Type of the canonical transformation generated by W

Consider a canonical transformation in which the new momenta pj are all constants of the motion α in particular is the constant of motion H, i.e., H(qj,pj)= α . Taking generating function to be W(q,p) = F, equations of transformation with W(q,p) as the generating function will be

pj = ∂W (q,P) /∂qj .......................1

Qj = ∂W/∂pj = ∂W(q,p) / ∂ αj......................2

H(qj,pj) = αi -> H (qj, ∂W/ ∂ qj) = αi

The new Hamiltonian is

K = H + ∂W/∂t = αi + ∂W (qj,αj/∂t = αi

Thus old and new Hamiltonian is equal. As Qj, new coordinates, do not occur in K, they are all cyclic. Thus W generates a canonical transformation in which all new coordinates are cyclic.

Using Hamilton's equation of motion, the new coordinates are given by

Thus out of all Qj, only Q1 emerges out involving time and is thus not a constant of motion. It is not necessary always to take αi and constants of integration αj in W as the new constant momenta.

This note is taken from Classical Mechanics, MSc physics, Nepal.