1. The problem statement, all variables and given/known data Show that the time reversal transformation given by Q = q, P = − p and T = − t, is canonical, in the sense that the form of the Hamiltonian equations of motion is preserved. However, it does not satisfy the invariance of the fundamental Poisson Bracket relations. This is an example when the two criteria are not equal. 2. Relevant equations 3. The attempt at a solution This I have done...I just ask you to check if the procedure is correct.( ' denotes d/dt ) Q'=(dQ/dT)=(dQ/dt)(dt/dT)= -(dQ/dt)= -[(∂Q/∂q)q' + (∂Q/∂p)p' + (∂Q/∂t)]= -[q']= -(∂H/∂p) Also, (∂K/∂P)=(∂H/∂p)(∂p/∂P)= -(∂H/∂p)...[we write (∂K/∂P)=(∂H/∂p) as Kamiltonian K is a function of Q,P,T and Hamiltonian H is a function of q,p,t] Thus, Q'= (∂K/∂P)...1st of Hamilton's canonical equations is proved. Similarly, P'= (dP/dT)= -(dP/dt)= -[(∂P/∂q)q' + (∂P/∂p)p' + (∂P/∂t)]= p'= -(∂H/∂q) Then, (∂K/∂Q)=(∂H/∂q)(∂q/∂Q)=(∂H/∂q) Thus, P'= -(∂K/∂Q) This shows that the given transformation leads (q,p) to canonically conjugate variables(Q,P) Evaluating the Poisson brackets it is easy to show that they do not satisfy fundamental Poisson bracket. Can anyone suggest why there is a mismatch between the two aspects?