dorsal/arxiv
View SchemaDiscrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top
| Authors | A. I. Bobenko, Yu. B. Suris |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9810018 |
| URL | https://arxiv.org/abs/solv-int/9810018 |
| DOI | 10.1007/s002200050642 |
| Journal | Commun. Math. Phys., 1999, V. 204, p. 147-188 |
Abstract
We develop the theory of discrete time Lagrangian mechanics on Lie groups, originated in the work of Veselov and Moser, and the theory of Lagrangian reduction in the discrete time setting. The results thus obtained are applied to the investigation of an integrable time discretization of a famous integrable system of classical mechanics, -- the Lagrange top. We recall the derivation of the Euler--Poinsot equations of motion both in the frame moving with the body and in the rest frame (the latter ones being less widely known). We find a discrete time Lagrange function turning into the known continuous time Lagrangian in the continuous limit, and elaborate both descriptions of the resulting discrete time system, namely in the body frame and in the rest frame. This system naturally inherits Poisson properties of the continuous time system, the integrals of motion being deformed. The discrete time Lax representations are also found. Kirchhoff's kinetic analogy between elastic curves and motions of the Lagrange top is also generalised to the discrete context.
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"abstract": "We develop the theory of discrete time Lagrangian mechanics on Lie groups,\noriginated in the work of Veselov and Moser, and the theory of Lagrangian\nreduction in the discrete time setting. The results thus obtained are applied\nto the investigation of an integrable time discretization of a famous\nintegrable system of classical mechanics, -- the Lagrange top. We recall the\nderivation of the Euler--Poinsot equations of motion both in the frame moving\nwith the body and in the rest frame (the latter ones being less widely known).\nWe find a discrete time Lagrange function turning into the known continuous\ntime Lagrangian in the continuous limit, and elaborate both descriptions of the\nresulting discrete time system, namely in the body frame and in the rest frame.\nThis system naturally inherits Poisson properties of the continuous time\nsystem, the integrals of motion being deformed. The discrete time Lax\nrepresentations are also found. Kirchhoff\u0027s kinetic analogy between elastic\ncurves and motions of the Lagrange top is also generalised to the discrete\ncontext.",
"arxiv_id": "solv-int/9810018",
"authors": [
"A. I. Bobenko",
"Yu. B. Suris"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1007/s002200050642",
"journal_ref": "Commun. Math. Phys., 1999, V. 204, p. 147-188",
"title": "Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top",
"url": "https://arxiv.org/abs/solv-int/9810018"
},
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