Spectral separation of variables from equivalent Lagrangian systems

TL;DR

This paper explores how equivalent quadratic Lagrangians lead to decoupled equations of motion through spectral decomposition, revealing conditions for variable separation and integrability in classical systems.

Contribution

It establishes a spectral condition for the separation of variables in equivalent quadratic Lagrangian systems, connecting spectral properties to decoupling of equations.

Findings

Equivalent quadratic Lagrangians impose a commutation condition between kinetic matrices and the potential Hessian.

Decoupling of equations occurs in block-separated form, fully when the spectrum is simple.

Classical integrable regimes are recovered in specific models like Sawada--Kotera and Hénon--Heiles.

Abstract

We investigate the dynamical equivalence of quadratic Lagrangians and its relation to separation of variables. We show that requiring two quadratic Lagrangians to generate the same Euler--Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space. The equations of motion then decouple into independent subsystems: generically in block-separated form, and completely when the spectrum is simple. Applications include the Sawada--Kotera system and an $n$-dimensional extension of the H\'{e}non--Heiles model, where the classical integrable parameter regimes are recovered.