On the Completeness of Trajectories for Some Mechanical Systems

Abstract

The classical tools which ensure the completeness of both, vector fields and second order differential equations for mechanical systems, are revisited. Possible extensions in three directions are discussed: infinite dimensional Banach (and Hilbert) manifolds, Finsler metrics and pseudo-Riemannian spaces, the latter including links with some relativistic spacetimes. Special emphasis is taken in the cleaning up of known techniques, the statement of open questions and the exploration of prospective frameworks.

Completeness of compact Lorentzian manifolds with abelian holonomy

“…It is well-known then that the sequence of velocities {γ ′ (t n )} n cannot converge in T M as γ ′ is the integral curve of the geodesic vector field on T M (see for example Prop. 3.28 and Lemma 1.56 in [9] or [13,Section 3]). Consider a normal (starshaped) neighbourhood U of p (see for example [10,14] for background results on linear connections).…”

“…However, when g is an indefinite semi-Riemannian metric of index ν (0 < ν < m), the corresponding orthogonal group O ν (T p M ) is non-compact and completeness may not hold; the Clifton-Pohl torus (see for instance [9,Example 7.16]) is a well-known example. There are some results that assure the completeness of a compact semi-Riemannian manifold, among them either to admit ν pointwise independent conformal Killing vector fields which span a negative definite subbundle of T M [12,11], or to be homogeneous [8] (local homogeneity is also enough in dimension 3 [4], and to be conformal to a homogeneous manifold is enough in any dimension [11]; see [13] for a review).…”

The Ehlers-Kundt conjecture about gravitational waves and dynamical systems

“…Known results. Even though the completeness of trajectories of dynamical systems is a very classical topic (see the book [2] or the survey [33]), as far as we know, the condition of harmonicity (which defines potential theory as well as divergence free gradient vector fields) has not been studied in this setting. Indeed, the progress along these decades has focused on other aspects of the conjecture rather than in the crude geodesic equation (or the dynamical system (4)), concretely: (a) All plane waves (gravitational or not) are geodesically complete, as the equation (4) reduces to a second order linear system of differential equations [17], [ [26] gave a local characterization of pp-waves, argued that a natural extension of the conjecture follows when a locally pp-wave metric is taken on a compact M , and proved that this extension becomes equivalent to the standard one on R 4 .…”

Compact affine manifolds with precompact holonomy are geodesically complete

“…It is well-known then that the sequence of velocities {γ ′ (t n )} n cannot converge in T M as γ ′ is the integral curve of the geodesic vector field on T M (see for example Prop. 3.28 and Lemma 1.56 in [9] or [13,Section 3]). Consider a normal (starshaped) neighbourhood U of p (see for example [10,14] for background results on linear connections).…”

“…However, when g is an indefinite semi-Riemannian metric of index ν (0 < ν < m), the corresponding orthogonal group O ν (T p M ) is non-compact and completeness may not hold; the Clifton-Pohl torus (see for instance [9,Example 7.16]) is a well-known example. There are some results that assure the completeness of a compact semi-Riemannian manifold, among them either to admit ν pointwise independent conformal Killing vector fields which span a negative definite subbundle of T M [12,11], or to be homogeneous [8] (local homogeneity is also enough in dimension 3 [4], and to be conformal to a homogeneous manifold is enough in any dimension [11]; see [13] for a review).…”