Here is a simple example of a canonical transformation

QUESTION:

Here is a simple example of a canonical transformation that illustrates how the Hamiltonian formalism lets one mix up the q's and the p's. Consider a system with one degree of freedom and Hamiltonian = H(q, p). The equations of motion are, of course, the usual Hamiltonian equations 4 = 83-781, and /3 = ag-f/aq. Now consider new coordinates in phase space defined as Q = p and P = q. Show that the equations of motion for the new coordinates Q and P are Q = 85f/a P and P = 83-co Q; that is, the Hamiltonian formalism applies equally to the new choice of coordinates where we have exchanged the roles of position and momentum.