Group theoretical formulation of free fall and projectile motion

TL;DR

This paper develops a group theoretical framework for describing free fall and projectile motion, providing a global perspective on the kinematic equations under constant acceleration, with educational benefits.

Contribution

It introduces a novel group theoretical formulation for constant acceleration kinematics, extending to projectile motion, and clarifies the structure of physical and unphysical solutions.

Findings

Group actions on phase space describe constant acceleration motion.

Group orbits relate to physical trajectories and unphysical solutions.

Method clarifies the global structure of free fall and projectile motion.

Abstract

In this work we formulate the group theoretical description of free fall and projectile motion. We show that the kinematic equations for constant acceleration form a one parameter group acting on a phase space. We define the group elements $\phi_t$ by their action on the points in the phase space. We also generalize this approach to projectile motion. We evaluate the group orbits regarding their relations to the physical orbits of particles and unphysical solutions. We note that the group theoretical formulation does not apply to more general cases involving a time dependent acceleration. This method improves our understanding of the constant acceleration problem with its global approach. It is especially beneficial for students who want to pursue a career in theoretical physics.