A system influenced by time dependent forces or which is represented in a rotating or non inertial reference frame has a Hamiltonian which depends explicitly on time,The rate of change of the Hamiltonian is given by
On using Hamilton's equations of motion this becomes
The value of the Hamiltonian is not conserved, however the area is.
Proof:and
not necessarily Hamiltonian. At time
the area of a region of phase space is given by
and at
Asincreases from
to
a point
in
is sent to a point
in
so that
can be considered as the initial condition on a trajectory
where
and the transformation from
to
is given by
The area at timeis given by
where
is the Jacobian matrix of the transformation.
We can expandand
in Taylor series:
Then the Jacobian can be written as
Consider
Henceon use of Hamilton's equations and area is preserved.