Consider the canonical transformation in Exercise 17.4.

(a) Find the proto-generating function f (q, p) which obeys eqn (18.1),

(c) Use f (q, p) to derive an F2(q, P) generating function. What are the conditions on the θi for this generating function to be valid? Use it to re-derive the canonical transformation in Exercise 17.4.

Exercise 17.4

A transformation q, p → Q, P is defined, for all i = 0,…, D, by

where θ0, θ1, θ2,…,θD are independent, constant parameters, and a is a given constant having appropriate units. (Note that no summation convention is being used here. For example, Q2 and P2 depend only on q2, p2, and the constant parameter θ2.) Use the Poisson-bracket condition to show that this transformation is canonical for any set of θi values.

 

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