
Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces
Author(s) -
André Marques,
F. Silva Leite
Publication year - 2022
Publication title -
journal of geometric mechanics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.511
H-Index - 17
eISSN - 1941-4897
pISSN - 1941-4889
DOI - 10.3934/jgm.2021033
Subject(s) - mathematics , euclidean space , tangent space , hyperbolic space , euclidean geometry , pure mathematics , kinematics , motion (physics) , affine transformation , space (punctuation) , mathematical analysis , geometry , classical mechanics , computer science , physics , operating system
This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [ 26 ] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.