On the Lagrangian Structure of the Discrete Isospectral and Isomonodromic Transformations

TL;DR

This paper demonstrates that discrete isospectral and isomonodromic systems, including the difference Painlevé V equation, have a Lagrangian structure, extending classical integrability concepts to a broader class of rational matrix functions.

Contribution

It generalizes the Lagrangian formulation of discrete integrable systems to rational matrix functions and explicitly derives the Lagrangian for the quadratic case, linking isomonodromic transformations to Lagrangian mechanics.

Findings

Established Lagrangian structure for discrete isospectral systems.

Derived explicit Lagrangian for quadratic rational matrix functions.

Connected isomonodromic transformations to difference Painlevé equations.

Abstract

We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the Moser-Veselov approach to integrability of discrete systems via the re-factorization of matrix polynomials to a more general class of matrix rational functions that have a simple divisor and, in the quadratic case, explicitly write the Lagrangian function for such systems. Next we show that if we let certain parameters in this Lagrangian to be time-dependent, the resulting Euler-Lagrange equations describe the isomonodromic transformations of systems of linear difference equations. It is known that in some special cases such equations reduce to the difference Painlev\'e equation. As an example, we show how to obtain the difference Painlev\`e V equation in this way, and hence we establish that this equation can be written in the Lagrangian form.