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

Ioan Bucataru; Matias F. Dahl

doi:10.3934/jgm.2009.1.159

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2009, Volume 1, Issue 2: 159-180. Doi: 10.3934/jgm.2009.1.159

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

Ioan Bucataru^1, and
Matias F. Dahl^2,

1.
Faculty of Mathematics, Al.I.Cuza University, Iasi, 700506
2.
Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, 02015 Helsinki

Received: April 2009

Revised: June 2009

Published: June 2009

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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.

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Mathematics Subject Classification: Primary: 58E30, 53C60; Secondary: 58B20, 53C22.

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