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.

References in corpus (21)

• The classical mechanics of non-conservative systems
• Contact Hamiltonian Mechanics
• Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems
• Geometry of Thermodynamic Processes
• Contact manifolds and dissipation, classical and quantum
• Quantization of Contact Manifolds and Thermodynamics
• New integrable (3+1)-dimensional systems and contact geometry
• A thermostat algorithm generating target ensembles
• From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy
• Dynamical Similarity
• Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$
• Contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics
• Contact flows and integrable systems
• Contact Geometry and Quantum Mechanics
• Variational order for forced Lagrangian systems
• Contact Geometry of the Pontryagin Maximum Principle
• On the geometric quantization of contact manifolds
• Modified Equations for Variational Integrators
• Integrable (3+1)-dimensional systems with rational Lax pairs
• Exact Baker-Campbell-Hausdorff formula for the contact Heisenberg algebra
• A contact covariant approach to optimal control with applications to sub-Riemannian geometry

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• Numerical integration in celestial mechanics: a case for contact geometry
• Four spacetime dimensional simulation of rheological waves in solids and the merits of thermodynamics
• Generalized nonconservative gravitational field equations from Herglotz action principle
• A Variational Integrator for the Discrete Element Method
• BullFrog: Multi-step perturbation theory as a time integrator for cosmological simulations
• Numerical integration of stochastic contact Hamiltonian systems via stochastic Herglotz variational principle
• Perturbation-theory informed integrators for cosmological simulations
• Formulation of Stochastic Contact Hamiltonian Systems
• New directions for contact integrators
• When action is not least for systems with action-dependent Lagrangians
• Geometric numerical integration of Liénard systems via a contact Hamiltonian approach