The Hamilton-Jacobi Theory, Action-Angle Variables

Problem:
Solve the Hamilton-Jacobi differential equation for a particle moving in a uniform gravitational field.
Solution:

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Problem:

The one dimensional motion of a mass m subject to a gravitational force F=-mgj is described by the relativistic Hamiltonian . At t=0 the mass is released from rest at y=0.

(a) Write the Hamilton-Jacobi equation and solve it for y(t) for t³0.

Hint: In setting up the Hamilton-Jacobi equation, the substitution made for the relativistic momentum p is the same as the one used for the non-relativistic momentum in the non-relativistic case.

(b) Show that in the non-relativistic limit, i.e., when gt<<c, the answer to part (a) reduces to -½gt2 as expected.

Solution:

Problem:

A particle of mass m moves in the potential and   elsewhere.

(a) Using the Hamilton-Jacobi equation and the characteristic function W, find t as a function of x; assume that x=0 and at t=0.

(b)  Sketch the motion of the particle in the p - x phase space. Show the trajectories, or phase space paths, for energy E>>a, E>a, E=a, and E<a.

Solution:

Problem:

Consider an one-dimensional harmonic oscillator of mass m.
(a) Write the Hamiltonian.
(b) Write the corresponding Hamilton-Jacobi equation.
(c)  Use the Hamilton-Jacobi method to obtain the motion of the oscillator

Solution:

Problem:

(a)  Use the Hamilton-Jacobi theory and separation of variables to examine the 3-D motion of a particle in a central force field.
(b)  Set up the action integrals and determine whether any of the variables are cyclic.

Solution: