A Geometric Setting for Systems of Ordinary Differential Equations

Abstract

Abstract. To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also 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.

Geometrical structures on the cotangent bundle

“…The notion of dynamical covariant derivative was introduced for the first time in the case of tangent bundle by Cariñena and Martinez [5] as a derivation of degree 0 along the tangent bundle projection (see also [3,4,7,16,17,27]). An extensive study and discussions about the dynamical covariant derivative which is associated to a second-order vector field (semispray) on T M can be found in [25].…”

“…Using the Frölisher-Nijenhuis bracket [9,27] we study the Jacobi endomorphism on the cotangent bundle. In the second section, using the notions of J -regular vector field and an arbitrary nonlinear connection we introduce the dynamical covariant derivative on the cotangent bundle and prove that the condition of compatibility with the adapted tangent structure, that is ∇J = 0, fix the nonlinear connection (see [4] for the tangent bundle). In Sec.…”

“…We introduce the almost complex structure using the Frölisher-Nijenhuis bracket and prove that its dynamical covariant derivative vanishes. These subjects can by found in the case of tangent bundle in a lot of papers (see for instance [3,4,7,8,11,[14][15][16][24][25][26]). …”

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

“…A geometric setting for semisprays. In this section, we use the Frölicher-Nijenhuis formalism [13] to associate a geometric setting to a given system of second order ordinary differential equations, [2,3,14,15].…”

“…We introduce now the dynamical covariant derivative, ∇, associated to a semispray, following the approach from [2,3]. For f ∈ C ∞ (T M ) and X ∈ X(T M ), we define…”