In the text, the Hamilton–Jacobi equation for S was obtained by seeking a contact transformation from the canonical coordinates (q, p) to the constants (α, β). Conversely, if S(qi, αi, t) is any complete solution of the Hamilton–Jacobi equation (10.3), show that the set of variables (Qi, pi) defined by Eqs. (10.7) and (10.8) are canonical variables, that is, that they satisfy Hamilton’s equations.
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