Summary.We study a discrete analog of the Lagrange-d'Alembert principle of nonhonolomic mechanics and give conditions for it to define a map and to be reversible. In specific cases it can generate linearly implicit, semi-implicit, or implicit numerical integrators for nonholonomic systems which, in several examples, exhibit superior preservation of the dynamics. We also study discrete nonholonomic systems on Lie groups and their reduction theory, and explore the properties of the exact discrete flow of a nonholonomic system.

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called "conformal". We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie-Poisson manifolds, and their natural quotients.

This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980's by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein [1974]) and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misio lek, Perlmutter and Ratiu [1998]). The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott-Virasoro group and the KdV equation. Contents 1 Introduction and Background 2 Commuting Reduction

A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge-Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and Störmer-Verlet schemes.

Energy drift is commonly observed in reversible integrations of systems of molecular dynamics. We show that this drift can be modelled as a diffusion and that the typical energy error after time T is O( √ T ).PACS numbers: 45.10.−b, 05.45.PqIn simulations of conservative systems, the energy H is usually monitored as a check on the calculation. In symplectic integration of Hamiltonian systems, it is known that the integrator is very close to the flow of a Hamiltonian system with Hamiltonian close to H, so that one can give conditions under which the energy error is bounded for exponentially long times [14]. However, in reversible integration [1-3, 5, 7, 8, 10], one typically sees the energy drift away from its initial value. In this letter we model this drift as a diffusion process, showing that the expected drift after time T is O( √ T ). There are several reasons why one might use a reversible integrator on a conservative system. First, if the system is Hamiltonian, a symplectic integrator might be prohibitively expensive. This occurs if one wants to adaptively vary the time step, which can be much cheaper to do reversibly than symplectically, or if the symplectic structure is noncanonical, perhaps as a result of a change of variables. See, e.g., the discussion of the Nosé-Hoover thermostat of molecular dynamics in [2].Second, if the system is not Hamiltonian but still has a first integral H, then a reversible integrator is the natural choice of geometric method. One can construct integrators which are reversible and preserve energy, but they are expensive, typically fully implicit in the dependent variables and in the (introduced) Lagrange multipliers. It is usually much cheaper to preserve just the reversibility, which is the dominant property characterizing the dynamics, and merely monitor the energy.We consider systems with phase space M and dynamicsẋ = f (x), reversible under the diffeomorphism R : M → M, i.e. R