1 Tuesday Week 1 Lecture Jeff Eldred Generating

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

2 Overview 1. Generating Functions 2. Derivation of Action Angle Coordinates 3. Action-Angle Example

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 Derivation of Action-Angle Coordinates

7 Action-Angles Coordinates

8 Action-Angles (1 D system) The formula for action is given by: But let’s derive it. Consider a 1 D 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 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: