Related papers: Geometric integrators for higher-order mechanics o…
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
A variational integrator for ideal magnetohydrodynamics is derived by applying a discrete action principle to a formal Lagrangian. Discrete exterior calculus is used for the discretisation of the field variables in order to preserve their…
Discrete Hamiltonian variational integrators are derived from Type II and Type III generating functions for symplectic maps, and in this paper we establish a variational error analysis result that relates the order of accuracy of the…
For integrable systems in the sense of multidimensional consistency (MDC) we can consider the Lagrangian as a form, which is closed on solutions of the equations of motion. For 2-dimensional systems, described by partial difference…
This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations of a generic autonomous system along a particular solution. The main result of this paper is a compact yet explicit and…
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector…
Symplectic schemes are powerful methods for numerically integrating Hamiltonian systems, and their long-term accuracy and fidelity have been proved both theoretically and numerically. However direct applications of standard symplectic…
We propose to use the properties of the Lie algebra of the angular momentum to build symplectic integrators dedicated to the Hamiltonian of the free rigid body. By introducing a dependence of the coefficients of integrators on the moments…
We consider systems of ordinary differential equations with known first integrals. The notion of a discrete tangent space is introduced as the orthogonal complement of an arbitrary set of discrete gradients. Integrators which exactly…
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…
Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional…
This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through…
This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of…
We present a class of non-standard numerical schemes which are modifications of the discrete gradient method. They preserve the energy integral exactly (up to the round-off error). The considered class contains locally exact discrete…
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid…
Using Dirac's approach to constrained dynamics, the Hamiltonian formulation of regular higher order Lagrangians is developed. The conventional description of such systems due to Ostrogradsky is recovered. However, unlike the latter, the…
We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the…
This paper presents a generalization of conventional sliding mode control designs for systems in Euclidean spaces to fully actuated simple mechanical systems whose configuration space is a Lie group for the trajectory-tracking problem. A…
Compatible discretizations, such as finite element exterior calculus, provide a discretization framework that respect the cohomological structure of the de Rham complex, which can be used to systematically construct stable mixed finite…
We develop an explicit, second-order, variational time integrator for full body dynamics that preserves the momenta of the continuous dynamics, such as linear and angular momenta, and exhibits near-conservation of total energy over…