Show that the volume element
dω˙=i=1∏3N(dqidpi)
of the phase space remains invariant under a canonical transformation of the (generalized) coordinates (q,p) to any other set of (generalized) coordinates (Q,P).
[Hint: Before considering the most general transformation of this kind, which is referred to as a contact transformation, it may be helpful to consider a point transformation—one in which the new coordinates Qi and the old coordinates qi transform only among themselves.]