Week 4
|
week 4 overview |
- 7. Discrete symmetries of dynamics
- 8. Discrete symmetry reduction of dynamics to a fundamental domain
- Homework 4
- Optional
What is a symmetry of laws of motion? The families of symmetry-related full state space cycles are replaced by fewer and often much shorter "relative" cycles, and the notion of a prime periodic orbit is replaced by the notion of a "relative" periodic orbit, the shortest segment that tiles the cycle under the action of the group. Discrete symmetries: a review of the theory of finite groups.
While everyone can visualize the fundamental domain for a 3-disk billiard, the simpler problem - symmetry reduction of 1d dynamics that is equivariant under a reflection, the most common symmetry in applications - seems to baffle everyone. So here is a step-by-step walk through to this simplest of all symmetry reductions.
|
Chapter 11 World in a mirror |
|
Chapter 11 slides |
|
|
Fundamental domain |
|
|
D. Lippolis : Discrete symmetry reduction - examples (60 min) |
|
|
D. Lippolis : slides |
 |
discrete symmetry reduction |
 |
Discussion forum for week 5 |
|
|
Handwritten: a 1-dimensional fundamental domain |
| "It does not say anyplace in the Bible that if equations of motion have a symmetry, solutions should have it too." |