Kinematics and symmetry

Abstract

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 μ of p and its periodic subgroup P(μ) are investigated and it is found that P(μ) is a subgroup of the centre of Sym μ. 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 μ 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(μ) of u is of great interest, each instantaneous axis C_t(μ) of μ being a totally geodesic submanifold of X of even codimension. The centrode C(μ) 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 μ (in particular, where μ 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 E^l 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 → R. The symmetry properties and the classification of smooth stable motions are studied in some detail.

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