On geometry and mechanics

TL;DR

This paper investigates the topological limits of projections of secant planes and hyperplanes to smooth surfaces and hypersurfaces, and demonstrates that every twice-differentiable space curve can satisfy the principle of stationary action.

Contribution

It establishes topological bounds for secant projections and proves the universality of space curves as solutions to stationary action principles.

Findings

Topological upper limits for secant projections to surfaces and hypersurfaces.

Every C^2 space curve can be realized as a stationary action solution.

Abstract

Our purpose in this article is first, following [14], to find the topological upper limits of projections of secant planes to $C^{1}$ surfaces and the topological upper limits of projections of secant hyperplanes to $C^{1}$ hypersurfaces and second to prove that every $C^{2}$ space curve can be the solution of the principle of stationary action.