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Tuesday Week 1 Lecture

This lecture by Jeff Eldred introduces generating functions and their importance in determining action-angle coordinates, including a step-by-step derivation and an example involving a triangle well potential.

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Tuesday Week 1 Lecture

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  1. 1 Tuesday Week 1 Lecture Jeff Eldred Generating Functions, Action-Angle

  2. 2 Overview Generating Functions Derivation of Action Angle Coordinates Action-Angle Example

  3. 3 Generating Functions

  4. Why use Generating Functions? • To make sure you are doing a change-of-coordinates / change-of-reference-frame correctly. • To help you determine if there is a change-of-coordinates that would simplify the problem. • To find the constants of motion which will help you identify and distinguish the stable and unstable regions of the phase-space. • As a theoretical tool to develop Liouville’s theorem and other general results about Hamiltonian systems.

  5. Types of Generating Functions q, Q independent q, P independent q, Q independent p, P independent Credit: Wikipedia “Generating Function (physics)”

  6. 6 Derivation of Action-Angle Coordinates

  7. Action-Angles Coordinates

  8. Action-Angles (1D system) The formula for action is given by: But let’s derive it. Consider a 1D Hamiltonian system: Invent a generating function to new coordinates: For J, phi to be action angle, we require: Which means we can calculate J from the angular freq.:

  9. Action-Angles (Time-Energy form) We can calculate J from the angular freq.: And we can calculate the angular freq. from: That gives us J, we can then calculate phi:

  10. Action-Angles (p-q form) We can calculate J from the angular freq.: We can rewrite this is the familiar form:

  11. Two methods for Action-Angles Method 1 (Position - Momentum): Method 2 (Time - Energy):

  12. 12 Action-Angle Example

  13. Example: Triangle Well Given Potential: Calculate the Action:

  14. Triangle Well (cont.) Calculate Generating Function: by using Calculate Angle: by using

  15. Triangle Well (final) After some algebra, Coordinates from Action-angle: by using Calculate Angular Freq. from Action: We have found the trajectories:

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