## Hamiltonian Systems

In general a Hamiltonian is a function of 'generalized coordinates' and 'generalized momenta' written Every Hamiltonian system has generalized coordinates and generalised momenta For a one dimensional system and The fixed points of a Hamiltonian system are the solutions to the simultaneous equations and Example: Find the fixed points of the system with Hamiltonian   We can factorise the quadratic in above to give The solutions of this quadratic are and The fixed points of this system are and 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 to for this linearisation so we solve This has solutions This implies that the fixed points of a Hamiltonian system are either a saddle or a centre.

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

The eigenvalues are purely imaginary and of the same sign if so the fixed point is a centre. In fact it is a maximum if and a minimum if For the example above and When Hence is a centre and since it is a maximum. Hence is a saddle.