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.

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