Hamilton’s equations

The Hamiltonian of a system is expresses in terms of the generalized coordinates and the generalized momenta of the system,

.

Hamilton’s canonical equations are

.

Symplectic notation:

Let

,

for a system with n degrees of freedom.  Let

,

where I is the   identity matrix.  Then

.

Properties of the matrix J:

.

Canonical transformations

In the Hamiltonian formalism we have 2n independent variables.  We denote these variables by q and p and write H(q,p,t).  We can also express H in terms of 2n different independent variables Q and P, where Q_i=Q_i(q,p,t) and P_i=P_i(q,p,t).

Canonical transformations, or contact transformations are a special class of transformations which preserve the structure of the canonical equations, i.e. for which there exists a function K(Q,P,t) such that

.

If we can find a generating function F, such that

then the transformation is canonical.  F must be a function of 2n independent variables and

.

Canonical transformations in symplectic notation

Let h denote a set of canonical variables, .  Let c denote a second set of variables. Let

.

The transformation is canonical if and only if

.

Poisson brackets

Let H=H(q,p,t) and f=f(q,p,t). H is the Hamiltonian and f is an arbitrary function.

Then

.

The values of all Poisson brackets are independent of the set of canonical variables they are expressed in.

A transformation is canonical if and only if [Q,P] = 1.