Related papers: Surrogate Lagrangians for Variational Integrators:…
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…
Hierarchies of Lagrangians of degree two, each only partly determined by the choice of leading terms and with some coefficients remaining free, are considered. The free coefficients they contain satisfy the most general differential…
The Dirac-Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories -- general relativity, electromagnetism, Yang Mills,…
We present the first method to directly use a learned continuous Lagrangian to forecast the dynamics of systems governed by partial differential equations, exploiting the inherent conservative structure to achieve stable long-range…
In this paper, we consider the problem of finding surrogate models for large-scale second-order linear time-invariant systems with inhomogeneous initial conditions. For this class of systems, the superposition principle allows us to…
It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincar\'e…
In this paper, we introduce a Lagrange multiplier approach to construct linearly implicit energy-preserving schemes of arbitrary order for general Hamiltonian PDEs. Unlike the widely used auxiliary variable methods, this novel approach does…
On the basis of the previous work by Tang \& Zhang (Appl. Math. Comput. 323, 2018, p. 204--219), in this paper we present a more effective way to construct high-order symplectic integrators for solving second order Hamiltonian equations.…
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…
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…
A consequent approach is proposed to construct symplectic force-gradient algorithms of arbitrarily high orders in the time step for precise integration of motion in classical and quantum mechanics simulations. Within this approach the basic…
We consider (3+1)-dimensional second-order evolutionary PDEs where the unknown $u$ enters only in the form of the 2nd-order partial derivatives. For such equations which possess a Lagrangian, we show that all of them have a symplectic…
In this paper, we develop variational integrators for the nonequilibrium thermodynamics of simple closed systems. These integrators are obtained by a discretization of the Lagrangian variational formulation of nonequilibrium thermodynamics…
Finite-dimensional non-canonical Hamiltonian systems arise naturally from Hamilton's principle in phase space. We present a method for deriving variational integrators that can be applied to perturbed non-canonical Hamiltonian systems on…
The dynamical motion of mechanical systems possesses underlying geometric structures, and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single…
We develop the technique of the variational approximation for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms can be discarded for various reasons. For…
We present high-order variational Lagrangian finite element methods for compressible fluids using a discrete energetic variational approach. Our spatial discretization is mass/momentum/energy conserving and entropy stable. Fully implicit…
We consider the question of existence of Hamiltonians for autonomous non-holonomic mechanical systems in this paper. The approach is elementary in the sense that the existence of a Hamiltonian for a given non-holonomic system is considered…
We extend the perturbative approach developed in an earlier work to deal with Lagrangians which have arbitrary higher order time derivative terms for both bosons and fermions. This approach enables us to find an effective Lagrangian with…
Variational inequality problems are recognized for their broad applications across various fields including machine learning and operations research. First-order methods have emerged as the standard approach for solving these problems due…