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arXiv · 1111.3887

Symmetry Reduction by Lifting for Maps

H. R. Dullin·H. E. Lomeli·J. D. Meiss
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Abstract

We study diffeomorphisms that have one-parameter families of continuous symmetries. For general maps, in contrast to the symplectic case, existence of a symmetry no longer implies existence of an invariant. Conversely, a map with an invariant need not have a symmetry. We show that when a symmetry flow has a global Poincaré section there are coordinates in which the map takes a reduced, skew-product form, and hence allows for reduction of dimensionality. We show that the reduction of a volume-preserving map again is volume preserving. Finally we sharpen the Noether theorem for symplectic maps. A number of illustrative examples are discussed and the method is compared with traditional reduction techniques.

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H. R. Dullin, H. E. Lomeli, J. D. Meiss. 2011-11-22. Symmetry Reduction by Lifting for Maps. https://doi.org/10.1088/0951-7715%2F25%2F6%2F1709

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