The principle of least action, also known as Hamilton's principle, is a fundamental principle of theoretical physics from which the laws of motion in many areas of physics can be derived. However, our current knowledge gives no indication as to why this principle is universally valid.
In the classical interpretation, Hamilton's principle states that the path taken by a physical system between two points in time is the one that requires the least change in action, as the system evolves from the initial to the final state. Richard Feynman adopted the prinicple of least action to quantum mechanics, through allowing particles to take take every possible path, with each path weighted by its respective action.
Action is a physical quantity with the unit J·s = kg·m2/s. In contrast to other physical quantities, there is no device measuring it directly. Instead, the quantity action is obtained by calculating it from other quantities. It is given by the product of energy (unit J) and time (unit s) or the product of momentum (unit kg·m/s) and distance (unit m).
The principle of least action therefore means that a particle chooses the path between a defined start and end point at which it arrives the fastest with the least possible energy expenditure or at which it takes the shortest distance with the least possible momentum. Instead of the term “action”, the term “effort” would probably have been more intuitive, because ultimately the principle of least action means nothing other than that nature always chooses the path which requires the least effort.
Knowledgeable readers may object that it would be more precise to speak of the principle of stationary action instead of least action. This has to do with the fact that usually in the practical application of the principle, a mathematical trick is used to obtain a stationary solution for the function of action. In purely mathematical terms, it is therefore not possible to say whether the action is minimal or maximal at the solution found. From a physical point of view, however, it is more obvious that in nature only those solutions are realized in which the expenditures of energy and time or distance and momentum are as small as possible.
Starting with the principle of least action, the laws of motion, i.e. the mathematically formulated rules for temporal change, can be derived in all areas of physics. These include, among others:
- Law of refraction of optics
- Lorentz equations of electrodynamics
- Newton's equations of motion in classical mechanics
- Schrödinger equation of quantum mechanics
- Equations of motion of general relativity
The question of why all laws of motion obey the principle of least action remains unanswered within the field of physics. The universal validity of the principle is a highly remarkable fact, particularly given that action is an abstract quantity that, unlike time or energy, cannot be directly measured. Instead of accepting it as a mere heuristic principle, it would be highly desirable to find an explanation for the fundamental role played by the abstract quantity action in formulating the laws of physics.
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