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Hamiltonian Systems

In general a Hamiltonian is a function of 'generalized coordinates'and 'generalized momenta'written

Every Hamiltonian system has generalized coordinatesand generalised momentaFor a one dimensional system

and

The fixed points of a Hamiltonian system are the solutionsto the simultaneous equations

and

Example: Find the fixed points of the system with Hamiltonian

We can factorise the quadratic inabove to give

The solutions of this quadratic areand

The fixed points of this system areand

The fixed points of a Hamiltonian system can only be a saddle or a centre, since the linearisation matrix is given by

The eigenvalues are the solution tofor this linearisation so we solve

This has solutionsThis implies that the fixed points of a Hamiltonian system are either a saddle or a centre.

The eigenvalues are real and have opposite sign ifso the fixed point is a saddle.

The eigenvalues are purely imaginary and of the same sign ifso the fixed point is a centre. In fact it is a maximum ifand a minimum if

For the example aboveand

When

Henceis a centre and sinceit is a maximum.

Henceis a saddle.