II. — On the geometry of dynamics

Abstract

To the pure mathematician of the present day the tensor calculus is a notation of differential geometry, of special utility in connection with multi-dimensional spaces; to the applied mathematician it is the backbone of the general theory of relativity. But when it is recognised that every problem in applied mathematics may be regarded as a geometrical problem (in the widest sense) and that the geometrical forms which many of these problems take are such that the tensor calculus can be directly applied, it is realised that the possibilities of this calculus in the field of applied mathematics can hardly be overestimated. It has a dual importance: first, by its help, known results may be exhibited in the most compact form; secondly, it enables the mathematician to exercise his most potent instrument of discovery, geometrical intuition. In the present paper we are concerned with the development of general dynamical theory with the aid of the tensor calculus. In view of the present close association of the tensor theory with the theory of relativity, it should be clearly understood that this paper only attempts to deal with the classical or Newtonian dynamics of a system of particles or of rigid bodies. The subject is presented in a semi-geometrical aspect, and the reader should visualise the results in order to realise the close analogy between general dynamical theory and the dynamics of a particle. Mathematicians display a strange reluctance in summoning to their assistance the power of visualisation in multidimensional space. They forget that they have studied the geometry of three dimensions largely through the medium of a schematic representation on a two dimensional sheet of paper. The same method is available in the case of any number of dimensions.