The Variational Principle and natural transformations II. Time dependent theory

In the previous Paper I, the Variational Principle has been presented as a principle of natural transformation between conserved quantities and infinitesimal in variances, which preserves the geometrical character of the various standard objects under point transformations. So as to establish the theory as easily as possible, Paper I (2) confined itself to the simple case of autonomous dynamical systems, and their 2n dimensional space. The aim of the present paper is to extend the work to the 2n + 1 dimensional theory, in which time is included as an extra variable, and time dependent systems can be handled. It will be seen that within a suitable framework the extension is straightforward, and there are some rather more satisfactory features of the 2n + 1 dimensional theory - even for autonomous systems when included in the new formalism - than in the previous 2n dimensional theory. The exposition will now be rather concise; readers are requested to refer to Paper I for fuller discussion, particularly of the underlying ideas.