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

Abstract

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-Kähler 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.

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

“…Using a very similar proof as the one of Theorems 4.2 and 4.3 in [5] and Theorems 3 and 4 in [8], we can state the following result. An important consequence of the previous proposition appears in dimension 2.…”

“…For the formal integrability of the system (D 1 ): d J θ = 0 and d h θ = 0, we can follow the results of [5,8], where the formal integrability of very similar systems was investigated. Using a very similar proof as the one of Theorems 4.2 and 4.3 in [5] and Theorems 3 and 4 in [8], we can state the following result.…”

Generalized Helmholtz conditions for non-conservative Lagrangian systems

“…For the formal integrability of the system (D 1 ): d J θ = 0 and d h θ = 0, we can follow the results of [5,8], where the formal integrability of very similar systems was investigated. Using a very similar proof as the one of Theorems 4.2 and 4.3 in [5] and Theorems 3 and 4 in [8], we can state the following result. An important consequence of the previous proposition appears in dimension 2.…”