Higher order geodesics in Lie groups

Abstract

For all n > 2, we study nth order generalisations of Riemannian cubics, which are second-order variational curves used for interpolation in semiRiemannian manifolds M. After finding two scalar constants of motion, one for all M, the other when M is locally symmetric, we take M to be a Lie group G with bi-invariant semi-Riemannian metric. The Euler-Lagrange equation is reduced to a system consisting of a linking equation and an equation in the Lie algebra. A Lax pair form of the second equation is found, as is an additional vector constant of motion, and a duality theory, based on the invariance of the Euler-Lagrange equation under group inversion, is developed. When G is semisimple, these results allow the linking equation to be solved by quadrature using methods of two recent papers; the solution is presented in the case of the rotation group SO(3), which is important in rigid body motion planning.

About simple variational splines from the Hamiltonian viewpoint

“…The bibliography is vast, see e.g. [85], [50], [49], [26], [46], [47], [77], [95], [43]. In this note we consider only the simplest case, namely, of splines having the tangent bundle as state space, the acceleration vector being the control.…”

Invariant Higher-Order Variational Problems II

“…Riemannian cubics are solutions of Euler-Lagrange equations for a certain secondorder variational problem in a finite-dimensional connected Riemannian manifold, to find a curve that interpolates between two points with given initial and final velocities, subject to minimal mean-square covariant acceleration. [Pop07,MSK10] and [Noa06b] for extensive references and historical discussions concerning Riemannian cubics, their higherorder generalizations, and related higher-order interpolation methods.…”

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

“…In [31], the author solves by quadratures the linking equation on SO(3) and SO(1, 2) of the Riemannian cubics. Finally, [34] studies n-th order generalizations of RCP introduced in [14]. To our knowledge, the first Hamiltonian description of the RCP problem has been considered in [15] (made in collaboration with one of the authors).…”

“…In Riemannian context, I 1 plays a role similar to the one played by the length of the velocity vector field in the theory of geodesics (see, for example [16]). Recently, in [3,30,31,34], the analysis of RCP from a variational point of view was carried out for locally symmetric manifolds and a second invariant was obtained:…”

“…The equation (1) [32]) and explored from a dynamical interpolation perspective in 1995 (see [17]). Interesting points related to this subject have been developed in the last few years, namely a geometric theory surprisingly close to the Riemannian theory of geodesics (see [2,3,4,12,14,15,16,19,30,31,34,35]). We recall, in particular, a result which says that if V denotes the velocity vector field of a cubic polynomial x, then…”