Mechanical Systems with Symmetry, Variational Principles, and Integration Algorithms

This paper studies variational principles for mechanical systems with symmetry and their applications to integration algorithms. We recall some general features of how to reduce variational principles in the presence of a symmetry group along with general features of integration algorithms for mechanical systems. Then we describe some integration algorithms based directly on variational principles using a discretization technique of Veselov. The general idea for these variational integrators is to directly discretize Hamilton’s principle rather than the equations of motion in a way that preserves the original systems invariants, notably the symplectic form and, via a discrete version of Noether’s theorem, the momentum map. The resulting mechanical integrators are second-order accurate, implicit, symplectic-momentum algorithms. We apply these integrators to the rigid body and the double spherical pendulum to show that the techniques are competitive with existing integrators

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature

Asynchronous Variational Integrators

We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system.A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorith

Generalized, energy-conserving numerical simulations of particles in general relativity. II. Test particles in electromagnetic fields and GRMHD

Direct observations of compact objects, in the form of radiation spectra, gravitational waves from VIRGO/LIGO, and forthcoming direct imaging, are currently one of the primary source of information on the physics of plasmas in extreme astrophysical environments. The modeling of such physical phenomena requires numerical methods that allow for the simulation of microscopic plasma dynamics in presence of both strong gravity and electromagnetic fields. In Bacchini et al. (2018) we presented a detailed study on numerical techniques for the integration of free geodesic motion. Here we extend the study by introducing electromagnetic forces in the simulation of charged particles in curved spacetimes. We extend the Hamiltonian energy-conserving method presented in Bacchini et al. (2018) to include the Lorentz force and we test its performance compared to that of standard explicit Runge-Kutta and implicit midpoint rule schemes against analytic solutions. Then, we show the application of the numerical schemes to the integration of test particle trajectories in general relativistic magnetohydrodynamic (GRMHD) simulations, by modifying the algorithms to handle grid-based electromagnetic fields. We test this approach by simulating ensembles of charged particles in a static GRMHD configuration obtained with the Black Hole Accretion Code (BHAC)

Exactly Conservative Integrators

Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems.Comment: 30 pages, postscript (1.3MB). Submitted to SIAM J. Sci. Comput

Variational Integrators and the Newmark Algorithm for Conservative and Dissipative Mechanical Systems

The purpose of this work is twofold. First, we demonstrate analytically that the classical Newmark family as well as related integration algorithms are variational in the sense of the Veselov formulation of discrete mechanics. Such variational algorithms are well known to be symplectic and momentum preserving and to often have excellent global energy behavior. This analytical result is veried through numerical examples and is believed to be one of the primary reasons that this class of algorithms performs so well. Second, we develop algorithms for mechanical systems with forcing, and in particular, for dissipative systems. In this case, we develop integrators that are based on a discretization of the Lagrange d'Alembert principle as well as on a variational formulation of dissipation. It is demonstrated that these types of structured integrators have good numerical behavior in terms of obtaining the correct amounts by which the energy changes over the integration run