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Canonical Transformations

Transformations of coordinates and momenta that preserve the form of Hamilton's equations, allowing for simplification of problems.
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The statement of the theorem

Let (q,p)(q, p) and (Q,P)(Q, P) be two sets of canonical coordinates on a phase space M\mathcal{M}. The transformation is canonical if the Poisson bracket structure is preserved, i.e., ∂Q∂q∂P∂p−∂Q∂p∂P∂q=1\frac{\partial Q}{\partial q} \frac{\partial P}{\partial p} - \frac{\partial Q}{\partial p} \frac{\partial P}{\partial q} = 1. Equivalently, the differential form dp∧dq\text{d}p \wedge \text{d}q is invariant, and the new Hamiltonian K(Q,P,t)K(Q, P, t) is related to the old Hamiltonian H(q,p,t)H(q, p, t) by K=H+∂F∂tK = H + \frac{\partial F}{\partial t}, where FF is the generating function.