×

On Henri Poincaré’s note “Sur une forme nouvelle des équations de la Mécanique”. (English) Zbl 1315.37034

The author presents and re-interprets in modern language the contents of [H. Poincaré, C. R. Acad. Sci., Paris 132, 369–371 (1901; JFM 32.0715.01)]. Poincaré proves that when a Lie algebra acts locally transitively on the configuration space of a Lagrangian mechanical system, the Euler-Lagrange equations are equivalent to a new system of differential equations, now defined on the product space of the configuration space with the Lie algebra.
He writes this new system, the Euler-Poincaré equations, in intrinsic form and shows how they can be expressed in terms of the Legendre and momentum maps. Indeed, the author does a good deal more than translate a bit of Poincaré’s work into modern terms. He finds that more recent work on Lagrangian reduction often makes more restrictive assumptions than those needed by Poincaré.
Along the way, the author discusses the use of the Euler-Poincaré equations for reduction – a process now often called Lagrangian reduction – and compares it with the Hamiltonian reduction process of Marsden and Weinstein.
Finally, he explains how and why a break in the symmetry of phase space gives rise to the semi-direct product of Lie groups.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37-03 History of dynamical systems and ergodic theory
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics

Biographic References:

Poincaré, Henri

Citations:

JFM 32.0715.01
PDF BibTeX XML Cite
Full Text: arXiv