In Goldstein's Classical Mechanics (2nd ed.) on section 9-1 page 382, there is a discussion about finding a canonical transformation (qi,pi)→(Qj(qi,pi,t),Pj(qi,pi,t)) from a given generating function F=F1(qi,Qi,t). The following is written on the page:
Equation (9-11) then takes the form,
pi˙qi−H=Pi˙Qi−K+dF1dt,=Pi˙Qi−K+∂F1∂t+∂F1∂qi˙qi+∂F1∂Qi˙Qi
Since the old and the new coordinates, qi and Qi , are separately independent, Eq. (9-13) can hold identically only if the coefficients of ˙qi and ˙Qi each vanish:
I don't understand the bold text above. How are qi and Qi separately independent? I thought Qi was, in general, a function of the original canonical coordinate variables qi, thereby making it explicitly dependent on qi (not independent). Could somebody explain how these two canonical coordinate variables are separately independent?
Answer
Let us suppress t-dependence in this answer for simplicity. A canonical transformation1 (q,p)⟶(Q(q,p),P(q,p))
References:
- H. Goldstein, Classical Mechanics.
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