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

2. Overview Generating Functions Derivation of Action Angle Coordinates2
Overview
Generating Functions
Derivation of Action Angle Coordinates
Action-Angle Example 2 2 2 2 2

3. 3
Generating Functions 3 3 3 3 3

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. 4 4 4 4

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

6. Derivation of Action-Angle Coordinates6
Derivation of Action-Angle Coordinates 6 6 6 6 6

7. Action-Angles Coordinates7 7 7 7

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.: 8 8 8 8

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: 9 9 9 9

10. 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

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

12. 12
Action-Angle Example 12 12 12 12 12

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

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

15. 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