  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.
