Symplectic, Presymplectic, Poisson, Dirac, ...,

Symplectic, Presymplectic, Poisson, Dirac, ...
Eduardo Martínez
University of Zaragoza
emf@unizar.es
Jaca (Spain), August 2003

Symplectic structures
Hamiltonian mechanics
ẋ = ∂H
∂p
ṗ = − ∂H
∂x .
Can be written in the form ż = JdH(z):
[ẋ ] [ ] [ 0 ∂H
]
In
=
∂p
ṗ −I n 0
∂H
∂x
A symplectic structure on a manifold M is a 2-form ω on M
such that it is
□ Closed: dω = 0.
□ Non degenerated: v ↦→ ω(v, ·) has inverse.
1

Pre-Symplectic structures
The nondegeneration condition fails:
– Singular Lagrangians
– Hamiltonian mechanics with constraints
– Pullback of a symplectic form to a submanifold.
A presymplectic structure on a manifold M is a 2-form ω on
M such that it is
□ Closed: dω = 0.
2

Poisson bracket
In both cases we can define a Poisson algebra
□ f ∈ C ∞ (M) ↦→ X f ∈ X(M) Hamiltonian vectorfield.
□ {f, g} = −ω(X f , X g ).
□ Hamilton equations ˙ f = {f, H}.
In the presymplectic case restrict to admissible functions.
f admissible ⇐⇒ ∃X ∈ X(M) s.t. i X ω = df.
3

Poisson structures
The Poisson bracket on the dual of a Lie algebra is not defined
by a presymplectic form.
Contravariant point of view: consider the inverse Λ of the symplectic
form
{f, g} = Λ(df, dg)
and ‘allow it to be degenerate’.
A Poisson structure on a manifold M is a 2-vector Λ on M such
that it is closed: [Λ, Λ] = 0.
Constraints are still a problem.
4

Dirac structures
A Dirac structure on a manifold M is a subbundle
such that it is
D ⊂ T ∗ M × M T M
□ Maximally isotropic for ( , ) +
(
(α, v), (β, w)
)
+
= 〈v, β〉 + 〈w, α〉.
□ Closed:
〈X 1 , L X2 α 3 〉 + 〈X 2 , L X3 α 1 〉 + 〈X 3 , L X1 α 2 〉 = 0.
for all (X i , α i ) sections of D.
5

□ The graph of a presymplectic form is a Dirac structure.
□ The graph of a Poisson tensor is a Dirac structure.
□ Restricts easily to submanifolds.
□ Implicit Hamilton equations:
(ẋ(t), dH(x(t))) ∈ D x(t) .
□ Defines a Poisson bracket of admissible functions.
f admissible ⇐⇒ ∃X ∈ X(M) s.t. (df, X) ∈ D.
6

□ The manifold M is foliated by presymplectic leaves.
□ The closure condition is equivalent to closure of the Courant
bracket
[(α, X), (β, Y )] = ( L X β − i Y dα, [X, Y ] )
i.e. (D, [ , ], ρ = pr 2
∣ ∣∣D
) is a Lie algebroid.
7

Lie Algebroids
A Lie algebroid structure on the vector bundle τ : E → M is given
by
□ a Lie algebra structure (Sec(E), [ , ]) on the set of sections
of E, and
□ a morphism of vector bundles ρ: E → T M over the identity,
such that
[σ, fη] = f[σ, η] + (ρ(σ)f) η,
where ρ(σ)(m) = ρ(σ(m)).
8

Examples
Tangent bundle
E = T M,
ρ = id,
[ , ] = bracket of vector fields.
Integrable subbundle
E ⊂ T M, integrable distribution
ρ = i, canonical inclusion
[ , ] = restriction of the bracket to vector fields in E.
9

Lie algebra
E = g → M = {e}, Lie algebra (fiber bundle over a point)
ρ = 0, trivial map (since T M = {0 e })
[ , ] = the bracket in the Lie algebra.
Atiyah algebroid
Let π : Q → M a principal G-bundle.
E = T Q/G → M, (Sections are equivariant vector fields)
ρ([v]) = T π(v) induced projection map
[ , ] = bracket of equivariant vectorfields (is equivariant).
10

Action algebroid
Let Φ: g → X(M) be an action of a Lie algebra g on M.
E = M × g → M,
ρ(m, ξ) = Φ(ξ)(m) value of the fundamental vectorfield
[ , ] = induced by the bracket on g.
11

Exterior differential
On 0-forms
df(σ) = ρ(σ)f
On p-forms (p > 0)
dω(σ 1 , . . . , σ p+1 ) =
∑p+1
= (−1) i+1 ρ(σ i )ω(σ 1 , . . . , ̂σ i , . . . , σ p+1 )
i=1
+ ∑ i

Morphism of a Lie algebroid
A linear bundle map Φ: E → E ′ over ϕ: M → M ′ is a Lie
algebroid morphism if pull-back commutes with d, i.e.
where
and
Φ ∗ ◦ d = d ◦ Φ ∗ ,
(Φ ∗ ω) m (a 1 , . . . , a p ) = ω ϕ(m) (Φ(a 1 ), . . . , Φ(a p )),
In particular, it follows that
Φ ∗ f = f ◦ ϕ.
ρ ◦ Φ = T ϕ ◦ ρ.
13

Hamiltonian Mechanics
Prolong the dual π : E ∗ → M to a Lie algebroid π 1 : T E ∗ → E ∗
T µ E ∗ = { (b, V ) ∈ E m × T µ E ∗ | T π(V ) = ρ(b) }.
and anchor is ρ 1 (b, V ) = V , and the bracket makes T π : (b, V ) ↦→
b a morphism.
There exists a canonical 1-form
θ 0 (b, V µ ) = 〈µ, b〉.
The 2-form ω 0 = −dθ 0 is regular and obviously closed.
Given a Hamiltonian H ∈ C ∞ (E ∗ ), the dynamical vector field is
X H = ρ 1 (σ H ), where σ H is the solution of
i σH ω 0 = dH.
14

Lagrangian Mechanics
Prolong τ : E → M to a Lie algebroid τ 1 : T E → E
T a E = { (b, V ) ∈ E m × T a E | T τ(V ) = ρ(b) }.
and anchor is ρ 1 (b, V ) = V , and the bracket makes T τ : (b, V ) ↦→
b a morphism.
Given a Lagrangian L ∈ C ∞ (E), there exists a 1-form
θ L (b, V a ) = ξ V a (b)L.
The 2-form ω L = −dθ L is obviously closed and it is regular iff
∂ 2 L/∂y∂y is regular.
The dynamical vector field is ρ 1 (Γ), where Γ is the solution of
i Γ ω L = dE L .
15

Symplectic Lie algebroids
A presymplectic form on a Lie algebroid E is a 2-form on E,
Ω: E ∧ E → R, which is closed with respect to the differential in
the Lie algebroid: dω = 0 .
A symplectic form on a Lie algebroid is a nondegenerated
presymplectic form on E.
(M, Ω) presymplectic manifold
↕
(T M, Ω) presymplectic Lie algebroid
16

Hamiltonian sections
Hamiltonian section σ f
C ∞ (M)
defined by an admissible function f ∈
i σf Ω = df.
Hamiltonian vectorfield X f = ρ(σ f ).
Poisson bracket of two admissible functions
{f, g} = −Ω(σ f , σ g ).
In particular
If Ω is symplectic, the base M is Poisson
17

Compatibility
A 2-form Ω is compatible with the Lie algebroid structure if
ker ρ ⊂ ker Ω.
Theorem 1 If a presymplectic form Ω is compatible, then the
leaves of the Lie algebroid are presymplectic manifolds. In that
case, the leaves are symplectic if and only if ker ρ = ker Ω.
Proof: ω L (v, w) = Ω(a, b) for ρ(a) = v and ρ(b) = w. Thus
Ω| EL = ρ ∗ ω L .
18

Theorem 2 If Ω is a compatible presymplectic form on E and
Φ: Ē → E is a morphism, then ¯Ω = Φ ∗ Ω is a compatible presymplectic
form on Ē.
Proof: If a ∈ ker ¯ρ then Φ(a) ∈ ker ρ because ρ(Φ(a)) =
T ϕ(¯ρ(a)).
Therefore Φ(a) ∈ ker Ω.
Thus ¯Ω(a, b) = Ω(Φ(a), Φ(b)) = 0 for all b, i.e. b ∈ ker ¯Ω.
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Quotient by the kernel
The kernel of Ω is a Lie subalgebroid:
i [σ1,σ 2]Ω = d σ1 i σ2 Ω = 0.
We can quotient by ker Ω
ker Ω
E
p
E/ ker Ω
τ
M M/ρ(ker Ω)
provided that some regularity properties are asumed.
20

The form Ω defines a form ¯Ω in the quotient
¯Ω [m] ([a], [b]) = Ω m (a, b)
Since p is a morphism, then ¯Ω is closed and by construction is
regular, therefore it is symplectic.
It follows that
M/ρ(ker Ω) is a Poisson manifold
21

Canonical form on a Dirac structure
Let D be an almost-Dirac structure on a manifold M. The canonical
2-form on D is the 2-form Ω defined by
Ω((α, v), (β, w)) = 〈v, β〉 = −〈w, α〉.
Proposition 1 The kernel of Ω is ker Ω = ker ρ ⊕ ker λ, where
λ: D → T ∗ M is the projection onto the first factor. In particular,
Ω is compatible with the Lie algebroid structure.
Proof:
a = (α, v) ∈ ker ρ iff v = 0. Thus
Ω(a, b) = 〈0, β〉 = 0
for all b = (w, β) ∈ D
22

Closed iff Closed
Theorem 3 An almost-Dirac structure D on a manifold M is
closed (i.e. it is a Dirac structure) if and only if
∑ (
)
ρ(σ i )Ω(σ j , σ k ) − Ω([σ i , σ j ], σ k ) = 0
ciclic(i,j,k)
for all σ i , σ j , σ k sections of D.
23

Proof:
∑
ciclic(i,j,k)
(
)
ρ(σ i )Ω(σ j , σ k ) − Ω([σ i , σ j ], σ k ) =
∑
ciclic(i,j,k)
∑
ciclic(i,j,k)
∑
ciclic(i,j,k)
(
)
ρ(σ i )〈ρ(σ j ), λ(σ k )〉 − 〈ρ([σ i , σ j ]), λ(σ k )〉 =
(
)
X i 〈X j , α k 〉 − 〈[X i , X j ], α k 〉
〈X j , L Xi α k 〉.
=
24

Dirac structures
are particular cases of
compatible presymplectic Lie
algebroids.
25

Gauged
Given a closed 2-form ω on the manifold M we can define the Lie
algebroid
E = { (α + ω(v), v) | (α, v) ∈ D }
with the Courant bracket. Notice that E is a Lie algebroid but
it is not a Dirac structure. It is said to be obtained by a gauge
transformation from D.
Nevertheless, there also exists a 2-form
which is closed.
¯Ω(a, b) = 〈ρ(a), λ(b)〉
26

The map Φ: D → E given by Φ(α, v) = (α + B(v), v) is an
isomorphism of Lie algebroids and we have Φ ∗ ¯Ω = Ω − ρ ∗ ω.
Therefore gauge equivalent Dirac structures have the same leaves
but the presymplectic structure on every leaf is different.
27

Twisted
Given a closed 3-form φ on the manifold M we can modify the
Courant bracket
[(α, X), (β, Y )] = (L X β − i Y dα + φ(X, Y, ·), [X, Y ]).
A φ-twisted Dirac structure is an almost Dirac structure which is
closed under the above bracket.
An almost Dirac structure D is a φ-twisted Dirac structure if and
only if
dΩ + ρ ∗ φ = 0.
28

Conclusions and open problems
□ Symplectic and presymplectic forms on Lie algebroids appears
naturally in the study of some dynamical systems, and
are as easy to use as ordinary presymplectic forms.
□ Most (all?) geometric constructions on Dirac structures
are but a consequence of the existence of the presymplectic
structure.
□ There is a well known constraint algorithm for presymplectic
forms.
□ The theory of symmetries for presymplectic forms leads to
the results by Blankenstein and Van der Schaft about symmetries
of Dirac structures.
29

□ A Dirac structure integrates to a presymplectic groupoid.
What is the integration of a compatible presymplectic Lie
algebroid?
□ Non-compatible symplectic forms are also of interest. What
is the integration of a non-compatible presymplectic Lie algebroid?
□ Jacobi versions?
30

In collaboration with Jesús Clemente-Gallardo, University of Coimbra.
The end
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