Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

Abstract

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, $J^1\pi$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian if and only if a special class, ${\Lambda }_S^1(J^1\pi )$, of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class ${\Lambda }_S^1(J^1\pi )$ of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in ${\Lambda }_S^1(J^1\pi )$, which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.

References in corpus (1)

• Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations

Cited by in corpus (3)

• On the definition and examples of cones and Finsler spacetimes
• Time-dependent kinetic energy metrics for Lagrangians of electromagnetic type
• Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations