
145Hamiltonian Formulation of Mechanics
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for any pair of functions F and G, where the q, p and the Q, P are related by a canonical transfor-
mation, such as Equation 5.21. The proof is straightforward, but it is tedious. We shall not pursue
it here.
It is easy to show that the fundamental Poisson brackets are canonical invariants. Suppose we
make the canonical transformation from a set of variables (q, p) to a new set of (Q, P). Now, any
canonical transformation preserves the form of Hamilton’s equations so that Equations 5.45 to 5.51
still hold for the new variables as do Equations 5.44a and 5.44b.
The Poisson-bracke ...