Continuous symmetries of Lagrangians and exact solutions of discrete equations

  title={Continuous symmetries of Lagrangians and exact solutions of discrete equations},
  author={V. A. Dorodnitsyn and Roman Kozlov and Pavel Winternitz},
  journal={Journal of Mathematical Physics},
One of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an… Expand
Invariance and first integrals of continuous and discrete Hamiltonian equations
The relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations is considered. The observation that canonical HamiltonianExpand
First integrals of difference Hamiltonian equations
In the present paper, the well-known Noether's identity, which represents the connection between symmetries and first integrals of Euler-Lagrange equations, is rewritten in terms of the HamiltonianExpand
Symmetries and exact solutions of discrete nonconservative systems
Based on the property of the discrete model entirely inheriting the symmetry of the continuous system, we present a method to construct exact solutions with continuous groups of transformations inExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems
This paper focuses on studying Lie symmetries and conserved quantities of discrete nonholonomic Hamiltonian systems. Firstly, the discrete generalized Hamiltonian canonical equations and discreteExpand
First variation formula and conservation laws in several independent discrete variables
Abstract This paper sets the scene for discrete variational problems on an abstract cellular complex that serves as discrete model of R p and for the discrete theory of partial differential operatorsExpand
One-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates: Symmetry classification, conservation laws, difference schemes
A comprehensive analysis of the one-dimensional gas dynamics equations of a polytropic gas in Lagrangian coordinates is performed, given by a single second-order partial differential equation, which results in conservation laws being obtained. Expand
Symmetries and Integrability of Difference Equations: Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals
Abstract In this chapter the relation between symmetries and first integrals of discrete Euler–Lagrange and discrete Hamiltonian equations is considered. These results are built on those forExpand
Discretization of partial differential equations preserving their physical symmetries
A procedure for obtaining a 'minimal' discretization of a partial differential equation, preserving all of its Lie point symmetries, is presented. 'Minimal' in this case means that the differentialExpand
On the Linearization of Second-Order Differential and Difference Equations
This article complements recent results of the papers (J. Math. Phys. 41 (2000), 480; 45 (2004), 336) on the symmetry classification of second-order ordinary difference equations and meshes, as wellExpand


Lie symmetries of multidimensional difference equations
A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. TheyExpand
Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics
Some remarks on a class of ordinary differential equations: the Riccati property, differential-algebraic and differential-geometric approach to the study of involutive symbols, and the bihamiltonian approach to integrable systems. Expand
Symmetries, exact solutions, and conservation laws
APPARATUS OF GROUP ANALYSIS: Lie Theory of Differential Equations: One Parameter Transformation Groups. Integration of Second-Order Ordinary Differential Equations. Group Classification ofExpand
Lie symmetries of finite‐difference equations
Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations areExpand
Symmetries and conditional symmetries of differential difference equations
Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined withExpand
Lie algebra contractions and symmetries of the Toda hierarchy
The Lie algebra L(Δ) of generalized and point symmetries of the equations in the Toda hierarchy is shown to be a semidirect sum of two infinite-dimensional Lie algebras, one perfect, the otherExpand
Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction
Abstract A method is given to derive the point symmetries of partial differential-difference equations. Applying the method to the Kacvan Moerbeke equation we find its symmetries form aExpand
Symmetry-adapted moving mesh schemes for the nonlinear Schrodinger equation
In this paper we consider symmetry-preserving difference schemes for the nonlinear Schrodinger equation where n is the number of space dimensions. This equation describes one-dimensional waves in nExpand
Symmetries of discrete dynamical systems involving two species
The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, inExpand
Lie point symmetries and commuting flows for equations on lattices
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie pointExpand