Power and beauty of the Lagrange equations

Abstract

The Lagrangian formulation of the equations of motion for point particles is
usually presented in classical mechanics as the outcome of a series of
insightful algebraic transformations or, in more advanced treatments, as the
result of applying a variational principle. In this paper we stress two main
reasons for considering the Lagrange equations as a fundamental description
of the dynamics of classical particles. Firstly, their structure can be
naturally disclosed from the existence of integrals of motion, in a way
that, though elementary and easy to prove, seems to be less popular
--or less frequently made explicit-- than others in
support of the Lagrange formulation. The second reason is that the Lagrange
equations preserve their form in \emph{any} coordinate system --
even in moving ones, if required. Their covariant nature makes them
particularly suited to deal with dynamical problems in curved spaces or
involving (holonomic) constraints. We develop the above and related ideas in
clear and simple terms, keeping them throughout at the level of intermediate
courses in classical mechanics. This has the advantage of introducing some
tools and concepts that are useful at this stage, while they may also serve
as a bridge to more advanced courses.