Action principles
[1] Action principles start with an energy function called a Lagrangian describing the physical system.
The accumulated value of this energy function between two states of the system is called the action.
The action depends on the energy function, and the energy function depends on the position, motion, and interactions in the system: variation of the action allows the derivation of the equations of motion without vectors or forces.
Several distinct action principles differ in the constraints on their initial and final conditions.
The names of action principles have evolved over time and differ in details of the endpoints of the paths and the nature of the variation.
These applications built up over two centuries as the power of the method and its further mathematical development rose.
Introductory study of mechanics, the science of interacting objects, typically begins with Newton's laws based on the concept of force, defined by the acceleration it causes when applied to mass:
This approach to mechanics focuses on a single point in space and time, attempting to answer the question: "What happens next?".
These approaches answer questions relating starting and ending points: Which trajectory will place a basketball in the hoop?
[7] The explanatory diagrams in force-based mechanics usually focus on a single point, like the center of momentum, and show vectors of forces and velocities.
The explanatory diagrams of action-based mechanics have two points with actual and possible paths connecting them.
Depending on the action principle, the two points connected by paths in a diagram may represent two particle positions at different times, or the two points may represent values in a configuration space or in a phase space.
An important result from geometry known as Noether's theorem states that any conserved quantities in a Lagrangian imply a continuous symmetry and conversely.
The only major exceptions are cases involving friction or when only the initial position and velocities are given.
When total energy and the endpoints are fixed, Maupertuis's least action principle applies.
For example, to score points in basketball the ball must leave the shooters hand and go through the hoop, but the time of the flight is not constrained.
[16] Consequently, the same path and end points take different times and energies in the two forms.
The solutions in the case of this form of Maupertuis's principle are orbits: functions relating coordinates to each other in which time is simply an index or a parameter.
When the physics problem gives the two endpoints as a position and a time, that is as events, Hamilton's action principle applies.
[3] Starting with Hamilton's principle, the local differential Euler–Lagrange equation can be derived for systems of fixed energy.
[2]: 128 Both Richard Feynman and Julian Schwinger developed quantum action principles based on early work by Paul Dirac.
Close to the path expected from classical physics, phases tend to align; the tendency is stronger for more massive objects that have larger values of action.
Analysis like this connects particle-like rays of geometrical optics with the wavefronts of Huygens–Fresnel principle.
[Maupertuis] ... thus pointed to that remarkable analogy between optical and mechanical phenomena which was observed much earlier by John Bernoulli and which was later fully developed in Hamilton's ingenious optico-mechanical theory.
Action principles can be directly applied to many problems in classical mechanics, e.g. the shape of elastic rods under load,[23]: 9 the shape of a liquid between two vertical plates (a capillary),[23]: 22 or the motion of a pendulum when its support is in motion.
[23]: 39 Quantum action principles are used in the quantum theory of atoms in molecules (QTAIM), a way of decomposing the computed electron density of molecules in to atoms as a way of gaining insight into chemical bonding.
The action principle is so central in modern physics and mathematics that it is widely applied including in thermodynamics,[26][27][28] fluid mechanics,[29] the theory of relativity, quantum mechanics,[30] particle physics, and string theory.
[33] Building on the early work of Pierre Louis Maupertuis, Leonhard Euler, and Joseph-Louis Lagrange defining versions of principle of least action,[34]: 580 William Rowan Hamilton and in tandem Carl Gustav Jacob Jacobi developed a variational form for classical mechanics known as the Hamilton–Jacobi equation.
[35]: 201 In 1915, David Hilbert applied the variational principle to derive Albert Einstein's equations of general relativity.
[37] Subsequently Julian Schwinger and Richard Feynman independently applied this principle in quantum electrodynamics.
