Dynamics over cocycle double cross-products

In this paper, we present the Euler–Lagrange and Hamilton’s equations for a system whose configuration space is a unified product Lie group [Formula: see text], for some [Formula: see text]. By reduction, then, we obtain the Euler–Lagrange-type and Hamilton’s-type equations of the same form for the quotient space [Formula: see text], although it is not necessarily a Lie group. We observe, through further reduction, that it is possible to formulate the Euler–Poincaré-type and Lie–Poisson-type equations on the corresponding quotient [Formula: see text] of Lie algebras, which is not a priori a Lie algebra. Moreover, we realize the [Formula: see text]th order iterated tangent group [Formula: see text] of a Lie group [Formula: see text] as an extension of the [Formula: see text]th order tangent group [Formula: see text] of the same type. More precisely, [Formula: see text] is the Lie algebra of [Formula: see text], [Formula: see text] for some [Formula: see text]. We thus obtain the [Formula: see text]th order Euler–Lagrange (and then the [Formula: see text]th order Euler–Poincaré) equations over [Formula: see text] by reduction from those on [Formula: see text]. Finally, we illustrate our results in the realm of the Kepler problem, and the nonlinear tokamak plasma dynamics.