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 inabove 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
Henceis a centre and since
it is a maximum.
Hence
is a saddle.