Symmetry reduction of discrete Lagrangian mechanics on Lie groups
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2000
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| _version_ | 1866911217092657152 |
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| author | Marsden, Jerrold E. Pekarsky, Sergey Shkoller, Steve |
| author_facet | Marsden, Jerrold E. Pekarsky, Sergey Shkoller, Steve |
| contents | For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $ω_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0004018 |
| institution | arXiv |
| publishDate | 2000 |
| record_format | arxiv |
| spellingShingle | Symmetry reduction of discrete Lagrangian mechanics on Lie groups Marsden, Jerrold E. Pekarsky, Sergey Shkoller, Steve Numerical Analysis Symplectic Geometry For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group $G$ of the canonical discrete Lagrange 2-form $ω_\mathbb{L}$ on $G \times G$. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton-Jacobi equation are made. The rigid body is discussed as an example. |
| title | Symmetry reduction of discrete Lagrangian mechanics on Lie groups |
| topic | Numerical Analysis Symplectic Geometry |
| url | https://arxiv.org/abs/math/0004018 |