2.15 Principle of Least Action
Some ancient philosophers noticed that many things in nature, such as rivers, always follow the shortest path available. In optics, Euclid noticed that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.
This principle is called the least action, and it is often credited to Pierre Louis Maupertuis (1698-1759 AD), because he wrote about it in 1744, but Leonhard Euler (1707-1783 AD) discussed this principle before, and it has also been attributed to Leibniz.
The principle of least action is applied to the action of a mechanical system to obtain the equations of motion for that system. It can also be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even General Relativity, or what is known as the Einstein-Hilbert action. It has also been applied to derive important equations in Quantum Mechanics and Quantum Field Theory, as we shall see in chapter III.
The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, Quantum Mechanics, particle physics, and string theory.