A binary operation , called Poisson brackets, is introduced for functions on the phase space:

In terms of Poisson brackets the Hamiltonian equations can be rewritten as

Exercise: prove, that for any function A(t,x,p) the following identity (Liouville equation) is valid:

Poisson brackets have the following properties:

1. Skew symmetry: .

2. Linearity: for constants .

3. Leibniz identity: .

4. Jacoby identity: .

In axiomatic way, the Poisson brackets can be defined as any binary operation, satisfying these 4 properties. Axioms 2,3 are equivalent to . Therefore, the Poisson brackets, being introduced on some complete set of variables , can be defined for any functions:

Particularly, in transformation to a new complete set, the matrix is multiplied by Jacoby matrices from left and right. Variables (x,p), for which (and consequently (1) is valid), are called canonical basis. There is a theorem (Darboux), which states that for any a transformation to canonical basis exists, at least locally in a vicinity of a given point . Such transformation is not unique, because non-trivial transformations exist, called canonical transformations, which conserve Poisson brackets. They can be applied after some canonical basis was found.

Exercise: for 2-dimensional phase space (x,p) find all canonical transformations.

Answer: arbitrary area conserving transformation of the plane.

  The described construction allows to think about Hamiltonian mechanics as a kind of geometry, introduced in the phase space. The matrix defines a transformation of any vector to orthogonal one: , ab(a)=0, due to skew symmetry of . The Hamiltonian H creates a field of gradients , which is transformed by to a field of velocities: . Subsequent integration gives the phase trajectories (note that all they automatically lye on the surface H=Const, Hamiltonian is conserved in evolution).

In this way Hamiltonian generates the phase flow. Moreover, any function on the phase space, being used as Hamiltonian, generates some phase flow. All variables in Hamiltonian theory have double role: as -valued functions on the phase space and as generators of some transformations.

Exercise: what transformations are generated by z-components of momentum and angular moment ?

Answer: shifts of along z and rotations of about z.

The matrix can be considered as skew symmetric metric in the phase space. Canonical transformations are motions (metric conserving transformations). While a symmetric metric defines a Riemann structure in the space, the skew symmetric metric gives a symplectic structure.

  Definition: Let (due to skew symmetry of this is possible only in even dimensional space). Let be inverse to . A differential 2-form

is called symplectic form on the phase space. More information on differential forms can be found in [1],[2]. Here we should only know that the operation (outer product) is skew symmetric: and operation d (outer derivative) has properties: , .

Note: form with zero outer derivative is called closed, form represented as an outer derivative is called exact. Exact form is necessarily closed. Converse is not always true, look to [2] for details.

Exercise: show that Jacoby identity is equivalent to closeness of the form: .

Sketch of the solution.

Due to Jacoby identity, a term , being summed over cyclic permutations of (ijk), gives zero. Multiplying this sum to and using the identity , obtain that , summed over the cyclic permutations of (ijk), is zero. On the other hand, the derivative of symplectic form:

Here is not changed in the cyclic permutations. Summing over them, obtain .

  The symplectic form was introduced for the following purpose. Let's consider a surface in the space. A metric, introduced in the whole space, induces a metric on the surface. For Riemann structure: if we are able to calculate angles and distances in the whole space, we can do this on the surface also. Formally, if the surface is specified parametrically as , the scalar product

In the same way the symplectic form introduced in the whole phase space induces a symplectic form on the surface:

Due to the specifics of our problems, we will often use this property.

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