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Castrillón López, Marco and García Pérez, P.L and Ratiu, T.S.
(2001)
*Euler-Poincaré reduction on principal bundles.*
Letters in Mathematical Physics, 58
(2).
pp. 167-180.
ISSN 0377-9017

PDF
Restringido a Repository staff only 159kB |

Official URL: http://link.springer.com/article/10.1023%2FA%3A1013303320765

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## Abstract

Let G be a Lie group and let L:TG→R be a Lagrangian invariant under the natural action of G on its tangent bundle. Then L induces a function l:(TG)/G≅g→R called the reduced Lagrangian, g being the Lie algebra of G. As is well known, the Euler-Lagrange equations defined by L for curves on G are equivalent to a new kind of equation for l for the reduced curves in the Lie algebra g. These equations are known as the Euler-Poincaré equations. In the paper under review, the authors extend the idea of the Euler-Poincaré reduction to a Lagrangian L:J1P→R defined on the first jet bundle of an arbitrary principal bundle π:P→M with structure group G. The Lagrangian is assumed to be invariant under the natural action of G on J1P. Let l:(J1P)/G→R be the reduced Lagrangian. It is known that the quotient manifold (J1P)/G can be identified with the bundle of connections of π:P→M. The reduced variational problem has a nice geometrical interpretation in terms of connections. The authors study the compatibility conditions needed for reconstruction. In this framework the Euler-Poincaré equations do not suffice to reconstruct the Euler-Lagrange equations. Some extra conditions must be imposed, namely, the vanishing of the curvature of the critical sections. In the case of matrix groups this result has already been obtained [M. Castrillón López, T. S. Ratiu and S. Shkoller, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2155–2164;]. In this paper the authors give a proof for general Lie groups. Moreover, they point out several facts concerning the reduced variational problem: its relation with the variational calculus with constraints, Noether's theorem for reduced symmetries, and the second variation formula.

Item Type: | Article |
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Uncontrolled Keywords: | calculus of variations; Euler–Poincaré equations; reconstruction; reduction; symmetries. |

Subjects: | Sciences > Mathematics > Mathematical analysis Sciences > Mathematics > Applied statistics |

ID Code: | 23938 |

Deposited On: | 13 Dec 2013 18:26 |

Last Modified: | 12 Dec 2018 15:13 |

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