Motion of Galilean particles with curvature and torsion

Abstract

This paper examines the motion of particles governed by an action that depends on the curvature and torsion of their trajectories in the Galilean 3-space $G_{3}.$ We derive the Euler-Lagrange equation corresponding to the action $H(\gamma )=\int\limits_{\gamma}f(\kappa ,\tau )ds$ in $G_{3}$. We present examples to clarify the solutions of the system, clearly explaining their properties and relevance. With examples specifically focusing on the natural Hamiltonian problem derived from the Frenet frame of the curve and a generalization of these natural Hamiltonians, we aim to illustrate their key features and underlying principles.