Preservation of Area by Time Independent Systems

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:andnot necessarily Hamiltonian. At timethe area of a region of phase space is given byand at

Asincreases fromtoa pointinis sent to a pointinso thatcan be considered as the initial condition on a trajectorywhereand the transformation fromtois given by

The area at timeis given bywhere is the Jacobian matrix of the transformation.

We can expandandin Taylor series:

Then the Jacobian can be written as


Henceon use of Hamilton's equations and area is preserved.