127 Vincent Hall, 206 Church St. S.E., MN 55455, Minneapolis, USA
School of Mathematics, University of Minnesota
Mari Beffa G., Olver P. J.
Poisson structures for geometric curve flows in semi-simple homogeneous spaces
2010, vol. 15, no. 4-5, pp. 532-550
We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.
Kim P., Olver P. J.
Geometric integration via multi-space
2004, vol. 9, no. 3, pp. 213-226
We outline a general construction of symmetry-preserving numerical schemes for ordinary differential equations. The method of invariantization is based on the equivariant moving frame theory applied to prolonged symmetry group actions on multi-space, which has been proposed as the proper geometric setting for numerical analysis. We explain how to invariantize standard numerical integrators such as the Euler and Runge–Kutta schemes; in favorable situations, the resulting symmetry-preserving geometric integrators offer significant advantages.
Olver P. J.
Moving Frames and Joint Differential Invariants
1999, vol. 4, no. 4, pp. 3-18
This paper surveys the new, algorithmic theory of moving frames developed by the author and M.Fels. The method is used to classify joint invariants and joint differential invariants of transformation groups, and equivalence and symmetry properties of submanifolds. Applications in classical invariant theory, geometry, and computer vision are indicated.