Contact variational integrators

Abstract

We present geometric numerical integrators for contact flows that stem from a discretization of Herglotz' variational principle. First we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle. Then we discuss the backward error analysis of our variational integrators, including the construction of a modified Lagrangian. Throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues.

Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids

“…Partly inspired by, and partly based on, the intensive development on symplectic schemes for reversible problems, remarkable research is done in recent years on dissipative systems, more on ones with finite degrees of freedom (including [2][3][4][5]) and less for continua (see, e.g., [6][7][8][9]).…”

Numerical integration in Celestial Mechanics: a case for contact geometry

“…When using the Hamiltonian integrator of the sixth order we specify with A, B, C or E if we use, respectively, the approximate coefficients from column A, B, C of Table 1 or the exact coefficients from (14). The second order variational integrator is the one from [38], while the fourth order variational integrator is the one introduced in Section 3.1. In all figures, the eccentricity of the trajectory is defined in the following sense: it is the eccentricity of the trajectory of an unperturbed Kepler problem with the same initial conditions.…”

“…This was originally published by Herglotz in a set of lecture notes [23], which might explain why it has received relatively little attention. A modern discussion of Herglotz' variational principle can be found for example in [20,38] (see also [17,24,25] for extensions to field theories).…”

“…One of the advantages of having a contact Hamiltonian structure is that many ideas for geometric integrators for symplectic systems can be carried over with relatively small effort. A study of variational integrators in the contact case has been started recently in [38]. Here we develop higher order variational and Hamiltonian integrators for contact systems.…”

“…To quantify the error at the initial step, we recall that the initial condition for the integration of the Lane-Emden equation is y(0) = 1, p(0) = 0, and that at the initial step we have x = τ . Thus (36)- (38) for the initial step give…”