Question

How to find generating function of canonical transformation?

Answer 1

Explanation:

For the generating function F1(q,Q), we have dF1/dq=p and p=Q/q

then

F1=Qln(q)+K1(Q)(1)

and also

dF1/dQ=−P(2)

and

P=ln(Q/q)−(1/2)ln(Q)(a)

substituting equation (a) in (2) and integrating

F1=−(1/2)(Qln(Q)−Q)+Qln(q)+K2(q)(3)

comparing (1) and (3) we get

F1=−(1/2)(Qln(Q)−Q)+Qln(q)

with

K2(q)=0 and K1(Q)=−(1/2)(Qln(Q)−Q)

(here d()/dq and d()/dQ actually stand for the partial derivatives with respect to q and Q)