Dynamics over cocycle double cross-products

Abstract

In this paper, we present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group G=M (sic)(gamma) H, for some gamma : M x M -> H. By reduction, then, we obtain the Euler-Lagrange-type and Hamilton's-type equations of the same form for the quotient space M congruent to G/H, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler-Poincar & eacute;-type and Lie-Poisson-type equations on the corresponding quotient m congruent to g/h of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the n(th) order iterated tangent group T((n))G of a Lie group G as an extension of the nth order tangent group T(n)G of the same type. More precisely, g is the Lie algebra of G, T((n))G congruent to g(x 2n-1-n)(sic)(gamma) T-n G for some gamma : g(x2n-1-n) x g(x2n-1-n) -> T(n)G. We thus obtain the nnth order Euler-Lagrange (and then the nnth order Euler-Poincar & eacute;) equations over T(n)G by reduction from those on T(T(n-1)G). Finally, we illustrate our results in the realm of the Kepler problem, and the nonlinear tokamak plasma dynamics.