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This work proposes a suite of numerical techniques to facilitate the design of structure-preserving integrators for nonlinear dynamics. The celebrated LaBudde-Greenspan integrator and various energy-momentum schemes adopt a difference…
A discrete theory for implicit nonholonomic Lagrangian systems undergoing elastic collisions is developed. It is based on the discrete Lagrange-d'Alembert-Pontryagin variational principle and the dynamical equations thus obtained are the…
We reconsider the variational integration of optimal control problems for mechanical systems based on a direct discretization of the Lagrange-d'Alembert principle. This approach yields discrete dynamical constraints which by construction…
We introduce a data-driven method for learning the equations of motion of mechanical systems directly from position measurements, without requiring access to velocity data. This is particularly relevant in system identification tasks where…
We have recently presented an extension of the standard variational calculus to include the presence of deformed derivatives in the Lagrangian of a system of particles and in the Lagrangian density of field-theoretic models. Classical…
Variational integrators are a special kind of geometric discretisation methods applicable to any system of differential equations that obeys a Lagrangian formulation. In this thesis, variational integrators are developed for several…
Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they…
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…
A simple variational Lagrangian is proposed for the time development of an arbitrary density matrix, employing the "factorization" of the density. Only the "kinetic energy" appears in the Lagrangian. The formalism applies to pure and mixed…
Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for…
This paper is a summary of the theory of discrete embeddings introduced in [5]. A discrete embedding is an algebraic procedure associating a numerical scheme to a given ordinary differential equation. Lagrangian systems possess a…
In this paper we provide a variational derivation of the Euler-Poincar\'e equations for systems subjected to external forces using an adaptation of the techniques introduced by Galley and others. Moreover, we study in detail the underlying…
Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation…
In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert…
The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper,…
Energy methods for constructing time-stepping algorithms are of increased interest in application to nonlinear problems, since numerical stability can be inferred from the conservation of the system energy. Alternatively, symplectic…
In this paper structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton-Pontryagin variational principle. From this principle one can derive a novel class of variational partitioned…
Energetic particle effects in magnetic confinement fusion devices are commonly studied by hybrid kinetic-fluid simulation codes whose underlying continuum evolution equations often lack the correct energy balance. While two different…
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…
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…