Facets of Brachistochronic Trajectories

TL;DR

This paper explores brachistochrone trajectories by establishing rules for conservative and central forces, enabling direct solution of motion equations through magnetic field replacement instead of traditional Euler-Lagrange methods.

Contribution

It introduces four rules as sufficient conditions for brachistochrone trajectories, including a novel approach using magnetic fields to replace wire constraints for direct integration.

Findings

Four rules for brachistochrone trajectories are formulated.

Magnetic fields can replace wire constraints in trajectory analysis.

Direct integration methods are enabled for solving motion equations.

Abstract

This paper studies brachistochrone trajectories. Four rules are formulated as sufficient conditions. Two rules apply for a general conservative force. Two rules apply for a central force. A central force allows wire replacement. The wire is replaced by appropriate magnetic field. This enables solving motion equations directly. We replace Euler Lagrange with direct integration.