Condition for a Transformation to Preserve Hamiltonian Form
The area of a region of the phase space diagram at timeis given in terms of the area at some earlier timeby
Differentiating this expression gives
is a solution curve of the systemsothen for a transformation
The second bracket equals 0 for Hamiltonian systems hence
Obviously ifis constant then the system is Hamiltonian in the nnew coordinatesbut ifwheneverdoes thenfor every Hamiltonian system but given a pointin phase space we may construct a functionsuch thatandand another function such thatandat that point thus we must have thator thatis constant.