Classical Mechanics (Kibble & Berkshire) · Problem 14.9

The general theory of small oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · variational-principle-hamiltons-equations

Problem

A variational principle for Hamilton's equations Consider the functional

J[\boldsymbol{q}(t),\,\boldsymbol{p}(t)] = \int_{t_0}^{t_1}\left(H(\boldsymbol{q},\boldsymbol{p},t) + \boldsymbol{q}\cdot\dot{\boldsymbol{p}} - \dot{\boldsymbol{q}}\cdot\boldsymbol{p}\right)dt

of the

2n

independent functions

q_1(t),\ldots,q_n(t)

p_1(t),\ldots,p_n(t)

. Show that the extremals of

J

satisfy Hamilton's equations with Hamiltonian

H