Related papers: Discrete-time Ruijsenaars-Schneider system and Lag…
It is shown that the Zakharov-Mihailov (ZM) Lagrangian structure for integrable nonlinear equations derived from a general class of Lax pairs possesses a Lagrangian multiform structure. We show that, as a consequence of this multiform…
In these notes, we present an alternative version of discrete Dirac mechanics using Dirac structures. We first establish a notion of 'continuous Dirac system' and then propose a definition of discrete Dirac system, proving that it is…
We investigate a compressible two-fluid Navier-Stokes type system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. We are…
We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler…
A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the…
In this work we introduce a category $LDP_d$ of discrete-time dynamical systems, that we call discrete Lagrange--D'Alembert--Poincar\'e systems, and study some of its elementary properties. Examples of objects of $LDP_d$ are nonholonomic…
Controlled Lagrangian and matching techniques are developed for the stabilization of relative equilibria and equilibria of discrete mechanical systems with symmetry as well as broken symmetry. Interesting new phenomena arise in the…
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie…
We study a type of forced discrete mechanical system $(Q,L_d,f_d)$ -- that we name of Routh type -- whose (discrete) time-flow preserves a symplectic structure on $Q\times Q$. That structure arises as the pullback via the forced discrete…
By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system…
This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle…
One approach to studying the dynamics of a singular Lagrangian system is to attempt to regularize it, that is, to find an equivalent and regular system. In the case of time-independent singular Lagrangians, an approach due to \textit{A.…
A fully discrete Lagrangian scheme for solving a family of fourth order equations numerically is presented. The discretization is based on the equation's underlying gradient flow structure w.r.t. the $L^2$-Wasserstein distance, and adapts…
We discuss structure-preserving time discretization for nonlinear port-Hamiltonian systems with state-dependent mass matrix. Such systems occur, for instance, in the context of structure-preserving nonlinear model order reduction for…
For several types of information relations, the induced rough sets system RS does not form a lattice but only a partially ordered set. However, by studying its Dedekind-MacNeille completion DM(RS), one may reveal new important properties of…
We consider a system consisting of one conservation law and one balance law with a time-dependent source term, and provide a comprehensive analysis of Riemann solutions, including the non-classical overcompressive delta shocks. The minimal…
We consider a hierarchy of classical Liouville completely integrable models sharing the same (linear) $r$--matrix structure obtained through an $N$--th jet--extension of $\mathfrak{su}(2)$ rational Gaudin models. The main goal of the…
The problem of rigid body pose estimation is treated in discrete-time via discrete Lagrange-d'Alembert principle and discrete Lyapunov methods. The position and attitude of the rigid body are to be estimated simultaneously with the help of…
In this paper, we propose the concept of $(\pm)$-discrete Dirac structures over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and incorporate discrete constraints using $(\pm)$-finite difference maps. Specifically,…
It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit…