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We discuss some families of integrable and superintegrable systems in -dimensional Euclidean space which are invariant under rotations. The invariant Hamiltonian is integrable with integrals of motion and an additional integral of motion , which are first- and fourth-order polynomials in momenta, respectively. [ABSTRACT FROM AUTHOR]

We discuss, in all generality, the reduction of a Hamilton — Jacobi theory for systems subject to nonholonomic constraints and invariant under the action of a group of symmetries. We consider nonholonomic systems subject to both linear and nonlinear constraints and with different positioning of such constraints with respect to the symmetries. [ABSTRACT FROM AUTHOR]

In this article, we first present the Kustaanheimo - Stiefel regularization of the spatial Kepler problem in a symplectic and quaternionic approach. We then establish a set of action-angle coordinates, the so-called LCF coordinates, of the Kustaanheimo - Stiefel regularized Kepler problem, which is consequently used to obtain a conjugacy relation between the integrable approximating 'quadrupolar' system of the lunar spatial three-body problem and its regularized counterpart. This result justifies the study of Lidov and Ziglin [14] of the quadrupolar dynamics of the lunar spatial three-body problem near degenerate inner ellipses. [ABSTRACT FROM AUTHOR]

We show that the n-dimensional MICZ-Kepler system arises from symplectic reduction of the 'Kepler problem' on the cone over the rotation group SO( n). As a corollary we derive an elementary formula for the general solution of the MICZ-Kepler problem. The heart of the computation is the observation that the additional MICZ-Kepler potential, | ϕ|/ r, agrees with the rotational part of the cone's kinetic energy. [ABSTRACT FROM AUTHOR]

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor. [ABSTRACT FROM AUTHOR]