Related papers: Multisymplectic variational integrators and space/…
Most numerical integration algorithms are not designed specifically for Hamiltonian systems and do not respect their characteristic properties, which include the preservation of phase space volume with time. This can lead to spurious…
We develop the equations of motion for full body models that describe the dynamics of rigid bodies, acting under their mutual gravity. The equations are derived using a variational approach where variations are defined on the Lie group of…
This paper presents a unified framework for studying dynamics and integration on $q$-cosymplectic manifolds. After outlining the geometric foundations of $q$-cosymplectic structures, we derive new results concerning integrable systems and…
We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the…
This paper develops a structure-preserving numerical integration scheme for a class of higher-order mechanical systems. The dynamics of these systems are governed by invariant variational principles defined on higher-order tangent bundles…
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 paper presents an analytical model and a geometric numerical integrator for a tethered spacecraft model that is composed of two rigid bodies connected by an elastic tether. This model includes important dynamic characteristics of…
Multi-symplectic integrators are typically regarded as a discretization of the Hamiltonian partial differential equations. This is due to the fact that, for generic finite-dimensional Hamiltonian systems, there exists only one independent…
We present novel geometric numerical integrators for Hunter--Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter--Saxton equation, the modified…
A variational formulation for accelerated optimization on normed vector spaces was recently introduced in Wibisono et al., and later generalized to the Riemannian manifold setting in Duruisseaux and Leok. This variational framework was…
This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base…
Symplectic integrators are the tool of choice for many researchers studying dynamical systems because of their good long-term energy conservation properties. For systems with a dominant central mass, symplectic integrators are also highly…
A symplectic, symmetric, second-order scheme is constructed for particle evolution in a time-dependent field with a fixed spatial step. The scheme is implemented in one space dimension and tested, showing excellent adequacy to experiment…
In recent years, two important techniques for geometric numerical discretization have been developed. In computational electromagnetics, spatial discretization has been improved by the use of mixed finite elements and discrete differential…
Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with…
Retraction maps are known to be the seed for all numerical integrators. These retraction maps-based integrators can be further lifted to tangent and cotangent bundles, giving rise to structure-preserving integrators for mechanical systems.…
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…
By the simple finite element method, we study the symplectic, multisymplectic structures and relevant preserving properties in some semi-linear elliptic boundary value problem in one-dimensional and two-dimensional spaces respectively. We…
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of…
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a…