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 $G=M\bowtie_{\gamma} H$, for some $\gamma:M\times M \to H$. By reduction, then, we obtain the Euler-Lagrange type and Hamilton's type equations of the same form for the quotient space $M\cong G/H$, 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 $\mathfrak{m}\cong \mathfrak{g}/\mathfrak{h}$ 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 $n$th order tangent group $T^nG$ of the same type. More precisely, $\mathfrak{g}$ being the Lie algebra of $G$, $T^{(n)}G \cong \mathfrak{g}^{\times \,2^n-1-n} \bowtie_\gamma T^nG$ for some $\gamma:\mathfrak{g}^{\times \,2^n-1-n} \times \mathfrak{g}^{\times \,2^n-1-n} \to T^nG$. We thus obtain the $n$th order Euler-Lagrange (and then the $n$th order Euler-Poincar\'e) equations over $T^nG$ by reduction from those on $T(T^{n-1}G)$. Finally, we illustrate our results in the realm of the Kepler problem, and the non-linear tokamak plasma dynamics.