(a) If each of the four types of generating functions exist for a given canonical transformation, use the Legendre transformation to derive relations between them.
(b) Find a generating function of the F4 type for the identify transformation and of the F3 type for the exchange transformation.
(c) For an orthogonal point transformation of q in a system of n degrees of freedom, show that the new momenta are likewise given by the orthogonal transformation of an n-dimensional vector whose components are the old momenta plus a gradient in configuration space.
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