Some elementary geometric aspects in extending the dimension of the space of instants (Q1362135)

The article investigates the possibility of an extension of dimension of the space \({\mathcal I}\) of time instants. The space \({\mathcal I}\) is partially ordered, and contains a totally ordered subset \(C({\mathcal I}) \subset {\mathcal I}\). A duration function dur: \(C({\mathcal I})\to\mathbb{R}_+\) is assumed, where \(\mathbb{R}_+\) is the set of the non-negative real numbers. The author develops a Hamilton-Jacobi theory in the three-dimensional space of instants, introducing a function analogous to Lagrangian. However, any totally ordered set can be made linearly ordered, i.e. the totally ordered subset \(C({ \mathcal I})\) of the article is a linearly ordered subset, which does not contain a subset anti-isomorphic with the set of non-negative integers. The main result of the article is that the totally ordered subset \(C({\mathcal I}) \subset{\mathcal I}\) is assumed to have a three-dimensional affine coordinate structure.