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.