Nonlinear Mechanics - Physics at Oregon State University

2.5. ACTION-ANGLE VARIABLES 27
[
r 2
( ) 2
dWr
+ 2mr
dr
2 V (r) − 2mα1r 2
]
+
( ) 2
dWψ
= 0 (2.40)
dψ
At this point we notice ∂W
∂ψ = pψ, which we know is constant. Why not
call it something like αψ? Then Wψ = αψψ. This is worth stating as a
general principle: if q is cyclic, Wq = αqq, where αq is one of the n constant
α’s appearing in (2.30).
∫
W =
√
dr 2m(α1 − V ) − α2 ψ /r2 + αψψ (2.41)
We can find r as a function of time by inverting the equation for β1, just as
we did in (2.37), but more to the point
βψ = ∂W
∫
αψdr
= − √
∂αψ r 2m(α1 − V ) − α2 ψ /r2
+ ψ (2.42)
Make the usual substitution,u = 1/r.
∫
du
ψ − βψ = − √
2m(α1 − V (r))/α2 ψ − u2
(2.43)
This is a new kind of equation of motion, which gives ψ = ψ(r) or r = r(ψ)
(assuming we can do the integral), i.e. there is no explicit time dependence.
Such equations are called orbit equations. Often it will be more useful to
have the equations in this form, when we are concerned with the geometric
properties of the trajectories.
2.5 Action-Angle Variables
We are pursuing a rout to chaos that begins with periodic or quasi-periodic
systems. A particularly elegant approach to these systems makes use of a
variant of Hamilton’s characteristic function. In this technique, the integration
constants αk appearing directly in the solution of the Hamilton-Jacobi
equation are not themselves chosen to be the new momenta. Instead, we
define a set of constants Ik, which form a set of n independent functions of
the α’s known as action variables. The coordinates conjugate to the J’s are
angles that increase linearly with time. You are familiar with a system that
behaves just like this, the harmonic oscillator!
q =
√ 2E
k sin ψ p = √ 2mE cos ψ