Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

TL;DR

This paper investigates the formal integrability of the inverse calculus of variations problem for time-dependent second-order differential equations, using advanced geometric methods to identify conditions related to curvature and classify specific cases.

Contribution

It proves the involutivity of the key operator and identifies the unique obstruction to integrability in terms of the curvature tensor, extending previous results to the nonautonomous setting.

Findings

Operator P is involutive.

Obstruction to integrability is curvature tensor R.

Classifies flat, isotropic, and 2D semisprays.

Abstract

We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-K\"ahler theorem. We consider a linear partial differential operator $P$ given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that $P$ is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor $R$ of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.