A generating function of the first type,

PHYSICS · Subjective · 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

A generating function of the first type, F1(q,Q,t)F_1(q,Q,t), satisfies dF1=∑kpk dqk−∑kPk dQk+(K−H) dt.\mathrm{d}F_1 = \sum_k p_k\,\mathrm{d}q_k - \sum_k P_k\,\mathrm{d}Q_k + (K-H)\,\mathrm{d}t. Define F3(p,Q,t)=F1(q,Q,t)−∑kqkpk,F_3(p,Q,t) = F_1(q,Q,t) - \sum_k q_k p_k, in which qkq_k is eliminated in favour of pkp_k using pk=∂F1/∂qkp_k = \partial F_1/\partial q_k. Derive the equations that F3F_3 produces for qkq_k and PkP_k, and the relation between KK and HH.

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