lie-Poisson bracket. Example: linear momentum. Take the phase space of the N –particle system, that is, T ∗ R 3 N . The additive group R 3 acts on it by v · ( q i , p i ) = ( q i + v , p i ) ⇒ ξ R 3 ( q i ) = ( q 1 , . . . , q N ; ξ, . . . , ξ ) .

J : T ∗ R 3 N → Lie( R 3 ) ≃ R 3 − � N ( q i , p i ) �− i =1 p i → which is the classical linear momentum . Indeed, by the general formula ofcotangent lifted ac- tions, we have N � � � p i · ξ. J ( q i , p i ) , ξ = i =1 Example: angular momentum. Let SO(3) act on R 3 and then, by lift, on T ∗ R 3 , that is, A · ( q , p ) = ( A q , A p ). → so (3) ∗ ≃ R 3 J : T ∗ R 3 − ( q , p ) �− → q × p . which is the classical angular momentum .

Let’s do it using the formula for cotangent lifted actions. If ξ ∈ R 3 , ˆ ξ v := ξ × v , for any v ∈ R 3 , ˆ ξ ∈ so (3), then � ξ R 3 ( v ) = d � e t ˆ � ξ v = ˆ ξ v = ξ × v � � t =0 dt so that � J ( q , p ) , ξ � = p · ξ R 3 ( q ) = p · ( ξ × q ) = ( q × p ) · ξ which shows that J ( q , p ) = q × p

Example: Momentum map of the cotangent lifted left and right translations. Let G act on itself on the left: L g ( h ) := gh . The infinitesimal generator of ξ ∈ g is � � G ( h ) := d � L exp tξ ( h ) = d � ξ L � � R h (exp tξ ) = T e R h ξ � � � t =0 � t =0 dt dt The infinitesimal generator of left translation is given by the tangent map of right translation: ξ L G ( h ) = T e R h ξ . The momentum map of the cotangent lift of left trans- lation J L : T ∗ G → g ∗ is hence given by � � α g , ξ L = � α g , T e R g ξ � = � T ∗ � J L ( α g ) , ξ � = G ( g ) e R g α g , ξ � Hence J L ( α g ) = T ∗ e R g α g .