Week 5
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Week 5 overview |
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They still do not get it! - The group theory course is now online |
- 9. Continuous symmetries of dynamics
- 10. Got a continuous symmetry? Freedom and its challenges
- Homework 5
- Optional
Symmetry reduction: If the symmetry is continuous, the interesting dynamics unfolds on a lower-dimensional "quotiented" system, with "ignorable" coordinates eliminated (but not forgotten). 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 has to be reexamined: it is replaced by the notion of a ``relative'' periodic orbit, the shortest segment that tiles the cycle under the action of the group.
Hilbert's invariant polynomials. Cartan's moving frames.
Whenever you have a continuous symmetry, you need to cut the orbit to pick out one representative for the whole family. For continuous spatial symmetries, this is achieved by slicing. And then there is dicing.
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Chapter 13 Slice & dice Read Sects. 13.1 to 13.3. |
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Moving frames |
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Phase of a relative periodic orbit, choice of moving frame |
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Slice |
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How to slice a continuous symmetry Long, 14 minutes take. |
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Slices are not sections! |
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continuous symmetry reduction |
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Discussion forum for week 5 |
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Why does everybody write a book on group theory? |
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Cross-sections, orbitfolds |
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Low dimensional slices; 2D flat heart |