Conjugate points for systems of second-order ordinary differential equations

Abstract

We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization (to the current setting) of locally symmetric spaces. Next, we study conjugate points along relative equilibria of Lagrangian systems with a symmetry Lie group. We end the paper with some examples and applications.

Jacobi Fields and Conjugate Points for a Projective Class of Sprays

“…In [11], we have shown that the existence of the parallel vector field in part (2) of Proposition 1 can be guaranteed by requiring the condition [∇Φ, Φ] = 0 on the spray. When restricted to an eigendistribution of Φ, this condition can be characterized as follows.…”

“…In view of the first condition in Proposition 1, we may reach an even simplier expression if λ is a first integral of S. If that is the case, all geodesics are constant along λ and the factor − S(λ+a) λ+a + P simplifies to −4P + P = −3P . For instance, locally symmetric sodes fall into that category (see [11], Section 5 for details). The condition on the projective factor that guarantees this property can be calculated as follows.…”

“…In the context of sodes the role of the curvature and the Levi-Civita connection is played by the so-called Jacobi endomorphism Φ and the covariant dynamical derivative ∇ (see Section 2 for their definition, and for most of the preliminaries). In a recent paper [11] we have discussed conjugate points for sodes and in this paper we will mainly rely on the following proposition: Proposition 1. [11] Let c be a base integral curve of a sode S, through m 0 = c(0).…”

“…In a recent paper [11] we have discussed conjugate points for sodes and in this paper we will mainly rely on the following proposition: Proposition 1. [11] Let c be a base integral curve of a sode S, through m 0 = c(0). If (1) Φ has an eigenfunction λ that remains constant and strictly positive along c, i.e.…”