Action-Angle

1-Dimensional CT  Define a 1-D generator S’. Time-independent H.Time-independent H. Require new conjugate variables to be constants of motion.Require new conjugate variables to be constants of motion.  Conjugate momentum is a constant J. Hamiltonian is constantHamiltonian is constant Conjugate position is cyclicConjugate position is cyclic Linear in timeLinear in time a frequency  is a constant, ie  from HJ units of action

Periodic System  A frequency suggests periodic motion. Assume q, p periodic.Assume q, p periodic. Period is Period is   Evaluate the action and coordinate over one period. The change in w in one period is 1The change in w in one period is 1 J is the actionJ is the action w is the anglew is the angle

Alternate Generators  Generating functions differ by a Legendre transformation.  The transformation can be expressed as type I. S is also periodic with period 1S is also periodic with period 1

Simple Oscillator  The oscillator H is constant and expressed in terms of p.  The action can be integrated  The generator can be defined from the action

Derived Variables  The angle can be derived from the generator.  The momentum and position also can be derived.

Physical View  The motion in phase space is harmonic. Amplitudes of q, p Area in phase space is the  times the action. Angle w repeats per cycle. p q J = E/ w  = t

Generating Function  The generating function S’ can be found by integration and substitution. The function S comes from the Legendre transformationThe function S comes from the Legendre transformation

Periodic Motion  Libration is motion that is bounded in the angle.  Rotation is motion covering all values of the angle. next q p q p