Demystifying the Lagrangian of classical mechanics

Abstract

The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack explanations of the most basic details that make Lagrangian mechanics so practical. In this paper, we detail the steps taken to arrive at the principle of stationary action, the Euler-Lagrange equations, and the Lagrangian of classical mechanics. These steps are: 1) we derive the Lagrange formalism purely mathematically from the problem of the minimal distance between two points in a plane, introducing the variational principle and deriving the Euler-Lagrange equation; 2) we transform Newton's second law into an Euler-Lagrange equation, proving that the Lagrangian is kinetic minus potential energy; 3) we explain why it is important to reformulate Newton's law. To do so, we prove that the Euler-Lagrange equation is astonishingly the same in any set of coordinates. We demonstrate that because of this feature the role that coordinates play in classical mechanics is much simpler and clearer in the Lagrangian as compared to the Newtonian formulation. This is important because the choice of coordinates is not relevant to physical reality, rather they are arbitrarily chosen to provide a convenient way of analyzing a physical system.