Communications in Mathematical Sciences

Volume 14 (2016)

Number 1

Bi-Jacobi fields and Riemannian cubics for left-invariant $SO(3)$

Pages: 55 – 68



Lyle Noakes (School of Mathematics and Statistics, The University of Western Australia, Crawley, WA, Australia)

Tudor S. Ratiu (Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland; and Mathematical Sciences Program, Skolkovo Institute of Science and Technology, Skolkovo, Moscow Region, Russia)


Bi-Jacobi fields are generalized Jacobi fields, and are used to efficiently compute approximations to Riemannian cubic splines in a Riemannian manifold $M$. Calculating bi-Jacobi fields is straightforward when $M$ is a symmetric space such as bi-invariant $SO(3)$, but not for Lie groups whose Riemannian metric is only left-invariant. Because left-invariant Riemannian metrics occur naturally in applications, there is also a need to calculate bi-Jacobi fields in such cases. The present paper investigates bi-Jacobi fields for left-invariant Riemannian metrics on $SO(3)$, reducing calculations to quadratures of Jacobi fields. Then left-Lie reductions are used to give an easily implemented numerical method for calculating bi-Jacobi fields along geodesics in $SO(3)$, and an example is given of a nearly geodesic approximate Riemannian cubic.


Lie group, Riemannian manifold, Jacobi field, trajectory planning, mechanical system, rigid body, nonlinear optimal control, Riemannian cubic

2010 Mathematics Subject Classification

Primary 34E05, 49K99, 53A99, 70E17. Secondary 34H05, 49S05, 70E60.

Published 16 September 2015