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1. If I know that ##H(q_i,p_i,t)## is a valid Hamiltonian for which the hamilton equations hold. Now we are given that ##Q_j(q_i,p_i)## and ##P_j(q_i,p_i)## are canonical transformations. This means that there is a function ##K(Q_j,P_j)##, the new hamiltonian, for which the Hamilton equations hold in the new variables.
QUESTION : What is this new function ##K##? Can I just substitute the transformations into the old ##H## to find ##K##? If not, when is this allowed?
Reason for confusion: My notes from class tell me this is allowed. However any source on the net I find says that in general canonical transformations do not preserve the Hamiltonian and thus ##H## and ##K## are not always equal.
QUESTION : What is this new function ##K##? Can I just substitute the transformations into the old ##H## to find ##K##? If not, when is this allowed?
Reason for confusion: My notes from class tell me this is allowed. However any source on the net I find says that in general canonical transformations do not preserve the Hamiltonian and thus ##H## and ##K## are not always equal.