Question

Consider the 2-dimensional phase space R^2. Find the most general canonical transformation of this phase space,of the form Q=Q(q,p)=f(q)+g(p)

P=P(q,p)=c[f(q)+h(p)] where f,g and h are differentiable real-valued functions

Answer 1

Given equations are,

$Q=Q(q,p)=f(q)+g(p)$

$P=P(q,p)=c[f(q)+h(p)]$

$\dot{Q}=Q(p)\frac{dQ(q)}{dq}+Q(q)\frac{dQ(p)}{dq}$

$=f'(q)+g'(p)$

$\dot{P}=P(p)\frac{d(P(p))}{dp}+P(q)\frac{dP(q)}{dp}$

$=c[f'(q)+h'(p)]$