Dynamics Over Homogeneous Spaces

We present the Euler-Lagrange and Hamilton's equations for a system whose configuration space is a unified product Lie group G=Mγ H, for some γ:M× 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 G/H, although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler-Poincar\'{e} type and Lie-Poisson type equations on the corresponding quotient m g/h of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the nth order iterated tangent group T⁽ⁿ⁾G of a Lie group G as an extension of the nth order tangent group TⁿG of the same type. More precisely, g being the Lie algebra of G, T⁽ⁿ⁾G g× \,2ⁿ-1-n _γ TⁿG for some γ:g× \,2ⁿ-1-n × g× \,2ⁿ-1-n → TⁿG. We thus obtain the nth order Euler-Lagrange (and then the nth order Euler-Poincar\'e) equations over TⁿG by reduction from those on T(Tⁿ⁻¹G). Finally, we illustrate our results in the realm of the Kepler problem, and the non-linear tokamak plasma dynamics.