**1. Introduction **

This is the first note of a series of notes concert on semiclassical analysis. Given the basic material on symplectic geometry. Including the following material,

- The case at a point, or we can look it as the case in .
- The standard material in symplectic geometry, i.e. Hamiltonian mechanics, two approach, global one concentrating on lie derivative, and a locally one concentrating on the power of Darboux theorem, i.e. the existence of a canonical coordinate.
- The basic facts on Poission bracket.
- The basic facts on Lagrange sub-manifold, and the involve of Liouville measure.

**2. Case of a point, or **

Let be a vector field, at once we have a vector field, we could consider the associated flow of it,

express the trajectory start from along the vector field.

Remark 1There . One the other hand, due to the locally existence theorem of ODE, if the regularity of is enough, then the solution exist and is uniqueness.

Definition 1or more convenient . We call the flow map or the exponential map generated by .

Lemma 2For flow map, we have following:

- for all .
- for all .
- for each time , the mapping is a diffeomorphism with

So it is a group action on , with units as diffeomorphism of . *Proof:*

This lemma is the direct corollary of the theory of ODE.

Now let us special to the case . In local coordinate we have , express position of particle, express momentum of particle.

Definition 3in define their symplectic product,

In a matrix form, coincide with a matrix

Following lemma given the basic property of .

Lemma 4The following basic property are true.

- ,
- the bilinear form is antisymmetric, and degenerate, i.e. if for all , then .
- , .

*Proof:*

- trivial calculate get this.
- trivial.
- , by basic linear algebra everything follows.

**3. Hamiltonian mechanics **

Definition 5Symplectic form: non-degenerate closed 2 form in a standard coordinate(Darboux coordinate, coordinate like ) looks like,

, map

is an isomorphism. is called the symplectic form.

There is locally coordinate for , i.e.,

So , roughly we have , this is of course not true, but morally true. Now let us give the definition of symplectic manifold and the relationship of Hamiltonian mechanics.

Definition 6We have the following definition,

- A symplectic manifold is a pair where is a smooth manifold and is a closed two-form on such that the map,
is an isomorphism, is called the symplectic form.

- If is symplectic, and is differentiable, the hamiltonian vector field of is the field on whose image under the previous map is . In other word, is characticed by the property,
- The flow of will be referenced to as the hamiltonian flow of .

Lemma 7If coordinate and the symplectic form

then,

*Proof:* , due to we have . So we have:

Assume . Then we have,

and also,

Combine with the definition of integral curve we derive the integral curves of i.e. such that , is given by 5.

Now we begin to proof the Newton second law for the force . We consider the 2-dimensional case at first. We have,

The high dimension case is similar, thanks to the linearity of and .

Remark 2Two make Newton’s second law to be true, the form 6 play a crucial role. Is there some generalization of this type of result to more general case, roughly speaking, it is reasonable to expect this could still be true if the hamiltonian function could be divide into potential energy part and kinetic energy part. And the describe of potential energy part is that it is given by a quadratic form.

Lemma 8In general, for any Hamilton field one has:

- , conservation of energy. In orther word, is everywhere tangent to the level sets of .
- , so the Hamiltonian flow of consists of automorphism of .

General speaking, to proof a theorem on manifold, there always have two choice, coordinate free proof and proof in a careful choose coordinate. If we choose to believe the Darboux theorem 9 is true, the meaning of it is that locally the symplectic manifold are the same.

*Proof:* If we believe the Darboux theorem 9 is true. then consider in a standard coordinate , we have,

So of course . In general case, i.e. coordinate free proof, . Use identity if lie derivative. The second thing is also easy to proof by look in a local canonical coordinate, involve the indentity of lie derivative.

Remark 3I need more understanding on the lie derivative, see wiki.

Theorem 9(Darboux theorem) Near any point there exist coordinate: usually called Darboux coordinates, such that the sympletic form has the form,

Remark 4This theorem means there do not exist local invariant in symplectic manifold.

*Proof:*

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Theorem 10If is any smooth manifold, then its cotangent bundle has a natural symplectic structure.

*Proof:* we have local coordinate on derive from , it is . Remember we have Riemann metric: on , the existence of Riemann metric involve a unit decomposition argument and bump function, I just recall it there. Now we move on, consider the relationship between .

Remark 5It need not be the case that non-degenerate non-degenerate. This case in the lemma is a example to show that could be the case: degenerate non-degenerate. We glue something together on the space pf differential operator to understand the topology of it but not deifferential structure or more refinement structure. Quntalization could be look as a way to glue, this could be down if there is a differential equation with some special condition (come from a flow take charge of it suffice).

Lemma 11(The proof of is non-degenerate) Let be local coordinates on . Define a coordinate system on by the condition:

Prove that in Darboux coordinate, and therefore .

*Proof:*

Theorem 12Let be a smooth Riemann manifold and let be one half of the square of the Riemann norm, so that in local coordinate,

Then the trajectorics of the hamiltonian flow of , projected down to , are geodesic aries in this fashion.

Newton’s second law+ energy vanish.

*Proof:*

So second variation formula describe of geodesic give us the fact that the trajective is geodesic.

Remark 6We could directly calculate in local coordinate.

**4. Poisson brackets **

is the Hamiltonian generating the dynamic is any smooth function on phase space (the symplectic manifold), then the rate of change of along the trajectraries of d is the function

Definition 13If is symplectic and , the poisson bracket of and is defined to be the function on .

Lemma 14In canonical (Darboux) coordinate where , one has,

In particular, .

*Proof:*

Theorem 15If is a symplectic manifold then is a Lie algebra.

*Proof:* Bilinearty, skew-symmetric come form,

could be proved by calculate under a local coordinate.