Geometric cubic polynomials on SO(3)
by
Margarida Camarinha
Mathematics Department, University of Coimbra
Coauthors: Abrunheiro, Lígia

Geometric cubic polynomials on a Riemannian manifold M are the smooth solutions of the fourth order differential equation

D4x/dt4+R( D2x/dt2, dx/dt)dx/dt=0.

This is the Euler-Lagrange equation of a second order variational problem with Lagrangian given by the norm squared covariant acceleration, which can be viewed as an extension of the minimizing acceleration problem in Euclidean space. The analysis of the problem is carried out on a constant curvature manifold or on a compact and connected Lie group. In particular, the case of the group of rotations in R³, SO(3), is studied. Existence and uniqueness conditions for cubic polynomials with initial and final positions and velocities are established. In SO(3), the Euler-Lagrange equation gives rise to the nonlinear system of third order differential equations in R³

d3y/dt3=(d2y/dt2)×y.

The reduction of the essential size and the separation of the variables of the system are obtained using two invariants along the cubic polynomials.

Date received: May 29, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calo-23.