Oana Constantinescu

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.

SUB-WEYL GEOMETRY AND ITS LINEAR CONNECTIONS

The aim of this paper is to study from the point of view of linear connections the data with M a smooth (n+p)-dimensional real manifold, an n-dimensional manifold semi-Riemannian distribution on M, the conformal structure generated by g and W a Weyl substructure: a map such that W(ḡ) = W(g) - du, ḡ = eug;u ∈ C∞(M). Compatible linear connections are introduced as a natural extension of similar notions from Weyl geometry and such a connection is unique if a symmetry condition is imposed. In the foliated case the local expression of this unique connection is obtained. The notion of Vranceanu connection is introduced for a pair (Weyl structure, distribution) and it is computed for the tangent bundle of Finsler spaces, particularly Riemannian, choosing as distribution the vertical bundle of tangent bundle projection and as one-form the Cartan form.

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyroscopic forces, these conditions, when expressed in terms of a multiplier matrix, reduce to those obtained in Mestdag et al. (Differential Geom. Appl. 29(1), 55–72, 2011). When the involved geometric structures are homogeneous with respect to the fibre coordinates, we show how one can further simplify the generalized Helmholtz conditions. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan{Kahler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.

Helmholtz conditions and symmetries for the time dependent case of the inverse problem of the calculus of variations

Abstract We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frolicher–Nijenhuis theory on the first jet bundle, J 1 π . We prove that a system of time dependent SODE, identified with a semispray S , is Lagrangian if and only if a special class, Λ S 1 ( J 1 π ) , of semi-basic 1 -forms is not empty. We provide global Helmholtz conditions to characterize the class Λ S 1 ( J 1 π ) of semi-basic 1 -forms. Each such class contains the Poincare–Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1 -form in Λ S 1 ( J 1 π ) , which is not a Poincare–Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.