Continuous symmetries of Lagrangians and exact solutions of discrete equations

Abstract

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 invariant Lagrangian formalism for scalar single-variable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two- and three- dimensional groups of Lagrangian symmetries are considered.

31 pages, submitted to Journal of Mathematical Physics

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• Lie symmetries of multidimensional difference equations
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Cited by in corpus (22)

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• Invariant conservative difference schemes for shallow water equations in Eulerian and Lagrangian coordinates
• Invariant difference schemes and their application to $SL(2,\mathbb{R})$ invariant ordinary differential equations
• Lie group classification of first-order delay ordinary differential equations
• Structure Preserving Discretizations of the Liouville Equation and their Numerical Tests
• Symmetry preserving discretization of ordinary differential equations. Large symmetry groups and higher order equations
• The Korteweg-de Vries equation and its symmetry-preserving discretization
• Conservative invariant finite-difference schemes for the modified shallow water equations in Lagrangian coordinates
• Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations
• On the Linearization of Second-Order Differential and Difference Equations
• One-dimensional flows of a polytropic gas: Lie group classification, conservation laws, invariant and conservative difference schemes
• Conformally invariant elliptic Liouville equation and its symmetry preserving discretization
• Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation
• First integrals of ordinary difference equations which do not possess a variational formulation
• Invariance and first integrals of continuous and discrete Hamiltonian equations