Let be a complete solution of the Hamilton-Jacobi equation

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

Action and Symmetry, Ch Hamilton-Jacobi theory and action-angle variables

Problem

Let S(q,α,t)S(q,\alpha,t) be a complete solution of the Hamilton-Jacobi equation, with pk=∂S/∂qkp_k=\partial S/\partial q_k and αk\alpha_k held fixed on the true trajectory determined by βk=∂S/∂αk=const\beta_k=\partial S/\partial\alpha_k=\text{const}. ((a)) Show that, evaluated along that true trajectory, dS/dt=L\mathrm{d}S/\mathrm{d}t = L, and hence that SS (up to an additive constant) is the action of the true path as a function of its endpoint. ((b)) A free particle of mass m=2.0m=2.0 kg leaves the origin at t=0t=0 and is found at x=6.0x=6.0 m at t=3.0t=3.0 s. Compute SS directly from the true path's action, and check it against S=mx2/2tS=mx^2/2t. Then verify that ∂S/∂x\partial S/\partial x gives the particle's momentum and ∂S/∂t\partial S/\partial t gives minus its energy.

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