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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.

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  1. School of Mathematics and Statistics (M019), University of Western Australia, 35 Stirling Highway, Crawley, 6009, Western Australia, Australia

    Tomasz Popiel

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  1. Tomasz Popiel
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Popiel, T. Higher order geodesics in Lie groups. Math. Control Signals Syst. 19, 235–253 (2007). https://doi.org/10.1007/s00498-007-0012-x

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