Related papers: Modified equations for variational integrators app…

A multi-agent system designed to achieve distance-based shape control with flocking behavior can be seen as a mechanical system described by a Lagrangian function and subject to additional external forces. Forced variational integrators are…

Making use of the modern techniques of non-holonomic geometry and constrained variational calculus, a revisitation of Ostrogradsky's Hamiltonian formulation of the evolution equations determined by a Lagrangian of order >= 2 in the…

Discrete variational methods show excellent performance in numerical simulations of mechanical systems. In this paper, we adapt discrete variational integrators for the case of mechanical systems with double-bracket dissipation. In…

The unique solvability and error analysis of the original Lagrange multiplier approach proposed in [8] for gradient flows is studied in this paper. We identify a necessary and sufficient condition that must be satisfied for the nonlinear…

We shall use the variational decomposition technique in order to calculate equations of motion and Noether energy-momentum complex for some classes of non-linear gravitational Lagrangians within the first-order (Palatini) formalism. In…

The 2-step staggered (also called leap-frog) time discretisation of linear 2nd-order Hamiltonian systems (typically linear elastodynamics in a stress-velocity form) is extended for a 3-step staggered discretisation applicable for systems…

Most real optimization problems are defined over a mixed search space where the variables are both discrete and continuous. In engineering applications, the objective function is typically calculated with a numerically costly black-box…

This work builds on the Volterra series formalism presented in [D. W. Dreisigmeyer and P. M. Young, J. Phys. A \textbf{36}, 8297, (2003)] to model nonconservative systems. Here we treat Lagrangians and actions as `time dependent' Volterra…

In this paper, we consider the linear programming (LP) formulation for deep reinforcement learning. The number of the constraints depends on the size of state and action spaces, which makes the problem intractable in large or continuous…

In this paper, we propose a variational Lagrangian scheme for a modified phase-field model, which can compute the equilibrium states for the original Allen-Cahn type model. Our discretization is based on a prescribed energy-dissipation law…

We construct all (2+1)-dimensional PDEs depending only on 2nd-order derivatives of unknown which have the Euler-Lagrange form and determine the corresponding Lagrangians. We convert these equations and their Lagrangians to two-component…

An alternative class of the Lagrangian called the multiplicative form is suc- cessfully derived for a system with one degree of freedom for both non-relativistic and relativistic cases. This new Lagrangian can be considered as a…

We present examples of Lax-integrable multi-dimensional systems of partial differential equations with higher local symmetries. We also consider Lagrangian deformations of these equations and construct variational bivectors on them.

In this paper, we present a variational integrator that is based on an approximation of the Euler--Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator. The…

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using…

In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…

In this work, we consider optimal control problems for mechanical systems on vector spaces with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an…

Embedding parameterized optimization problems as layers into machine learning architectures serves as a powerful inductive bias. Training such architectures with stochastic gradient descent requires care, as degenerate derivatives of the…

In the framework of Lagrangian formulation, some q-deformed physical systems are considered. The q-deformed Legendre transformation is obtained for the free motion of a non-relativistic particle on a quantum line. This is subsequently…

In this paper, we develop a structure-preserving discretization of the Lagrangian framework for electromagnetism, combining techniques from variational integrators and discrete differential forms. This leads to a general family of…