Higher order geodesics in Lie groups

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Abstract

For all n > 2, we study nth order generalisations of Riemannian cubics, which are second-order variational curves used for interpolation in semi-Riemannian 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.

Original languageEnglish
Pages (from-to)235-253
Number of pages19
JournalMathematics of Control, Signals, and Systems
Volume19
Issue number3
DOIs
Publication statusPublished - Aug 2007
Externally publishedYes

Keywords

  • Geodesic
  • Lax equation
  • Lie group
  • Lie quadratic
  • Riemannian cubic
  • Riemannian polynomial

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