Kinematics and symmetry

This thesis is concerned with the study of Kinematics and Symmetry. It begins with an examination of motions in a general metric space X, and gives a complete discussion of the equivalence problem. A symmetry of a motion u in X is therefore a self-equivalence. The symmetry group Sym u of p and its periodic subgroup P(p) are investigated and it is found that P(u) is the centre of Sym u. The symmetry group of individual trajectories of u is shown to be closed in I*(X) x R (where. I*(X) is the identity component of the isometry group I(X)) and is isomorphic to {0}, Z or R. Some special types of symmetries including group motion, where the path p is a homomorphism, are examined. Special attention is given to smooth motions in a smooth connected Riemannian n-manifold X. In this context, the centrode C(u) of u is of great interest, each instantaneous axis Ct(u) of p being a totally geodesic submanifold of X of even codimension. The centrode C(p) is a 1-parameter family of such axes. The rest of the thesis is devoted to the case where X .-is Euclidean n-space En . The structure of I*(X).= E+(n) is exploited to exhibit more properties of the group Sym u (in particular, where u is translational or spherical). Group motions are studied in the low dimensions n - 1,2 and 3. A complete discussion is presented for the symmetry groups that can occur in plane motion.The study of Kinematics in El is reduced to the study of real-valued continuous functions of a real variable. In particular, stable smooth motions correspond to stable Morse functions f : R. -0..R. The symmetry properties and the classification of smooth stable motions are studied in some detail.