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

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

3 Generating Functions 3 3 3 3 3

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. 4 4 4 4

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

Derivation of Action-Angle Coordinates 6 Derivation of Action-Angle Coordinates 6 6 6 6 6

Action-Angles Coordinates 7 7 7 7

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.: 8 8 8 8

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: 9 9 9 9

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

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

12 Action-Angle Example 12 12 12 12 12

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

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

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