Ioan Bucataru

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

We use Frolicher-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.

Metric nonlinear connections

Abstract For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange metric. For the particular case when the system of SODE and the metric structure are Lagrangian, we prove that the canonical nonlinear connection of the Lagrange space is the only nonlinear connection which is metric and compatible with the symplectic structure of the Lagrange space. For this particular case, the metric tensor determines the symmetric part of the canonical nonlinear connection, while the symplectic structure determines the skew-symmetric part of the nonlinear connection.

A GEOMETRIC SETTING FOR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

To a system of second-order ordinary differential equations one can assign a canonical nonlinear connection that describes the geometry of the system. In this paper, we develop a geometric setting that also allows us to assign a canonical nonlinear connection to a system of higher-order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third- or fourth-order ordinary differential equations.

PROJECTIVE AND FINSLER METRIZABILITY: PARAMETERIZATION-RIGIDITY OF THE GEODESICS

In this work we show that for the geodesic spray

Invariant Properties for Finding Distance in Space of Elasticity Tensors

S

Projective Metrizability and Formal Integrability

of a Finsler function

A Complete Lift for Semisprays

F

FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM

the most natural projective deformation

Nonlinear connections for nonconservative mechanical systems

\widetilde{S}=S -2 \lambda F\mathbb C

Sprays metrizable by Finsler functions of constant flag curvature

leads to a non-Finsler metrizable spray, for almost every value of