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
The fixed points of a Hamiltonian system are the solutionsto the simultaneous equations
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
Henceis a centre and sinceit is a maximum.
Henceis a saddle.