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# Action-Angle

Action-Angle. Define a 1-D generator S’. Time-independent H . Require new conjugate variables to be constants of motion. Conjugate momentum is a constant J . Hamiltonian is constant Conjugate position is cyclic Linear in time. 1-Dimensional CT. units of action. a frequency.

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## Action-Angle

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1. Action-Angle

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

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

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

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

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

7. Physical View • The motion in phase space is harmonic. • Amplitudes of q, p • Area in phase space is the w times the action. • Angle w repeats per cycle. p J = E/n w= nt q

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

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

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