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Moving Frames: Difference and Differential-Difference Lagrangians

Lewis C. White, Peter E. Hydon

TL;DR

The paper develops moving frame theory for $P\Delta E$ and $D\Delta E$ with one continuous variable, enabling invariant calculus of variations and equivariant Noether conservation laws. It emphasizes that the differential–difference setting requires prolongation-preserving group actions and introduces projectable moving frames, ensuring the invariant derivative operator $\mathcal{D}$ commutes with discrete shifts. Concrete instantiations include a Toda-type equation and a method-of-lines semi-discretization of the nonlinear Schrödinger equation, with invariant Euler–Lagrange equations and equivariant conservation laws derived directly from invariant Lagrangians. This framework provides a systematic approach to invariant discretizations and conserved quantities for mixed continuous-discrete systems and can be extended to more complex differential–difference structures and higher-dimensional lattices.

Abstract

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the nonlinear Schrödinger equation.

Moving Frames: Difference and Differential-Difference Lagrangians

TL;DR

The paper develops moving frame theory for and with one continuous variable, enabling invariant calculus of variations and equivariant Noether conservation laws. It emphasizes that the differential–difference setting requires prolongation-preserving group actions and introduces projectable moving frames, ensuring the invariant derivative operator commutes with discrete shifts. Concrete instantiations include a Toda-type equation and a method-of-lines semi-discretization of the nonlinear Schrödinger equation, with invariant Euler–Lagrange equations and equivariant conservation laws derived directly from invariant Lagrangians. This framework provides a systematic approach to invariant discretizations and conserved quantities for mixed continuous-discrete systems and can be extended to more complex differential–difference structures and higher-dimensional lattices.

Abstract

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the equivariant formulation of the conservation laws arising from Noether's theorem. The differential-difference theory is not merely an amalgam of the differential and difference theories, but has additional features that reflect the need for the group action to preserve the prolongation structure. Projectable moving frames are introduced; these cause the invariant derivative operator to commute with shifts in the discrete variables. Examples include a Toda-type equation and a method of lines semi-discretization of the nonlinear Schrödinger equation.
Paper Structure (16 sections, 12 theorems, 167 equations, 3 figures)

This paper contains 16 sections, 12 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A brief summary of the building-blocks
  3. Difference prolongation space
  4. The difference variational calculus
  5. Variational point symmetries and Noether's theorem
  6. Moving frames
  7. Discrete moving frames
  8. Difference moving frames
  9. Difference frames and invariants
  10. Differential invariants and syzygies
  11. The invariant formulation of the Euler--Lagrange equations
  12. Conservation laws
  13. Differential-difference structure
  14. Differential-difference variational calculus
  15. Examples
  16. ...and 1 more sections

Key Result

Theorem 2.5

Suppose that a Lagrangian $\mathrm{L}$ has a variational symmetry with characteristic $\bf{Q} \neq {\bf 0}$. Then the system of Euler--Lagrange equations has the following conservation law:

Figures (3)

  • Figure 1: (a) The total space for scalar O$\Delta$Es is $\mathcal{T}=\mathbb{Z} \times \mathbb{R}$; (b) the total space for scalar P$\Delta$Es with two independent variables is $\mathcal{T} = \mathbb{Z}^{2} \times \mathbb{R}$.
  • Figure 2: Moving frame defined by a cross-section; $\mathcal{O}(z)$ denotes the group orbit through $z$.
  • Figure 3: A graph on (ordered) slices in the total space $\mathcal{T} = \mathbb{R} \times \mathbb{Z} \times \mathbb{R}$.

Theorems & Definitions (30)

  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: difference Noether's theorem
  • Definition 2.6: moving frame
  • Definition 2.7: discrete moving frames
  • Definition 3.1
  • Definition 3.2: discrete Maurer--Cartan invariants
  • Definition 3.3
  • ...and 20 more