Journal of Geometry and Symmetry in Physics

On Henri Poincaré’s Note “Sur une forme nouvelle des équations de la Mécanique”

Charles-Michel Marle

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Abstract

We present in modern language the contents of the famous note published by Henri Poincaré in 1901, in which he proves that, when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the well known Euler-Lagrange equations are equivalent to a new system of differential equations defined on the product of the configuration space with the Lie algebra. We write these equations, called the Euler-Poincaré equations, in an intrinsic form, without any reference to a particular system of local coordinates, and prove that they can be conveniently expressed in terms of the Legendre and momentum maps. We discuss the use of the Euler-Poincaré equation for reduction (a procedure sometimes called Lagrangian reduction by modern authors), and compare this procedure with the well known Hamiltonian reduction procedure (formulated in modern terms in 1974 by J. Marsden and A. Weinstein). We explain how a break of the symmetry in the phase space produces the appearance of a semi-direct product of Lie groups.

Article information

Source
J. Geom. Symmetry Phys., Volume 29 (2013), 1-38.

Dates
First available in Project Euclid: 26 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.jgsp/1495764098

Digital Object Identifier
doi:10.7546/jgsp-29-2013-1-38

Mathematical Reviews number (MathSciNet)
MR3113556

Zentralblatt MATH identifier
1315.37034

Citation

Marle, Charles-Michel. On Henri Poincaré’s Note “Sur une forme nouvelle des équations de la Mécanique”. J. Geom. Symmetry Phys. 29 (2013), 1--38. doi:10.7546/jgsp-29-2013-1-38. https://projecteuclid.org/euclid.jgsp/1495764098


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