Lagrangian Constraints and Differential Thomas Decomposition

TL;DR

This paper presents an algorithmic method using differential Thomas decomposition to identify algebraic constraints in singular polynomial Lagrangian models, including domain-specific singularities.

Contribution

It introduces a novel application of differential Thomas decomposition to compute constraints and detect singular domains in polynomial Lagrangian systems.

Findings

Successfully computes algebraic constraints for singular models.

Detects domains where the Lagrangian becomes singular.

Provides a systematic approach for analyzing polynomial Lagrangian systems.

Abstract

In this paper we show how to compute algorithmically the full set of algebraically independent constraints for singular mechanical and field-theoretical models with polynomial Lagrangians. If a model under consideration is not singular as a whole but has domains of dynamical (field) variables where its Lagrangian becomes singular, then our approach allows to detect such domains and compute the relevant constraints. In doing so, we assume that the Lagrangian of a model is a differential polynomial and apply the differential Thomas decomposition algorithm to the Euler-Lagrange equations.