Condition for a Transformation to Preserve Hamiltonian Form
The area of a region of the phase space diagram at time
is given in terms of the area at some earlier time
by
Differentiating this expression gives
or
is a solution curve of the system
so
then for a transformation
The second bracket equals 0 for Hamiltonian systems hence
Obviously if
is constant then the system is Hamiltonian in the nnew coordinates
but if
whenever
does then
for every Hamiltonian system but given a pointin phase space we may construct a function
such that
and
and another function such that
and
at that point thus we must have that
or thatis constant.
