[Paper Review] Matched pairs of discrete dynamical systems

Oğul Esen, Serkan Sütlü|arXiv (Cornell University)|Sep 3, 2018

Nonlinear Waves and Solitons35 references4 citations

TL;DR

This paper develops a geometric framework for discrete dynamical systems on matched pairs of Lie groupoids, decomposing the Euler-Lagrange equations into components from two interacting subsystems with additional terms arising from their mutual actions. The key contribution is the derivation of discrete Euler-Lagrange equations on matched pair Lie groups, illustrated via the trivial groupoid and SL(2,C) using Iwasawa decomposition.

ABSTRACT

Matched pairs of Lie groupoids and Lie algebroids are studied. Discrete Euler-Lagrange equations are written for the matched pairs of Lie groupoids. As such, a geometric framework to analyse a discrete system by decomposing it into two mutually interacting subsystems is established. Two examples are provided to illustrate this strategy; the discrete dynamics on the trivial groupoid, and the discrete dynamics on the special linear group.

Motivation & Objective

• To establish a geometric framework for analyzing discrete dynamical systems by decomposing them into two mutually interacting subsystems via matched pair structures.
• To extend the theory of matched pair dynamics from continuous to discrete settings using Lie groupoids and Lie algebroids.
• To derive explicit discrete Euler-Lagrange equations on matched pair Lie groupoids, incorporating mutual action terms.
• To demonstrate the framework through concrete examples: the trivial groupoid and SL(2,C) via Iwasawa decomposition.
• To lay the foundation for future work on matched pair Lagrangian dynamics on Lie algebroids.

Proposed method

• Utilizes matched pair theory of Lie groupoids and Lie algebroids, generalizing the semi-direct product construction.
• Applies the discrete variational principle on Lie groupoids, computing directional derivatives of the Lagrangian along left and right invariant vector fields.
• Derives discrete Euler-Lagrange equations on the matched pair Lie groupoid as a sum of individual subsystem equations plus interaction terms.
• Employs the Iwasawa decomposition of SL(2,C) into SU(2) and a simply connected subgroup K to decompose the dynamics.
• Computes the mutual actions via derived operators such as $\mathfrak{a}_B^*$, $\mathfrak{b}_A^*$, and $A^*\triangleright$, representing coadjoint actions and adjoint representations.
• Uses the transpose of the action maps to express the interaction terms in the final Euler-Lagrange equations.

Experimental results

Research questions

• RQ1 How can the discrete Euler-Lagrange equations on a matched pair Lie groupoid be decomposed into equations from its constituent subsystems?
• RQ2 What are the additional terms that arise from the mutual actions between two interacting discrete dynamical systems in a matched pair structure?
• RQ3 How does the matched pair decomposition of SL(2,C) into SU(2) and K affect the structure of the discrete Euler-Lagrange equations?
• RQ4 Can the discrete dynamics on the trivial groupoid be fully reconstructed from its action groupoid and coarse groupoid components with mutual action corrections?
• RQ5 What is the role of the Iwasawa decomposition in enabling the matched pair formulation of discrete dynamics on SL(2,C)?

Key findings

• The discrete Euler-Lagrange equations on a matched pair Lie groupoid are expressed as a sum of the individual subsystem equations, enriched by mutual action terms.
• For the trivial groupoid, the dynamics are decomposed into equations from the action groupoid and the coarse groupoid, with interaction terms arising from their mutual actions.
• On SL(2,C), the matched pair decomposition into SU(2) and K yields discrete Euler-Lagrange equations that explicitly include terms from the adjoint and coadjoint actions of SU(2) on K and vice versa.
• The interaction terms are encoded via the transpose operators $\mathfrak{a}_B^*$, $\mathfrak{b}_A^*$, and $A^*\triangleright$, derived from the group actions.
• The final Euler-Lagrange equation on SL(2,C) is a nonlinear system involving $\mathop{\rm Ad}^*_{A_k}\Phi_k$, $T^*r_{B_k}$, and $\mathop{\rm Rot}_{A_{k+1}}^*$, reflecting the nontrivial coupling.
• The framework successfully generalizes continuous matched pair dynamics to the discrete setting, preserving geometric structure and enabling modular analysis of complex systems.