Geometric Generalisations of SHAKE and RATTLE

A geometric analysis of the Shake and Rattle methods for constrained Hamiltonian problems is carried out. The study reveals the underlying differential geometric foundation of the two methods, and the exact relation between them. In addition, the geometric insight naturally generalises Shake and Rattle to allow for a strictly larger class of constrained Hamiltonian systems than in the classical setting. In order for Shake and Rattle to be well defined, two basic assumptions are needed. First, a nondegeneracy assumption, which is a condition on the Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy assumption, which is a condition on the geometry of the constrained phase space. Non-trivial examples of systems fulfilling, and failing to fulfill, these assumptions are given

Multi-symplectic discretisation of wave map equations

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions

Symplectic integrators for index one constraints

We show that symplectic Runge-Kutta methods provide effective symplectic integrators for Hamiltonian systems with index one constraints. These include the Hamiltonian description of variational problems subject to position and velocity constraints nondegenerate in the velocities, such as those arising in sub-Riemannian geometry and control theory.Comment: 13 pages, accepted in SIAM J Sci Compu

A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic

Variational collision integrator for polymer chains

The numerical simulation of many-particle systems (e.g., in molecular dynamics) often involves constraints of various forms. We present a symplectic integrator for mechanical systems with holonomic (bilateral) and unilateral contact constraints, the latter being in the form of a nonpenetration condition. The scheme is based on a discrete variant of Hamilton’s principle in which both the discrete trajectory and the unknown collision time are varied (cf. [Fetecau et al., 2003, SIAM J. Applied Dynamical Systems, 2, pp. 381–416]). As a consequence, the collision event enters the discrete equations of motion as an unknown that has to be computed on-the-fly whenever a collision is imminent. The additional bilateral constraints are e ciently dealt with employing a discrete null space reduction (including a projection and a local reparametrisation step) which considerably reduces the number of unknowns and improves the condition number during each time-step as compared to a standard treatment with Lagrange multipliers. We illustrate the numerical scheme with a simple example from polymer dynamics, a linear chain of beads, and test it against other standard numerical schemes for collision problems

Geometry of discrete-time spin systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features, that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on $T^*\mathbf{R}^{2n}$ for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with K\"ahler geometry, this provides another geometric proof of symplecticity.Comment: 17 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.333

Order conditions for sampling the invariant measure of ergodic stochastic differential equations on manifolds

We derive a new methodology for the construction of high order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge-Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.Comment: 40 page