Symplectic Structures

The Symplectic Form

Symplectic Manifold

Let be an even dimensional smooth manifold. A symplectic structure on is a closed non-degenerate differential 2-form , called the symplectic form, such that , and for all nonzero , there exists so that . The pair is called a symplectic manifold. We call a symplectic manifold exact if the symplectic form is exact.

e.g.

• Suppose is a coordinate of . Then is the standard symplectic form on , with exact form . Clearly, by definition of the exterior derivative.
• The cotangent bundle of a smooth manifold has a natural exact symplectic structure. In fact, there is a canonical 1-form on , which can be defined as follows: suppose is the natural projection that maps any covector to the base point where it is defined. Then the canonical differential 1-form is defined by It turns out that is a symplectic form on . In fact, the cotangent bundle is identified as the phase space in classical mechanics.

Symplectomorphism

A symplectomorphism is a diffeomorphism between symplectic manifolds and such that .

Proposition

Given a diffeomorphism , there is a canonical diffeomorphism such that , where and are the canonical 1-forms on and respectively. Thus is a symplectomorphism.

Hamiltonian Vector Fields

A Riemannian metric on a manifold establishes an isomorphism between the space of tangent vectors and the space of cotangent vectors (i.e. 1-forms). A symplectic structure establishes a similar isomorphism.

Symplectic Duality

Let be a symplectic manifold. To each vector , we associate a 1-form by the relation Then the correspondence is an isomorphism between and . The isomorphism is called the symplectic duality, denoted as .

Proof This is straightforward from the construction.

Hamiltonian Vector Field

Let be a symplectic manifold. The Hamiltonian vector field associated with a smooth function , called the Hamiltonian function, is the vector field defined by

Equivalent Definition of Hamiltonian Vector Field

Given a smooth function on a symplectic manifold , the Hamiltonian vector field is the unique vector field such that

Proof is the Hamiltonian vector field iff $$X_{H}=\iota^{-1} (\d H) \iff \d H=\iota(X_{H})=\omega(\cdot, X_{H}).$$$\square$

e.g.

• Suppose is a coordinate for , let the standard symplectic form be . Then the vector field defined by the Hamilton’s equations of motion: is the Hamiltonian vector field associated with the Hamiltonian function . We can show this by checking that the vector field defined by the Hamilton’s equations of motion, say satisfies the equation Indeed, for any vector field , we have
• Consider the symplectic structure on the torus induced by the standard symplectic structure on , with coordinates and consider the vector field . The the flow of this vector field preserves the symplectic form: But this vector field is not a Hamiltonian vector field, because if we assume is Hamiltonian, then we have it follows that which means for some constant . Note that has no global chart, so can’t be defined smoothly globally. In detail, at some point , the value of at must be the same for all charts that cover . Since is injective, these charts must be the same. However, has no global chart, so this is impossible. Alternatively, observe that is strictly decreasing, so for some appropriate points , under the chart at and at , we might have and both hold. This is a contradiction, thus is not Hamiltonian.

Conserved Quantities

Proposition

Suppose is a Hamiltonian vector field associated with a Hamiltonian function on a symplectic manifold . Then and .

Proof By Cartan’s formula, we have since . By the definition of Hamiltonian vector field, we know that . Therefore, we have For the second part, we have $$\L_{X_{H}}H=i_{X_{H}}(\d H)=\d H(X_{H})=\omega(X_{H},X_{H})=0.$$$\square$

Remark

The second equation means that the Hamiltonian is conserved along the flow of the Hamiltonian vector field . In a natural Lagrangian system, the Hamiltonian is the total energy of the system, and thus it is conserved. This is a manifestation of the conservation of energy in classical mechanics.

Symplectic Manifolds Have a Natural Volume Form

If is a symplectic manifold, then the th wedge of is a volume form, and also the flow of preserves this volume form.

Proof Clearly is a -form on . To see that it is a volume form, we need to check that it is non-degenerate. This follows from the fact that is non-degenerate. To see that the flow of preserves this volume form, we show that : Note that since . Therefore, we have Moreover, so by induction, we have . Thus .

Proposition

There are no compact exact symplectic manifolds.

Proof In fact, there is a unique vector field , such that . So that Hence,

Action of a Loop

Given an oriented closed loop in an exact symplectic manifold , the action of is defined as

Proposition

The flow of any Hamiltonian vector field preserves the action.

Proof To show this, suppose is a flow of a Hamiltonian vector field , we claim that