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June  2009, 1(2): 159-180. doi: 10.3934/jgm.2009.1.159

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, Romania
2. 

Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, 02015 Helsinki, Finland

Received  April 2009 Revised  June 2009 Published  July 2009

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: projective metrizability, Finsler metrizability., Poincaré-Cartan 1-form, inverse problem, Helmholtz conditions.
Mathematics Subject Classification: Primary: 58E30, 53C60; Secondary: 58B20, 53C2.
Citation: Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
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