Nonlinear Mechanics - Physics at Oregon State University

Chapter 3
Abstract Transformation
Theory
So, one-dimensional problems are simple. Given the restrictions listed in
the previous section, their phase space trajectories are circles. How does
this generalize to problems with two or more degrees of freedom? A brief
answer is that, given a number of conditions that we must discuss carefully,
the phase space trajectories of a system with n degrees of freedom, move
on the surface of an n-dimensional torus imbedded in 2n dimensional space.
The final answer is a donut! In order to prove this remarkable assertion and
understand the conditions that must be satisfied, we must slog through a
lot of technical material about transformations in general.
3.1 Notation
Our first job is to devise some compact notation for dealing with higher
dimensional spaces. I will show you the notation in one dimension. It will
then be easy to generalize. Recall Hamilton’s equations of motion.
˙p = − ∂H
∂q
We will turn this into a vector equation.
( )
q
η =
p
(
0
J =
−1
1
0
The equations of motion in vector form are
˙q = ∂H
∂p
)
∇ =
( ∂
∂q
∂
∂p
)
(3.1)
˙η = J · ∇H (3.2)
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