Higher order geodesics in Lie groups (Q998620)

A generalization of geodesics in semi-Riemannian manifolds is presented, namely a generalization of the standard functional whose critical points are geodesics is introduced and its critical points are called \(n\)-geodesics. These curves are used in applications for interpolation by variational curves in problems of trajectory planning for rigid body motion. In the paper, the special situation when the manifold is a Lie group with a bi-invariant metric is considered in more detail. The system of differential equations describing \(n\)-geodesics is reduced to a system of two differential equations: a first order equation called ``linking equation'' and an equation of order \(2n-1\) in the Lie algebra. The Lax pair form of the second one is found. When the group is semisimple, the linking equation can be solved. The example of the group SO(3) is presented.