# American Institue of Mathematical Sciences

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

 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.
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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