Take a canonical transformation with generating function

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Hamiltonian mechanics and canonical structure · Hamiltonian mechanics and canonical structure

Action and Symmetry, Ch Canonical transformations, generating functions and symplectic structure

Problem

Take a canonical transformation with generating function F1(q,Q,t)F_1(q,Q,t), so that pk=∂F1/∂qkp_k=\partial F_1/\partial q_k, Pk=−∂F1/∂QkP_k=-\partial F_1/\partial Q_k and K=H+∂F1/∂tK=H+\partial F_1/\partial t. ((a)) For a system with two degrees of freedom x,yx,y, consider the infinitesimal transformation generated by F2=xPx+yPy+ϵGF_2=xP_x+yP_y+\epsilon G with G=xpy−ypxG=xp_y-yp_x. Derive δx,δy,δpx,δpy\delta x,\delta y,\delta p_x,\delta p_y to first order in ϵ\epsilon, and check that each equals ϵ{f,G}\epsilon\{f,G\} for the corresponding ff. ((b)) The loop integral ∮∑kpk dqk\oint\sum_k p_k\,\mathrm dq_k around a closed path in phase space is a canonical invariant. Using the exactness of dF1\mathrm dF_1 at fixed time, explain why this integral keeps its value under the transformation of part (a) for any closed path, while K=H+∂F1/∂tK=H+\partial F_1/\partial t shows that HH itself need not.

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