Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 2913.10.3.2 Lagrange–Poincaré DynamicsEuler–Poincaré EquationsLet G be a Lie group and let L : T G → R be a left–invariant Lagrangian.Let l : g → R be its restriction to the identity. For a curve g(t) ∈ G, letξ(t) = g(t) −1 · ġ(t); that is, ξ(t) = T g(t) L g(t) −1ġ(t). Then the following areequivalent [Marsden and Ratiu (1999)]:(1) g(t) satisfies the Euler–Lagrangian equations for L on G;(2) The variational principle holds,∫δ L(g(t), ġ(t)) dt = 0for variations with fixed endpoints;(3) The Euler–Poincaré equations hold:d ∂ldt ∂ξ = δlAd∗ ξδξ ;(4) The variational principle holds on g,∫δ l(ξ(t)) dt = 0,using variations of the form δξ = ˙η + [ξ, η], where η vanishes at theendpoints.Lagrange–Poincaré EquationsHere we follow [Marsden and Ratiu (1999)] and drop Euler–Lagrangianequations and variational principles from a general velocity phase–spaceT M to the quotient T M/G by an action of a Lie group G on M. If Lis a G−invariant Lagrangian on T M, it induces a reduced Lagrangian lon T M/G. We introduce a connection A on the principal bundle M →S = M/G, assuming that this quotient is nonsingular. This connectionallows one to split the variables into a horizontal and vertical part. Letinternal variables x α be coordinates for shape–space S = M/G, let η a becoordinates for the Lie algebra g relative to a chosen basis, let l be theLagrangian regarded as a function of the variables x α , ẋ α , η a and let Cdbabe the structure constants of the Lie algebra g of G.If one writes the Euler–Lagrangian equations on T M in a local principalbundle trivialization, with coordinates x α on the base and η a in the fibre,