The present lecture is devoted to the description and use of such a principle, known as hamiltons principle or the principle of least action. You calculate the action and just differentiate to find the minimum. In classical mechanics, maupertuiss principle states that the path followed by a physical system. Recall that we defined the lagrangian to be the kinetic energy less potential energy. Maupertuis principle of least action is the following. You sayoh, thats just the ordinary calculus of maxima and minima. Hence it is one of the most fundamental and important principles of mechanics and mathematical physics. If there occurs some change in nature, the amount of action necessary for this change must be as small as possible. A similar principle has been postulated by hamilton for a material particle following the laws of classical physics. The rst is naturally associated with con guration space, extended by time, while the latter is the natural description for working in phase space. It is a special case of the more generally stated principle of least action. So our principle of least action is incompletely stated. It is shown that these principles are equivalent to the equations of motion resulting from a principle of least action. Hamiltons principle is one of the variational principles in mechanics.
We will start with an analysis of fermats principle for light. The principle of least action in dynamics damtp university of. Hamiltons principle of stationary action was introduced in two papers published by hamilton in \1834\ and \1835. Chapter 2 lagranges and hamiltons equations in this chapter, we consider two reformulations of newtonian mechanics, the lagrangian and the hamiltonian formalism. It isnt that a particle takes the path of least action but that it smells all the paths in the neighborhood and chooses the one that has the least action by a method analogous to the one by which light chose the shortest time. The action is an integral over time of a combination of energy contri butions, and the equation of motion that is derived from the principle of least. I want to describe a mathematical construction, also called the principle of least action, which is probably the best example showing that maupertuis was onto something. As the title says, it is on the principle of least action in physics. All the laws of mechanics can be derived by using the hamiltons principle. This article discusses the history of the principle of least action. The action is then defined to be the integral of the lagrangian along the path, s t0 t1 l t t0 t1 k u t. This is a book not a paper, oxford university press. Metaphysics of the principle of least action1 arxiv.