Some problems in Geometric Mechanics - Geometric Mechanics ...

XIV Encuentro de Invierno Geometría, Mecánica y Teoría de Control 

Zaragoza, 6-7 February, 2012 

Some problems in Geometric Mechanics 

Geometric Mechanics, Classical field theories and Poisson structures 

Research group ULL 

http://www.alojamientoavanzado.com/digeme 


Some problems in Geometric Mechanics

General research proposal 

To study, from a geometric perspective, different methods to simplify how obtain 

the trajectories of a mechanical system (subject or not to non-holonomic 

constraints) 

solutions of dynamic equations in first order classical field theories 

Several geometric methods 

Reduction Theory 

Hamilton-Jacobi Theory 

Invariant volumes 

Geometric integrators 

Geometric tools 

Poisson structures and generalizations 

Multisymplectic structures 



Research lines 

Reduction Theory 

Reduction of symplectic Lie algebroids 

Reduction of the Jacobi structure on the sphere of the dual bundle to a Lie algebroid endowed with 

a bundle metric 

Reduction and reconstruction of non-autonomous hamiltonian systems and symplectic principal 

R-bundles 

Introduction of the notion of a multi-Poisson structure as the Poisson version of a multisymplectic 

structure. Application to the reduction of first-order classical field theories 

Marsden-Weinstein multisymplectic reduction. Application to the reduction of first-order classical 

field theories 

Hamilton-Jacobi Theory 

Hamilton-Jacobi theory for hamiltonian systems with respect to linear Poisson structures 

Hamilton-Jacobi theory and complete integrability for (generalized) non-holonomic mechanical 

systems 

Hamilton-Jacobi theory for first-order classical field theories in the Lie algebroid setting 

Tulczyjew’s triple for first-order classical field theories in the Lie algebroid 

setting 

Study of invariant volume forms for non-holonomic mechanical systems 

Introduction of the intrinsic notion of Poisson symmetric space 

Local description on discrete mechanics on Lie groupoids. The exact discrete 

lagrangian on a Lie groupoid 



eduction theory 

Autonomous hamiltonian systems 

The configuration space: 

Q 

The phase space of momenta: 

T ∗ Q 

Hamiltonian function: H : T ∗ Q → R 

Structure: The canonical symplectic structure Ω Q ∈ Ω 2 (T ∗ Q) 

Hamiltonian vector field: H Ω Q 

H ∈ X(T ∗ Q), i Ω H Q 

Ω Q = dH 

H 

Solutions of Hamilton equations: integral curves of H Ω Q 

H 

In the presence of a group of symmetries 

⇓ 

The system can be reduced to a system with less freedom degrees 

Reduction and reconstruction processes ⇒ the solutions of Hamilton equations 

dp 

dt = − ∂H 

∂q , 

dq 

dt = ∂H 

∂p 



ν ∈ g ∗ φ ν : G ν × J −1 (ν) → J −1 (ν), G ν = {g ∈ G/ Ad ∗ ν = ν} 

reduction theory 

Marsden-Weinstein reduction Theorem 

ingredients: 

(M, Ω) symplectic manifold 

φ : G × M → M a free and proper symplectic action of a Lie group G 

J : M → g ∗ an Ad ∗ -equivariant momentum map associated with φ 

ξ M = H Ω J ξ 

, J(φ g (x)) = Ad ∗ g −1 (J(x)), for any ξ ∈ g, x ∈ M 

⇓ 

(J −1 (ν)/G ν, Ω ν) is a symplectic manifold 

π ∗ ν Ων = i ∗ ν Ω 

π ν : J −1 (ν) → J −1 (ν)/G ν, 

i ν : J −1 (ν) → M 



eduction theory 

Marsden-Weinstein reduction Theorem by stages 

G = H 1 × H 2 

Some Applications 

Reduce by the subgroup H 1 at a time and then to reduce by H 2 

⇕ 

Reduce by G 

underwater vehicle dynamics and stability (Leonard and Marsden [1997]) 

Applications to compressible fluids (Holm and Kupershmidt [1983]) 

H closed and normal subgroup of G 

Reduce by H and then by a group related with G/H 

⇕ 

Reduce by G 



two applications of symplectic reduction 

cotangent reduction 

Kirillov-Kostant-Souriau Theorem 



cotangent reduction 

(M, Ω) = (T ∗ Q, Ω Q ) 

φ : G × Q → Q a free and proper action 

T ∗ φ : G × T ∗ Q → T ∗ Q free and proper action 

J : T ∗ Q → g ∗ Ad ∗ -equivariant hamiltonian momentum map 

J(α q)(ξ) = α q(ξ Q (q)) 

ν ∈ g ∗ 

⇓ 

(J −1 (ν)/G ν, (Ω Q ) ν) reduced symplectic manifold 

+ 

α ν ∈ Ω 1 (Q) 

G ν-invariant 1-form with values in J −1 (ν) 

⇓ 

ϕ αν : (J −1 (ν)/G ν, (Ω Q ) ν) → (T ∗ (Q/G ν), Ω Q/Gν − B αν ) embedding 

ϕ αν 

is a symplectomorphism ⇐⇒ g = g ν 





The coadjoint orbits in the dual space of the Lie algebra of a Lie group admit a 

symplectic structure 

ingredients 

G a Lie group with Lie algebra g 

The free and proper action l : G × G → G, 

The symplectic manifold (T ∗ G ∼ = G × g ∗ , Ω G ) 

The symplectic action T ∗ l : G × G × g ∗ → G × g ∗ 

l(g, h) = l g (h) = gh. 

The Ad ∗ -equivariant momentum map J T ∗G : G × g ∗ → g ∗ 

(J T ∗G ) −1 (ν) = {(g, Ad ∗ g ν)|g ∈ G} ≃ G 

⇓ Marsden-Weinstein symplectic reduction theorem 

(J T ∗G ) −1 (ν)/G ν ∼ = G/Gν ≃ O ν 



Poisson reduction 


Poisson manifold (M, {·, ·}) 

φ : G × M → M a free and proper canonical Poisson action 

⇓ 

M/G a Poisson manifold 

{f 1 , f 2 } r ◦ π = {f 1 ◦ π, f 2 ◦ π} , 

where π : M → M/G is the canonical projection and f 1 , f 2 ∈ C ∞ (M/G) 

M = T ∗ Q 

T ∗ φ : G × T ∗ Q → T ∗ Q with φ : G × M → M free and proper action 

T ∗ Q/G → Q/G linear Poisson manifold 

Q = G and φ = l : G × G → G 

⇓ 

the symplectic leaves of T ∗ G/G are the coadjoint orbits 



Linear Poisson structures on a vector bundle 

To extend the reduction process for linear Poisson structures on vector bundles 

A → M vector bundle 

{Λ A ∗ ∈ V 2 (A ∗ ) linear Poisson structure on A ∗ } ↔ {([[·, ·]], ρ) Lie algebroid structure on A} 

[[·, ·]] : Γ(A) × Γ(A) → Γ(A) Lie bracket 

ρ : A → TM vector bundle morphism (anchor map) 

[[X , fY ]] = f [[X , Y ]] + ρ(X )(f )Y , ∀X , Y ∈ Γ(A), ∀f ∈ C ∞ (M) 

⇓ 

(A, [[·, ·]], ρ) Lie algebroid 

Schouten bracket [·, ·] : Γ(∧ p A) × Γ(∧ p′ A) → Γ(∧ p+p′ −1 A) 

The differential of A d A : Γ(∧ k A ∗ ) → Γ(∧ k A ∗ ) 



Symplectic Lie algebroids 

(A → M, [[·, ·]], ρ) Lie algebroid of rank 2n + Ω ∈ Γ(∧ 2 A ∗ ) 

d A Ω = 0, (Ω) n ≠ 0 

⇓ 

M Poisson manifold 

example: 

(τ : A → M, [[·, ·]], ρ) Lie algebroid of rank n 

T A A ∗ → A ∗ vector bundle 

(T A A ∗ ) a ∗ = {(a, v) ∈ A τ ∗ (a ∗ ) × T a ∗ A ∗ /ρ(a) = T a ∗ τ ∗ (v)} 

T A A ∗ → A ∗ is a Lie algebroid of rank 2n ≡ the cover of fiberwise linear Poisson manifold 

λ A ∈ Γ((T A A ∗ ) ∗ ), Ω A = −d T A A ∗ λ A 

⇓ 

T A A ∗ → A ∗ is a symplectic Lie algebroid 



eduction of symplectic lie algebroids 

Reduction process for Lie algebroids 

Reduction process for symplectic Lie algebroids in the presence of a momentum 

map 

Application to the symplectic cover of a fiberwise linear Poisson manifold 

J.C. Marrero, E. Padrón, M. Rodríguez-Olmos: Reduction of a symplectic Lie algebroid with momentum map and 

its application to fiberwise linear poisson structures Preprint 2011. 



eduction of lie algebroids 


(A → M, [[·, ]], ρ, Ω) symplectic Lie algebroid 

Φ : G × A → A action by complete lifts associated with φ : G × M → M 

ψ : g → Γ(A) Lie algebra anti-morphism such that Φ ∗ : G × A ∗ → A ∗ action by Poisson automorphisms 

ξ A ∗ = H Ω̂ψ(ξ) 

⇓ 

Φ T : TG × A → A an affine action of TG ∼ = G × g over A 

Φ T ((g, ξ), a x ) = Φ g (a x + ψ(ξ)(x)) 

⇓ 

A/TG is a Lie algebroid over M/G and ˜π : A → A/TG is a Lie algebroid epimorphism 





(A → M, [[·, ·]], ρ, Ω) symplectic Lie algebroid 

Φ : G × A → A action by complete lifts 

J : M → g ∗ momentum map 

Adg−1 ∗ (J(x)) = J(φg (x)), ∀x ∈ M, ∀g ∈ G 

Φ ∗ g (Ω) = Ω and i ψ(ξ) Ω = d A J ξ , for all g ∈ G and ξ ∈ g 

⇓ 

J T : A → g ∗ × g ∗ , 

J T (a) = ((TJ ◦ ρ)(a), J(τ(a))) 

TG × A → A affine action 

(A ν = (J T ) −1 (0, ν)/TG ν → J −1 (ν)/G ν, Ω ν) symplectic Lie algebroid 



the particular case of T A A ∗ 


(T A A ∗ → A ∗ , Ω A ) symplectic Lie algebroid 

Φ : G × A → A action by complete lifts with respect to ψ : g → Γ(A) 

⇓ 

(Φ, T Φ ∗ ) : G × T A A ∗ → T A A ∗ 

ψ T : g → Γ(T A A ∗ ) 

J A ∗ : A ∗ → g ∗ hamiltonian momentum map 

J A ∗ (α x )(ξ) = α x (ψ(ξ)(x)) 

((T A A ∗ ) ν, Ω ν) → (T A 0,ν 

A ∗ 0,ν , Ω A 0,ν 

− (pr 1 ) ∗ B ν) symplectic Lie algebroid embedding 

isomorphism if and only if g = g ν 

A 0,ν = A/TG ν → M/G ν 




new objectives: 

To apply this reduction process to new examples 

Bundle version reduction of the symplectic cover of a fiberwise linear Poisson 

manifold 

Reduction by stages of symplectic Lie algebroids 



Linear Poisson manifolds on a vector bundle with a fiber metric 

The cotangent sphere bundle of a Riemannian manifold (Q, g) of dimension n 

S n−1 (T ∗ Q) = T ∗ 1 Q = {α ∈ T ∗ Q/g(α, α) = 1} → Q 

i S n−1 (T ∗ Q) 

: S n−1 (T ∗ Q) → T ∗ Q, 

λ ∈ Ω 1 (T ∗ Q) Liouville 1-form 

θ = −i ∗ S n−1 (T ∗ Q) (λ) ∈ Ω1 (S n−1 (T ∗ Q)) 

θ ∧ (dθ) n is a volume form on T ∗ Q 

Unit sphere of a real Lie algebra with scalar product (g, [·, ·], < ·, · >) of dimension n 

S n−1 (g ∗ ) = {α ∈ g ∗ / < α, α >= 1} 

Λ g ∗ Lie-Poisson structure on g ∗ 

¯Λ = Λ g ∗ − ∆ g ∗ ∧ i αg ∗ Λ g ∗ ∈ V2 (g ∗ ), Ē = i αg ∗ Λ g ∗ ∈ X(g∗ ), 

(g ∗ , (¯Λ, Ē)) Jacobi manifold 

[¯Λ, ¯Λ] = 2Ē ∧ ¯Λ, [Ē, ¯Λ] = 0 

⇓ 

(S n−1 (g ∗ ), (¯Λ |S n−1 (g ∗ ) 

, Ē |S n−1 (g ∗ ) 

)) Jacobi manifold 



(A ∗ , Λ A ∗ ) linear Poisson manifold with τ : A → M + g fiber bundle metric on A ∗ 

⇓ 

S n−1 (A ∗ ) → Q Jacobi manifold ? 

T A S n−1 (A ∗ ) 

T A i 

✲ 

T 

A A 

∗ 

❄ 

S n−1 (A ∗ ) 

i 

✲ 

❄ 

T A S n−1 (A ∗ ) → S n−1 (A ∗ ) subalgebroid of T A A ∗ → A ∗ 

A 

∗ 

θ A = −(T A i) ∗ (λ A ) contact structure 

Ω A = −d T A A ∗ λ A symplectic structure 

θ A ∧ (d A θ A ) n ≠ 0 

⇓ 

S n−1 (A ∗ ) Jacobi manifold 

A ∗ linear Poisson manifold 

i ∗ T A A ∗ = Ti(T A S 1 (A ∗ )) ⊕ i ∗ (< ∆ >), ∆ ∈ Γ(T A A ∗ ) 



The Jacobi structure on the sphere of a linear Poisson manifold on a vector 

bundle with fiber metric 

D. Chinea, J.C. Marrero, E. Padrón: The Jacobi structure on the sphere of a linear Poisson manifold on a vector 

bundle with fiber metric, work in progress 

New Objetives 

Give an explicit description of the Jacobi structure on S n−1 (A ∗ ) for known 

examples of Lie algebroids A 

Give a contact reduction procedure for contact Lie algebroids in the presence of 

momentum map, apply this result to the particular case of T S n−1 (A ∗ ) 

Under certain regularity conditions, study the manifold S n−1 (A ∗ )/E Is it again a 

Poisson manifold? 

R 2n symplectic manifold ⇒ S n−1 (R 2n ) contact manifold ⇒ S n−1 (R 2n )/E ∼ = CP 

n−1 symplectic manifold 

A ∗ linear Poisson manifold ⇒ S n−1 (A ∗ ) Jacobi manifold ⇒ S n−1 (A ∗ )/E Poisson manifold 



Hamiltonian Systems 

Non-autonomous hamiltonian systems 

The configuration space: 

The phase space of momenta: 

extended version: T ∗ M 

π : M → R surjective submersion 

restricted version: V ∗ π V x π = T x π for all x ∈ M 

Structure: µ π : T ∗ M → V ∗ M 

µ π : T ∗ M → V ∗ π is a principal R-bundle 

ψ π : R × T ∗ M → T ∗ M, 

ψ π(s, α x ) = α x + sπ ∗ (dt)(x) 

ψ π is symplectic 

µ : (A, Ω) → V is a symplectic principal R-bundle if Ω is a symplectic structure on A 

such that the associated principal action ψ : R × A → A is symplectic 

µ : (A, Ω) → (V ∼ = A/R, Λ) Poisson morphism 



canonical actions and momentum maps 

φ : G × A → A is a canonical action on the symplectic principal R-bundle 

µ : (A, Ω) → V 

φ is a symplectic action 

φ g ◦ ψ s = ψ s ◦ φ g 

for any g ∈ G, s ∈ R 

the 1-form ζ µ = i Zµ Ω is basic with respect to φ, i.e ζ µ(ξ A ) = 0 for any ξ ∈ g 

J : A → g ∗ Ad ∗ -equivariant momentum map for φ 

⇓ 

+ 

φ V : G × V → V Poisson action 

φ V (g, v) = µ(φ g (a))for any g ∈ G, v ∈ V 

J V : V → g ∗ Ad ∗ -equivariant momentum map for φ v 

J V (v) = J(a), with a ∈ µ −1 (v) 



eduction process of symplectic R-principal bundle 


µ : (A, Ω) → V symplectic R-principal bundle 

φ : G × A → A canonical action with φ V : G × V → V is free and proper 

J : A → g ∗ Ad ∗ -equivariant momentum map 

ν ∈ g ∗ ⇓Marsden-Weinstein Th. 

⇓Poisson reduction Th. 

(A ν = J −1 (ν)/G ν, Ω ν) V ν = (J V ) −1 (ν)/G ν 

⇓ 

µ ν : (A ν, Ω ν) → V ν is a symplectic principal R-bundle 

{·, ·} ν Poisson on V ν ≡ the Poisson bracket induced by µ ν : A ν → V ν 



standard symplectic principal R-bundle reduction theorem 

φ : G × M → M free and proper action 

π : M → R a G-invariant surjective submersion 

ν ∈ g ∗ 

λ ν ∈ Ω 1 (M) a G ν-invariant 1-form with values in J −1 

ν (ν ′ ) 

Then ˜π ν : M/G ν → R is a surjective submersion and there is a symplectic principal 

R-bundle embedding 

((T ∗ M) ν, (Ω M ) ν) 

ϕ λν 

(T ∗ (M/G ν), Ω M/Gν − B λν ) 

(µ π) ν 

 

µ ˜πν 

 

(V ∗ π) ν 

ϕ V λν 

V ∗˜π ν 

B λν ∈ Ω 2 (T ∗ (M/G ν )) magnetic term associated with λ ν 

ϕ λν 

is a symplectic principal R-bundle isomorphism ⇔ g = g ν 



Kirillov-Kostant-Souriau theorem (version for principal G-bundle 

with base space R 


φ : G × M → M free and proper action with principal G-bundle projection 

π : M → M/G ≃ R 

⇓ 

T ∗ φ : G ×T ∗ M → T ∗ M + Ad ∗ -equivariant momentum map J : T ∗ M → g ∗ 

⇓ 

µ π : T ∗ M → V ∗ π be the standard symplectic R-bundle 

ν ∈ g ∗ 

⇓ symplectic principal R-bundle reduction theorem 

(µ π) ν : ((T ∗ M) ν, (Ω M ) ν) → (V ∗ π) ν ∼ = M/Gν 

The Poisson structure on M/G ν 

The space of orbits M/G ν of the action of G ν on M admits a Poisson structure 

{·, ·} M/Gν and the symplectic leaf of M/G ν passing through the point [x] is 

symplectomorphic to O ν 

Research group ULL Some problems in Geometric Mechanics

Non-autonomous hamiltonian systems 

I. Lacirasella, J. C. Marrero, E. Padrón: Reduction of symplectic R-bundles, Preprint 2012, arXiv:1201.4690 

To develop the bundle version of the reduction of the standard symplectic 

principal R-bundle 

To reconstruct the dynamics of a non-autonomous hamiltonian systems on a 

symplectic principal R-bundle from reduced dynamics. 

To study the reduction theory by stages of a symplectic principal R-bundle 



multisymplectic geometry 

Classical field theories ⇔ multisymplectic formulation 

multisymplectic structure Ω ∈ Ω k (M), dΩ = 0 

i X Ω = 0 ⇒ X = 0, 

∀X ∈ X(M) 

example 

M = ∧ k T ∗ Q 

λ M (α)(X 1 , . . . , X k ) = α(T π(X 1 ), . . . , T π(X k )), 

π : ∧ k T ∗ Q → Q. 

Ω M = −dλ M es una forma multisimpléctica en Q 



polisympletic structures 

k-polisymplectic structure on M 

⇕ 

(ω 1 , . . . , ω k ) closed non-degenerate 2-forma R k -valued 

k∑ 

¯ω = ω A ⊗ e A 

A=1 

{e 1 , . . . , e k } canonical base of R k . 

Example 1: 

(T 1 k )∗ Q := T ∗ Q⊕ (k . . . ⊕T ∗ Q, 

ω A = (π k,A 

Q )∗ ω, 

ω canonical symplectic form on T ∗ Q and π k,A 

Q : T ∗ Q⊕ . (k . . ⊕T ∗ Q → T ∗ Q the 

projection 

((T 1 k )∗ Q, ω 1 , . . . , ω k ) polisympletic manifold 




G a Lie group and g its Lie algebra 

g ∗ × (k . . . ×g ∗ 

Coad k : G × g ∗ × (k . . . ×g ∗ → g ∗ × (k . . . ×g ∗ 

(g, ν 1 , . . . , ν k ) ↦→ (Coad (g, ν 1 ), . . . , Coad (g, ν k )) 

(ν 1 , . . . , ν k ) ∈ g ∗ × (k . . . ×g ∗ coadjoint k-orbits 

O (ν1 ,...,ν k ) = {Coad k (g, ν 1 , . . . , ν k ) | g ∈ G} , 

is a polisymplectic manifold 

ω A ν : = (pr A ) ∗ ω νA , 

pr A : : O (ν1 ,...,ν k ) → O νA , (ν 1 , . . . , ν k ), ↦→ ν A canonical projection and ω νA sympletic 

form of O νA 

Research group ULL Some problems in Geometric Mechanics

We return to the symplectic and Poisson cases 

Ω symplectic structure 

⇕ 

♭ ω : TM → T ∗ M, ♭ ω(v x ) = i vx ω x vector bundle isomorphism 

♭ ω([X , Y ]) = L X ♭ ω(Y ) − L Y ♭ ω(X ) − d(♭ ω(X )(Y )), ∀X , Y ∈ X(M) 

(M, Λ) Poisson manifold 

⇓ 

♯ Λ : T ∗ M → TM, ♯ Λ (α x ) = i αx Λ 

D = Im♯ Λ integrable distribution whose leaves are symplectic manifolds 

(T ∗ M, [[·, ·]], ♯ Λ ) Lie algebroid 

[[α, β]] = L ♯Λ (α)β − L ♯Λ (β)α − d(β(♯ Λ (α))) 

{ ♯Λ : T 

Λ Poisson structure ⇔ 

∗ → TM vector bundle homomorphism 

♯ Λ ([[α, β]]) = [♯ Λ (α), ♯ Λ (β)] ∀α, β ∈ Ω 1 (M) 




¯ω k-poly-symplectic structure on M 

⇕ 

♭ ω : TM → ♭ ω (TM) ⊆ (Tk 1 )∗ M := T ∗ M⊕ . k . . ⊕T ∗ M ♭ ω (v x ) = (i vx (ωx 1 ), . . . , ivx (ωk x )) 

isomorphism vector bundle from TM to a sub-bundle of T ∗ M⊕ . k . . ⊕T ∗ M 

Skew-symmetry: 

♭ ω (v x )(v x ) = (0, . . . , 0) 

(v x ∈ T x M, x ∈ M). 

Non degenerate condition: ♭ ω vector bundle monomorphism 

Ker (♭ ω ) = {0} 

Integrability condition: 

♭ ω ([X , Y ]) = L X ♭ ω (Y ) − L Y ♭ ω (X ) − d(♭ ω (X )(Y )), 

X , Y ∈ X(M) 



Poli-Poisson structures 

(S, ♯) poli-Poisson structure 

⇕ 

S vector sub-bundle of (Tk 1)∗ M 

♯ : S → TM vector bundle morphism 

Skew-symmetry: α i (♯(α)) = 0, α = (α 1 , . . . , α k ), i ∈ {1, . . . , k} 

Non degenerate condition: If α(♯(β)) = 0 ∀β ∈ S ⇒ ♯(α) = 0 

Integrability condition: α, β ∈ Γ(S) 

( 

) 

[♯(α), ♯(β)] = ♯ L ♯(α) β − L ♯(β) α − d(β(♯(α))) 




Polisymplectic manifold ⇒ Poli-Poisson manifold 

S = ♭ ω (TM) and ♯ = ♭ −1 

ω 

Poli-Poisson manifold of order k = 1 ≡ Poisson manifold 

properties: 

(M, S, ♯) poli-Poisson manifold⇒ D = Im♯ is integrable 

L a leaf of F D ⇒ polisymplectic structure 



polisympletic reduction 

ingredients: 

(M, ¯ω = (ω 1 , . . . , ω k )) a poly-simplectic manifold 

Φ : G × M → M a proper and free action such that Φ ∗ g ω i = ω i , 

∀g ∈ G 

π : M → M/G 

Im (♭ ¯ω ) ( ∩ [(V π) ◦ × . (k . . ×(V π) ◦ ] is a vector sub-bundle 

(♭ ¯ω ) −1 [((V π) ⊥ ) ◦ × . (k . . ×((V π) ⊥ ) ◦ ] ∩ [(V π) ◦ × . (k 

) 

. . × ∩ (V π) ◦ ] ∩ Im (♭ ¯ω ) 

(V π) ⊥ = 

⇓ 

k⋂ 

(ω ♭ i )−1 ((V π) ◦ ) 

i=1 

M/G is a poli-Poisson manifold 

D. Iglesias, J.C. Marrero, M. Vaquero: On poly-Poisson structures, work in progress 

⊂ V π 



Kirillov-Kostant-Souriau Theorem (version for polisymplectic 

setting) 

G Lie group 

G acts on the poly-symplectic manifold (T 1 k )∗ G and satisfies the above conditions 

((T 1 k )∗ G)/G ∼ = g ∗ × k) . . . ×g ∗ 

the leaves of the poly-symplectic foliation are the k-coadjoint orbits 

J.C. Marrero, N. Román Roy, M. Salgado, S. Vilarino: Reduction of polisymplectic structures, work in progress 



multisymplectic setting 

New objetives 

To introduce the notion of multi-Poisson structure. Application to the reduction 

of first order classical field theories. 

To develop the multisymplectic reduction. Application to the reduction of first 

order classical field theories 



Poisson reduction 

Thanks!!!!!

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