Power and beauty of the Lagrange equations

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.