Hamilton's Principle, also known as the least action principle, is a very important principle in Physics. It forms the basis of the Lagrangian and Hamiltonian formations of Classical Mechanics, and, with the generalization to the path integral formulation in quantum systems, Hamilton's Principle appears to be one of the most fundamental truths in nature.

The principle is directly inspired by ideas from Christian theology. In the renaissance and before, Christian scholars believed that things would evolve in the universe in such a way as to minimize the 'action' it took to move them along, this minimization of 'effort' being proof of the wisdom of God. Thus, Hamilton's principle and vague ideas similar to it are often known as the Principle of Least Action. Of course those who articulated this, such as Pierre Louise-Moreau de Maupertius in 1747, didn't have a firm idea of what this 'action' was. Ideas were further developed by Joseph Lagrange and Karl Freidrich Gauss, but the principle was stated most completely by by Sir William Rowan Hamilton (1805-1865), an Irish Mathematician, in 1835.

Hamilton's Principle says that a system will evolve in such a way as to minimize the time integral of the kinetic energy minus the potential energy or:

∫ T - U dt will be minimized.
where the integral is from some initial time to some later time.

In more precise symbols, this can be expressed as

δ ∫ L dt = 0

where the δ ( something ) = 0 notation just expresses that something is to be minimized and L is the Lagrangian function, defined as L = T - U.

The minimization condition of Hamilton's principle leads directly to Lagrange's equations of motion and the formulation of Lagrangian mechanics, and eventually to the 'canonical equations' of motion involving the Hamiltonian, which are very general and powerful formulations of Classical Mechanics.

In quantum systems, where the behavior of individual or aggregate microscopic particles is relevant, it turns out that Hamilton's principle still holds, albeit with a twist. The path integral formulation of quantum mechanics, developed by Richard Feynman in the 1940s, states that a quantum system has an amplitude to evolve on a particular path in phase space1 that is given by

∫ D e-(i/h) L

This is almost the same as for classical systems, for mathematically the overwhelmingly likely path in the limit as h->0 is the one that minimizes the exponent, that path being the path of stationary phase, so the most likely path is indeed the one that minimizes the lagrangian. However, quantum mechanically the system has some small amplitude to evolve on adjacent paths in phase space.

Thus, Hamilton's principle seems to be one of the most fundamental thruths in our universe.

1 Phase Space is simply the space containing all possible evolutions of the system according to the variables we use to describe it.