Week 6
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Week 6 overview |
- 11. Slice and dice
- 11. Qualitative dynamics for pedestrians
- Homework 6
- Optional
Actions of a Lie group on a state trace out a manifold of equivalent states, or its group orbit. Symmetry reduction is the identification of a unique point on a group orbit as the representative of this equivalence class. Thus, if the symmetry is continuous, the interesting dynamics unfolds on a lower-dimensional `quotiented', or `reduced' state space M/G. In the method of slices the symmetry reduction is achieved by cutting the group orbits with a set of hyperplanes, one for each continuous group parameter, with each group orbit of symmetry-equivalent points represented by a single point, its intersection with the slice. Moving frames give us a great deal of freedom - we discuss how to choose a frame The most natural of all moving frames: the comoving frame, the frame for space cowboys.
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Chapter Slice & dice Read Sects. 13.4 and 13.5. The rest is optional. |
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Overheads |
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Symmetry reduction NBB |
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Symmetry reduced equations of motion |
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Sections and slices are local, good up to a border |
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A spatial Fourier expansion NBB |
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First Fourier mode slice NBB |
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In-slice time NBB |
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Comoving frames As of 2 Mar 2015, exists only as a video, this is not yet written up in ChaosBook The most natural of all moving frames: the comoving frame, the frame for space cowboys. |
Qualitative properties of a flow partition the state space in a topologically invariant way.
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Symbolic dynamics |
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Discussion forum for week 6 |
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Master Slicer Certificate |
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Ring of Fire Visualize the O(2) equivariance of Kuramoto-Sivashinsky (AKA Ring of Fire) |