Principle of Least Action
The Principle of Least Action (PLA), or more generally the Principle of Stationary Action,[1] is the most fundamental principle in physics: the most efficient path of motion always occurs, such that a particle of fixed overall potential and kinetic energy follows the path that minimizes its action. This principle governing motion applies universally, including subatomic particles, planets, and light. This principle is remarkably productive in modern physics because it does not depend on vectors or forces, and tends to be independent of coordinate systems. The Parable of the Vineyard Workers is Biblical scientific foreknowledge by foreshadowing this powerful principle of modern physics. The Coase theorem in Economics is a similar insight, as is Romans 8:28
This principle predicts behavior based on energy, not forces, and on actions rather than points. A central equation to this approach integrates over time, for a particular path, the difference between kinetic and potential energies.
| “ | The Principle of Least Action says that, in some sense, the true motion is the optimum out of all possible motions, The idea that the workings of nature are somehow optimal, suggests that nature is working in an efficient way, with minimal effort, to some kind of plan.[2] | ” |
This principle assigns an "action" (usually denoted by "S") to be the difference in kinetic and potential energy for all possible paths a particle could take, as integrated over time, and the one "chosen" by the particle is the path that minimizes the action. Stated another way, particles take the path of least resistance in their motion.
Variations on this principle include the following insights that:
- a multi-body system will come to rest at a position that minimizes overall potential energy.
- light takes the shortest path to travel between two points
This principle is a physical extension of the Euclidean theorem that the shortest distance between two points is a straight line.
The doctoral thesis of physicist Richard Feynmann applied the Principle of Least Action to quantum mechanics, and he later built on that work to generate "Feynman diagrams," for which he won a shared Nobel Prize.
| “ | Basically, without outside intervention, objects travel along the path of least resistance and least change. This is called the principle of least action. We know it applies in our everyday world, and now—thanks to a new study—we know it applies in the quantum world as well.[3] | ” |
Contents
History
This principle was first proposed in 1744 by the French cosmologist Pierre-Louis Moreau de Maupertuis (1698-1759, after settling in Basel, Switzerland), who declared in Essai de cosmologie (1750; “Essay on Cosmology”) that “in all the changes that take place in the universe, the sum of the products of each body multiplied by the distance it moves and by the speed with which it moves is the least [that is] possible.”[4]
Mathematical rigor was added to this concept in 1835 by the Irish mathematician and scientist William Rowan Hamilton.
The brilliant and colorful Richard Feynmann devoted one of his famous lectures to this, as "entertainment" separate from his other lectures.[5]
Quotes
| “ | The minimum principle that unified the knowledge of light, gravitation, and electricity of Hamilton's time no longer suffices to relate these fundamental branches of physics. Within fifty years of its creation, the belief that Hamilton's principle would outlive all other physical laws of physics was shattered. Minimum principles have since been created for separate branches of physics... but these are not only restricted... but seem to be contrived... The hope of revising the principle so that it will achieve the unification... still drives mathematicians. This is the problem to which... Einstein devoted the last years of his life. Stripped of the theological associations, the belief of a minimum principle still activates physical science. ...
A single minimum principle, a universal law governing all processes in nature, is still the direction in which the search for simplicity is headed, with the price of simplicity now raised from a mastery of differential equations to a mastery of the calculus of variations.[6] |
” |
See also
- Lagrangian
- Stationary Action explained
- Medium.com explanation of the Principle of Least Action
- Richard Feynman
References
- ↑ https://physics.stackexchange.com/questions/144356/when-is-the-principle-of-stationary-action-not-the-principle-of-least-action
- ↑ N.S. Manton, University of Cambridge
- ↑ https://www.popularmechanics.com/science/a43981125/principle-of-least-action-demonstrated-in-quantum-realm/
- ↑ https://www.britannica.com/biography/Pierre-Louis-Moreau-de-Maupertuis#ref9124
- ↑ https://www.feynmanlectures.caltech.edu/II_19.html#:~:text=%E2%80%9CSo%2C%20for%20a%20conservative%20system,minimum%E2%80%94maybe%20it's%20a%20maximum.
- ↑ Morris Kline, Mathematics and the Physical World (1959) Ch. 25: From Calculus to Cosmic Planning, at p. 442. https://en.wikiquote.org/wiki/Principle_of_least_action
