Some elementary geometric aspects in extending the dimension of the space of instants
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A local geometric construction is proposed on thepartially ordered set of instantsI. A totally ordered subsetC(I)⊂I is assumed to have 3-dimensional affine coordinate structure,without a specified metric, called the τ-space ofC(I). Guided by a strong analogy with analytical mechanics theT-configuration space (θ, τα), θ a real parameter, is constructed where-upon 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 withtemporally equidistant hypersurfaces through which pass a congruence of extremal curves to the fundamental integral.
KeywordsAnalytical Mechanic Real Parameter Geometric Construction Geometric Aspect Coordinate Structure
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