46 research outputs found
Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
We use Fr\"olicher-Nijenhuis theory to obtain global Helmholtz conditions,
expressed in terms of a semi-basic 1-form, that characterize when a semispray
is locally Lagrangian. 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
Projective Metrizability and Formal Integrability
The projective metrizability problem can be formulated as follows: under what
conditions the geodesics of a given spray coincide with the geodesics of some
Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective
metrizability problem for a spray in terms of a first-order partial
differential operator and a set of algebraic conditions on semi-basic
1-forms. We discuss the formal integrability of using two sufficient
conditions provided by Cartan-K\"ahler theorem. We prove in Theorem 4.2 that
the symbol of is involutive and hence one of the two conditions is always
satisfied. While discussing the second condition, in Theorem 4.3 we prove that
there is only one obstruction to the formal integrability of , and this
obstruction is due to the curvature tensor of the induced nonlinear connection.
When the curvature obstruction is satisfied, the projective metrizability
problem reduces to the discussion of the algebraic conditions, which as we show
are always satisfied in the analytic case. Based on these results, we recover
all classes of sprays that are known to be projectively metrizable: flat
sprays, isotropic sprays, and arbitrary sprays on 1- and 2-dimensional
manifolds. We provide examples of sprays that are projectively metrizable
without being Finsler metrizable