Semi-basic 1-forms and Helmholtz conditions for the inverse
problem of the calculus of variations
Ioan Bucataru - Faculty of Mathematics, Al.I.Cuza University, Iasi, 700506, Romania (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.
Received: April 2009; Revised: June 2009; Available Online: July 2009.
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