Some Elementary Geometric Aspects in Extending the Dimension of the Space of Instants

Abstract

A local geometric construction is proposed on the partially ordered set of instants I. A totally ordered subset C(I) ⊂ I is assumed to have 3dimensional affine coordinate structure, without a specified metric, called the τ-space of C(I).Guided by a strong analogy with analytical mechanics the T-configuration space (θ, τ ^α), θ a real parameter, is constructed whereupon the usual Hamilton-Jacobi theory establishes a simple geometrical construction, viz., the complete figure from the calculus of variations. The duration function, dur:C(I) →R is associated with temporally equidistant hypersurfaces through which pass a congruence of extremal curves to the fundamental integral.