Decide if the energy surfaces in phase space are bounded

[jack476]

Homework Statement

From Classical Mechanics, Gregory, in the chapter on Hamilton's equations of motion:

14.13: Decide if the energy surfaces in phase space are bounded for the following cases:

i.) The two-body gravitation problem with E<0
ii.) The two-body gravitation problem viewed from the zero-momentum frame with E<0
iii.) The three-body gravitation problem viewed from the zero-momentum frame with E<0. Does the Solar System have the recurrence property?

Homework Equations

Hamilton's equations, Lioville's theorem, Poincare's recurrence theorem.

The Attempt at a Solution

The trouble here is that the chapter has not explained what an "energy surface in phase space" is or how one is to judge whether or not it's bounded. Can someone please help me understand what that means?

 

[Gene Naden]

It appears that the energy surfaces are the surfaces satisfying $H(P,Q)=E$ where $H$ is the Hamiltonian, $P$ and $Q$ are the canonically conjugate variables and $E$ is a constant (the energy).

They are bounded if $P$ and $Q$ don't go out to infinity for a given value of $E$.