Related papers: A Geometric Approach to Structure-Preserving Integ…
In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes…
We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational…
In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In…
We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference…
In this paper we develop a geometric version of the Hamilton-Jacobi equation in the Poisson setting. Specifically, we "geometrize" what is usually called a complete solution of the Hamilton-Jacobi equation. We use some well-known results…
Geometric representation learning in preserving the intrinsic geometric and topological properties for discrete non-Euclidean data is crucial in scientific applications. Previous research generally mapped non-Euclidean discrete data into…
We study the dynamics of contact mechanical systems on Lie groups that are invariant under a Lie group action. Analogously to standard mechanical systems on Lie groups, existing symmetries allow for reducing the number of equations. Thus,…
In this paper, we introduce local expressions for discrete Mechanics. To apply our results simultaneously to several interesting cases, we derive these local expressions in the framework of Lie groupoids, following the program proposed by…
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard…
We present a framework to systematically address topological phases when finer partitionings of bands are taken into account, rather than only considering the two subspaces spanned by valence and conduction bands. Focusing on…
In this paper, we present a stochastic augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints.We investigate the…
This article develops a unified and intrinsic framework for the theory of Sobolev spaces on vector bundles over Riemannian manifolds. The analytical core of our approach is an explicit higher-order geometric integration by parts formula,…
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…
We propose a coordinate-invariant geometric formulation of the GENERIC stochastic differential equation, unifying reversible Hamiltonian and irreversible dissipative dynamics within a differential-geometric framework. Our construction…
We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the…
The study of transport and mixing processes in dynamical systems is particularly important for the analysis of mathematical models of physical systems. We propose a novel, direct geometric method to identify subsets of phase space that…
Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…
A tutorial introduction to projective geometric algebra (PGA), a modern, coordinate-free framework for doing euclidean geometry. PGA features: uniform representation of points, lines, and planes; robust, parallel-safe join and meet…
Contact Hamiltonian systems extend symplectic Hamiltonian mechanics to dissipative settings while retaining geometric structure. We develop a structure-preserving splitting framework for contact Hamiltonian systems on $J^1(\mathbb{R}^n)$…
In this paper, we discuss the geometric integration of hamiltonian systems on Poisson manifolds, in particular, in the case, when the Poisson structure is induced by a Lie algebra, that is, it is a Lie-Poisson structure. A Hamiltonian…