The Motion in Phase Space

We want to understand the dynamics in phase space of the motion of the particle with initial position x0 and velocity v0 under the influence of the gradient force.

At any time the particle has velocity v(t)=dx/dt. Assuming the particle has constant mass equal to 1, Newton's Second Law says that the acceleration dv/dt must equal the force. So we can write the equations of motion as the system

dx/dt = v
dv/dt = -grad f(x).

Question #2:

Use the Maple command

fieldplot([v,-grad f(x)], x=-7..7, v=-3..3);

where you have replaced -grad f(x) with its calculated value, to plot the phase space vector field for this system.

Analytically find the equilibria for this system for all x and v. Classify each as a source, sink, saddle, or center, based on the geometry of your plot.
Explain why it is plausible on these grounds alone that the phase space vector field might be Hamiltonian.

Next: By George, it's Hamiltonian!
Previous: One-Dimensional Gradient Motion
Up: Introduction

Bob Thurman;thurman@geom.umn.edu>

Last modified: Tue Apr 2 13:29:28 1996