Moment Map - Symplectit Geometry

7/28/2019 Moment Map - Symplectit Geometry

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Lie Groups Actions

Lie Algebra

Exponential Map and Inifinitesimal Action

Adjoint and Coadjoint Representation

Symplectic Notions. Hamiltonian Systems

Examples

Momentum Maps in Symplectic Geometry

Iulian Danciu

West University of TimisoaraMAGS

May 24, 2013

Iulian Danciu Momentum Maps in Symplectic Geometry

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Lie Groups Actions

Lie Algebra




Examples

1 Lie Groups ActionsExamples

2 Lie Algebra

3 Exponential Map and Inifinitesimal Action

4 Adjoint and Coadjoint Representation

5 Symplectic Notions. Hamiltonian Systems

6 ExamplesLinear MomentumAngular MomentumU(n ) acts on Cn


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Lie Groups Actions

Lie Algebra




Examples

Examples

Lie Group Actions

This section present a brief review of the theory of Lie groups actionson a manifold.

Definition 1Let M be a manifold and G be a Lie group. A left action of G on M is a smooth mapping φ : G × M → M such that

1) φ(e , m ) = m , e ∈ G, (∀) m ∈ M ,

2) φ(g , φ(h , m )) = φ(gh , m ), (∀) g , h ∈ G and m ∈ M ,

where e denotes the identity of G.

We will often use the notation g · m := φ(g , m ) := φg (m ).


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Lie Groups Actions

Lie Algebra




Examples

Examples


2 Lie Algebra

3






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Lie Groups Actions

Lie Algebra




Examples

Examples

Examples

Example 1

An example of left action of a Lie group G on itself is given by lefttranslation Lg : G → G , h → gh .

We saw that an example ol left action of a Lie group G on itself isgiven by left translation Lg . Similarly, the right translationR g : G → G , h → hg define a right action. The inner automorphismAdg ≡ I g : G → G given by I g := R g −1 ◦ Lg defines a left action of G onitself called conjugation.


Li G A ti

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Lie Groups Actions

Lie Algebra




Examples

Lie Algebra

Let g be the Lie algebra of G , that is the tangent space on G at theidentity element, endowed with bracket [, ] induced via left invariant

vector fields. The dual of the Lie algebra is denoted by g

∗

. For eachξ ∈ g and l ∈ g∗, we sometimes use the following notation:

< l , ξ >:= l (ξ ).

Also, the Jacobi identity is verified:

[u , [v , w ]] + [v , [w , u ]] + [w , [u , v ]] = 0. (1)


Lie Groups Actions

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Lie Groups Actions

Lie Algebra




Examples

Exponential Map and Infinitesimal Action

We will denote exponential map with exp and it is defined to be afunction from g to G :

exp : g → G .

Every ξ ∈ g induces a vector field ξ M on M usually called theinfinitesimal action, which is defined by:

ξ M (x ) :=

d

dt

t =0 exp(t ξ ) · x .


Lie Groups Actions

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Lie Groups Actions

Lie Algebra




Examples


For ξ ∈ g and g ∈ G the Ad-operation is defined as follows

Adξ =d

dt

t =

0

g · exp(t ξ ) · g −1,

which can be interpreted as a map from g to itself, and is convenientto denote it by Adg (ξ ) = g ξ g −1.Its adjoint, Ad

∗

g : g∗ → g∗ is given by

The differential at the identity of the conjugation map defines a linear

left action of a Lie group G on its Lie algebra g, called the adjointrepresentation of G on g, that is:

Adg := T e I g : g → g.


Lie Groups Actions

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Lie Groups Actions

Lie Algebra




Examples

Symplectic Notions

Definition 2

A symplectic manifold is a pair (M , ω), where M is a manifold and ω

is a closed nondegenerate two form on M.

Since ω is a differential two-form, hence skew-symmetric, thedimension of M is always even. Let (M , ω) be a symplectic manifold.A group action φ of G - sometimes called G -action - is calledsymplectic if the symplectic form is invariant under each pullback:

φ∗g ω = ω, for all g ∈ G . (2)


Lie Groups Actions

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Lie Groups Actions

Lie Algebra




Examples

Moment Map - Symplectic Case

A moment map for a symplectic G -action, with Lie group G on M , is asmooth map µ : M → g

∗ such that the following properties hold.

i) < d µ(x )v , ξ >= ωx (ξ M (x ), v ) for all x ∈ M , v ∈ T x M and ξ ∈ g

where ξ M is the infinitesimal action of ξ .

ii) µ(φg (x )) = Ad∗

g −1 µ(x ) holds for all x ∈ M and g ∈ G .


Lie Groups Actions

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p

Lie Algebra




Examples

Hamiltonian System

Definition 3

A Hamiltonian dynamical system is a triple (M , ω, H ), where (M , ω) is a symplectic manifold and H ∈ C ∞(M ) is the Hamiltonian function

of the system.

To each Hamiltonian system we can associate a Hamiltonian vector

field M H ∈ TM , defined by the identity

i M H ω = dH . (3)


Lie Groups Actions

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p

Lie Algebra




Examples

Definition 4

The action φ of G on (M , ω) is called Hamiltonian if it is symplectic and there exists a moment map µ : M → g

∗ for it.

Definition 5

Let g be a Lie algebra acting on the manifold M. Suppose that for any ξ ∈ g the vector field ξ M is Hamiltonian, with function H ∈ C ∞(M )such that ξ M = M H . The map µ : M → g

∗ is defined by the relation

H (m ) =< µ(m ), ξ >

for all ξ ∈ g and m ∈ M, is called momentum map of the g-action.


Lie Groups Actions

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn


2 Lie Algebra

3

Exponential Map and Inifinitesimal Action4 Adjoint and Coadjoint Representation




Lie Groups Actions

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Linear Momentum

Now we will discuss about 1-particle system, with configuration spaceM = R3. Let R3 act on M by translations, i.e., φ : R3 × M → M isgiven by

x ,

q 1, q 2, q 3, p 1

, p 2

, p 3

=

q 1 + x , q 2 + x , q 3 + x , p 1

, p 2

, p 3

The infinitesimal generator corresponding to ξ ∈ g = R3 is:

ξ M (q 1, q 2, q 3) =d

dt

t =

0

φ(t ξ, (q 1, q 2, q 3)) (4)

=d

dt

t =0

(q 1 + t ξ, q 2 + t ξ, q 3 + t ξ )

= (ξ , ξ , ξ ) ∈ T q 1,q 2,q 3R3.


Lie Groups Actions

Li Al b

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Linear Momentum

On the other hand, by definition of the canonical symplectic structure,any candidate H has a Hamiltonian vector field given by

M H (q j , p j ) =

∂ H (ξ )

∂ p j ,−

∂ H (ξ )

∂ q j

.

Then, M H = ξ M implies that

∂ H (ξ )

∂ p j = ξ and

∂ H (ξ )

∂ q j = 0, , 1 ≤ j ≤ 3.

Solving these equations we get:

H (ξ )(q j , p j ) =

3

j =1

p j

· ξ i.e. J (q j , p j ) =

3 j =1

p j . (5)

This expression is called the linear momentum of the 1-particle

system.Iulian Danciu Momentum Maps in Symplectic Geometry

Lie Groups Actions

Lie Algebra

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn


2 Lie Algebra






Lie Groups Actions

Lie Algebra

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Angular Momentum

Let SO(3) act on the configuration space M = R3 by φ(A, q ) := Aq .The Lie algebra so(3) of SO(3) may be identified with R3 as follows.

We define the vector space isomorphismˆ: R3

→ so(3) called the hatmap, by

v = (v 1, v 2, v 3) → v =

0 −v 3 v 2

v 3 0 −v 1−v 2 v 1 0

.

Note thatv · w = v × w .


Lie Groups ActionsLie Algebra

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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Angular Moment

The infinitesimal generator corresponding to ξ ∈ g = so(3) is:

ξ M (q , p ) =d

dt

t =0

(exp(t ξ )q , exp(t ξ )p ) (6)

= (ξ p , ξ q )

= (ξ × q , ξ × p ) .



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Lie Algebra




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Angular Moment

As in the previous example, to find the momentum map, we solve

∂ H (ξ )

∂ p = ξ q and −

∂ H (ξ )

∂ q = ξ p . (7)

A solution is given by

H (ξ ) = (ξ q ) · p = (ξ × q ) · p = (q × p ) · ξ,

so thatJ (q , p ) = q × p . (8)



Li M t

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g




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn


2 Lie Algebra







Linear Momentum

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g




Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

U(n ) acts on Cn

Let U(n ) act on the space M = Cn by φ(A, z ) := Az . The Lie algebrafor U(n ) is:

u(n ) = {A ∈ L (Cn

,Cn

) :< Ax , y >= − < x , Ay >}. (9)

The first step is to find out the infinitesimal generator correspondingto ξ ∈ g = u(n ).

ξ M (q , p ) =

d

dt

t =0 (exp(t ξ )z ) (10)

= ξ z ∈ Cn .



Linear Momentum

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Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

U(n ) acts on Cn

We want to check that

i ξM ω = dz H , for all ξ ∈ g.

where

H (z ) =1

2< z , ξ z > .

d z H ξ · w =

1

2 < w , ξ z > +

1

2 < z , ξω > (11)

= −1

2< ξ w , z > −

1

2< ξ z , w >



Linear Momentum

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Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

U(n ) acts on Cn

= −1

2< ξ z , w > −

1

2< ξ w , z > (12)

= −12

< ξ z , w > + 12

< ξ z , w >

= −1

2

< ξ z , w > −< ξ z , w >

= −Im < ξ z , w > .

i ξM ω = ω (ξ M (z ), w ) = −Im < ξ z , w >

So the moment map < J , ξ >= 12

< z , ξ z >



E i l M d I ifi i i l A iLinear Momentum

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Examples

Linear Momentum

Angular Momentum

U(n ) acts on Cn

Bibliography

Jerrold E. Marsden., Tudor S. Ratiu, Introduction to Mechanicsand Symmetry, Springer - Verlag, New - York, 1994.


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