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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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**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**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**Alternate Generators**• Generating functions differ by a Legendre transformation. • The transformation can be expressed as type I. • S is also periodic with period 1**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**The angle can be derived from the generator.**The momentum and position also can be derived. Derived Variables**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**Generating Function**• The generating function S’ can be found by integration and substitution. • The function S comes from the Legendre transformation**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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