## 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 language | English |
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Pages (from-to) | 235-253 |

Number of pages | 19 |

Journal | Mathematics of Control, Signals, and Systems |

Volume | 19 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 2007 |

Externally published | Yes |

## Keywords

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