The Principle of Least Action

In their never-ending search for general principles, from which various laws of Physics could be derived, physicists, and most notably theoretical physicists, have often made use of variational techniques. These usually involved the minimisation of certain quantities. A typical example is Fermat’s principle of least time where light, propagating between two distant points in space, is postulated to follow the path for which the propagation time is a minimum. A similar principle has been postulated by Hamilton for a material particle following the laws of Classical Physics. The present lecture is devoted to the description and use of such a principle, known as Hamilton’s principle or the principle of least action. We will start with an analysis of Fermat’s principle for light. Once the philosophy behind this idea has been well grasped, we will step into Classical Mechanics and attempt to use the same argument to describe the propagation of a material particle between two distant points in space. In this way the concept of action and of lagrangian will be automatically introduced. The evolution in time of a moving particle will follow from a variational principle and be described by Euler-Lagrange equations. On more general mechanical systems the same technique can be used if generalised coordinates are appropriately introduced; these coordinates take into account the presence of constraints and embed them quite naturally in the formalism developed. Few examples will conclude the lecture.