Variational Methods: The Action Principle

The chapter is devoted to the variational methods of Analytic Mechanics. After a quick review of basic facts in the calculus of variations, we show that the Lagrangian equations of motion (that we deduced from the d’Alembert principle) follow more intrinsically from a variational principle, namely, the minimal action principle of Hamilton. As an aside we discuss the second variation and the Jacobi fields. We briefly comment on the other variational principles of mechanics, in particular, the Maupertius minimal action principle, and their relation with the Fermat principle in geometric optics. In the last section, we introduce the Lagrange method of the undetermined multipliers as an efficient tool to analyze mechanical systems which are subjected to more general types of constraints (such as the non-holonomic ones) and also as a mean to compute explicitly the constraint forces.