Week 15
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Have: evolution law. Want: invariant predictions |
- 29. Discrete symmetry factorization / 26 April 2022 /
- 30. Continuous symmetry factorization / 28 April 2022 bonus lecture /
- Homework 15
- Optional
3 disk pinball symmetries suffice to illustrate all that is needed to factorized spectral determinants for any system with a discrete symmetry: character. Discrete symmetry tiles the state space, and dynamics can be reduced to dynamics on the fundamental domain, together with a finite matrix that keeps track of the tile the full state space trajectory lands on. We need some group theory (one needs to underatand the projection to irreducible representations) and illustrate how different classes of periodic orbits contribute to different invariant subspaces for the 3-disk pinball. Read sects. 25.2 - 25.6.
Trace formulas relate short time dynamics (unstable periodic orbits) to long time invariant state space densities (natural measure). A trace formula for a partially hyperbolic (N+ 1)-dimensional compact manifold invariant under a global continuous symmetry is derived. In this extension of “periodic orbit” theory there are no or very few periodic orbits - the relative periodic orbits that the trace formula has support on are almost never eventually periodic.
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Chapter 26 Continuous symmetry factorization This chapter needs a major revision, the videos are more up-to-date than the chapter itself |
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Group averaging |
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The Great Orthonormality Theorem |
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Average over space and time |
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Discrete time averaging |
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Continuous time averaging |
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If discrete symmetry, average over it |
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If continuous symmetry, average over it |
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discrete factorization Due 28 April 2015 (note!) |
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Discussion forum for week 15 |
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John F Gibson solves the Navier-Stokes |