|week 4 overview|
- 7. Discrete symmetries of dynamics
- 8. Discrete symmetry reduction of dynamics to a fundamental domain
- Homework 4
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|
|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."|