The Journal of Geometric Mechanics (JGM)

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

Pages: 159 - 180, Volume 1, Issue 2, June 2009      doi:10.3934/jgm.2009.1.159

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Ioan Bucataru - Faculty of Mathematics, Al.I.Cuza University, Iasi, 700506, Romania (email)
Matias F. Dahl - Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, 02015 Helsinki, Finland (email)

Abstract: We use Frölicher-Nijenhuis theory to obtain global Helmholtz conditions, expressed in terms of a semi-basic 1-form, that characterize when a semispray is a Lagrangian vector field. We also discuss the relation between these Helmholtz conditions and their classic formulation written using a multiplier matrix. When the semi-basic 1-form is 1-homogeneous (0-homogeneous) we show that two (one) of the Helmholtz conditions are consequences of the other ones. These two special cases correspond to two inverse problems in the calculus of variation: Finsler metrizability for a spray, and projective metrizability for a spray.

Keywords:  Poincaré-Cartan 1-form, Helmholtz conditions, inverse problem, projective metrizability, Finsler metrizability.
Mathematics Subject Classification:  Primary: 58E30, 53C60; Secondary: 58B20, 53C22.

Received: April 2009;      Revised: June 2009;      Available Online: July 2009.