Rolling problems on spaces of constant curvature. (English) Zbl 1136.49028

Jurdjevic, Velimir; Zimmerman, Jason

Bullo, Francesco (ed.) et al., Lagrangian and Hamiltonian methods for nonlinear control 2006. Proceedings from the 3rd IFAC workshop, Nagoya, Japan, July 19–21, 2006. Berlin: Springer (ISBN 978-3-540-73889-3/pbk). Lecture Notes in Control and Information Sciences 366, 221-231 (2007).

Summary: The rolling sphere problem on \(\mathbb E^n\) consists of determining the path of minimal length traced by the point of contact of the oriented unit sphere \(\mathbb S^n\) as it rolls without slipping between two boundary points of \(\mathbb E^n\times \text{SO}_n\). This problem is extended to the following cases of rolling: \(\mathbb H^n\) on \(\mathbb E^n\), \(\mathbb S_\rho^n\) on \(\mathbb S_\sigma^n\), and \(\mathbb H_\rho\) on \(\mathbb H_\sigma^n\), where \(\sigma\neq\rho\) are the radii of the spheres or hyperboloids. The term “rolling” is generalized to an isometric sense: the length of a curve is measured using the Riemannian metric of the stationary manifold while the orientation of the rolling object is described by a matrix from its isometry group. These problems constitute left-invariant optimal control problems on Lie groups, whose Hamiltonian equations reveal certain integrals of motion and show, on the level of Lie algebras, that all of the above problems are governed by a single set of equations.
For the entire collection see [Zbl 1119.93006].

MSC:

49N90	Applications of optimal control and differential games
70H05	Hamilton’s equations
70Q05	Control of mechanical systems

Keywords:

Hamiltonian systems; isometric; sub-Riemannian; rolling sphere problem

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