Action principles are fundamental to physics, fromclassical mechanics throughquantum mechanics,particle physics, andgeneral relativity.[1] Action principles start with an energy function called aLagrangian describing the physical system. The accumulated value of this energy function between two states of the system is called theaction. Action principles apply thecalculus of variation to 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. Quantum action principles generalize and justify the older classical principles by showing they are a direct result of quantum interference patterns. Action principles are the basis forFeynman's version of quantum mechanics, general relativity andquantum field theory.
The action principles have applications as broad as physics, including many problems in classical mechanics but especially in modern problems of quantum mechanics and general relativity. These applications built up over two centuries as the power of the method and its further mathematical development rose.
This article introduces the action principle concepts and summarizes other articles with more details on concepts and specific principles.
Action principles are "integral" approaches rather than the "differential" approach ofNewtonian mechanics.[2]: 162 The core ideas are based on energy, paths, an energy function called the Lagrangian along paths, and selection of a path according to the "action", a continuous sum or integral of the Lagrangian along the path.
Introductory study of mechanics, the science of interacting objects, typically begins withNewton's laws based on the concept offorce, defined by the acceleration it causes when applied tomass:F =ma. This approach to mechanics focuses on a single point in space and time, attempting to answer the question: "What happens next?".[3] Mechanics based on action principles begin with the concept ofaction, an energy tradeoff betweenkinetic energy andpotential energy, defined by the physics of the problem. These approaches answer questions relating starting and ending points: Which trajectory will place a basketball in the hoop? If we launch a rocket to the Moon today, how can it land there in 5 days?[3] The Newtonian and action-principle forms are equivalent, and either one can solve the same problems, but selecting the appropriate form will make solutions much easier.
The energy function in the action principles is not the total energy (conserved in an isolated system), but theLagrangian, the difference between kinetic and potential energy. The kinetic energy combines the energy of motion for all the objects in the system; the potential energy depends upon the instantaneous position of the objects and drives the motion of the objects. The motion of the objects places them in new positions with new potential energy values, giving a new value for the Lagrangian.[4]: 125
Using energy rather than force gives immediate advantages as a basis for mechanics. Force mechanics involves 3-dimensionalvector calculus, with 3 space and 3 momentum coordinates for each object in the scenario; energy is a scalar magnitude combining information from all objects, giving an immediate simplification in many cases. The components of force vary with coordinate systems; the energy value is the same in all coordinate systems.[5]: xxv Force requires an inertial frame of reference;[6]: 65 once velocities approach thespeed of light,special relativity profoundly affects mechanics based on forces. In action principles, relativity merely requires a different Lagrangian: the principle itself is independent of coordinate systems.[7]
The explanatory diagrams in force-based mechanics usually focus on a single point, like thecenter 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.[8] These diagrammatic conventions reiterate the different strong points of each method.
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 aconfiguration space or in aphase space. The mathematical technology and terminology of action principles can be learned by thinking in terms of physical space, then applied in the more powerful and general abstract spaces.
Action principles assign a number—the action—to each possible path between two points. This number is computed by adding an energy value for each small section of the path multiplied by the time spent in that section:[8]
where the form of the kinetic energy (KE) and potential energy (PE) expressions depend upon the physics problem, and their value at each point on the path depends upon relative coordinates corresponding to that point. The energy function is called a Lagrangian; in simple problems it is the kinetic energy minus the potential energy of the system.
In classical mechanics, a system moving between two points takes one particular path; other similar paths are not taken. Each conceivable path corresponds to a value of the action. An action principle predicts or explains that the particular path taken has a stationary value for the system's action: similar paths near the one taken have very similar action value. This variation in the action value is key to the action principles.
In quantum mechanics, every possible path contributes an amplitude to the system's behavior, with the phase of each amplitude determined by the action for that path (phase =action/ħ). The classical path emerges because:[citation needed]
When the scale of the problem is much larger than thePlanck constantħ (the classical limit), only the stationary action path survives the interference.
The symbolδ is used to indicate the pathvariations so an action principle appears mathematically as
meaning that at thestationary point, the variation of the actionS with some fixed constraintsC is zero.[9]: 38 For action principles, the stationary point may be a minimum or asaddle point, but not a maximum.[10] Elliptical planetary orbits provide a simple example of two paths with equal action – one in each direction around the orbit; neither can be the minimum or "least action".[2]: 175 The path variation implied byδ is not the same as a differential likedt. The action integral depends on the coordinates of the objects, and these coordinates depend upon the path taken. Thus the action integral is afunctional, a function of a function.
An important result from geometry known asNoether's theorem states that any conserved quantities in a Lagrangian imply a continuous symmetry and conversely.[11] For examples, a Lagrangian independent of time corresponds to a system with conserved energy; spatial translation independence implies momentum conservation; angular rotation invariance implies angular momentum conservation.[12]: 489 These examples are global symmetries, where the independence is itself independent of space or time; more generallocal symmetries having a functional dependence on space or time lead togauge theory.[13] The observed conservation ofisospin was used byYang Chen-Ning andRobert Mills in 1953 to construct a gauge theory formesons, leading some decades later tomodern particle physics theory.[14]: 202
Action principles apply to a wide variety of physical problems, including all of fundamental physics. The only major exceptions are cases involving friction or when only the initial position and velocities are given.[3] Different action principles have different meaning for the variations; each specific application of an action principle requires a specific Lagrangian describing the physics. A common name for any or all of these principles is "the principle of least action". For a discussion of the names and historical origin of these principles seeaction principle names.
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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.[3] Maupertuis's least action principle is written mathematically as the stationary conditionon theabbreviated action(sometimes writtenS0), wherep = (p1,p2,…,pN) are the particle momenta or the conjugate momenta ofgeneralized coordinates, defined by the equationwhereL(q,q̇,t) is theLagrangian. Some textbooks write[15]: 76 [9]: 356 (δW)E = 0 asΔS0, to emphasize that the variation used in this form of the action principle differs fromHamilton's variation. Here the total energyE is fixed during the variation, but not the time, the reverse of the constraints on Hamilton's principle.[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 areorbits: functions relating coordinates to each other in which time is simply an index or a parameter.[16]
For time-invariant system, the action relates simply to the abbreviated actionW on the stationary path as[9]: 434 for energyE and time differenceΔt =t2 −t1. For a rigid body with no net force, the actions are identical, and the variational principles become equivalent toFermat's principle of least time:[9]: 360
When the physics problem gives the two endpoints as a position and a time, that is asevents,Hamilton's action principle applies. For example, imagine planning a trip to the Moon. During your voyage the Moon will continue its orbit around the Earth: it is a moving target. Hamilton's principle for objects at positionsq(t) is written mathematically asThe constraintΔt =t2 −t1 means that we only consider paths taking the same time, as well as connecting the same two pointsq(t1) andq(t2). TheLagrangian is the difference between kinetic energy and potential energy at each point on the path.[17]: 62 Solution of the resulting equations gives theworld lineq(t).[3] Starting with Hamilton's principle, the local differentialEuler–Lagrange equation can be derived for systems of fixed energy. The action in Hamilton's principle is theLegendre transformation of the action in Maupertuis's principle.[18]
The concepts and many of the methods useful for particle mechanics also apply to continuous fields. The action integral runs over a Lagrangian density, but the concepts are so close that the density is often simply called the Lagrangian.[19]: 15
For quantum mechanics, the action principles have significant advantages: only one mechanical postulate is needed, if a covariant Lagrangian is used in the action, the result is relativistically correct, and they transition clearly to classical equivalents.[2]: 128
BothRichard Feynman andJulian Schwinger developed quantum action principles based on early work byPaul Dirac. Feynman's integral method was not a variational principle but reduces to the classical least action principle; it led to hisFeynman diagrams. Schwinger's differential approach relates infinitesimal amplitude changes to infinitesimal action changes.[2]: 138
When quantum effects are important, new action principles are needed. Instead of a particle following a path, quantum mechanics defines a probability amplitudeψ(xk,t) at one pointxk and timet related to a probability amplitude at a different point later in time:whereS(xk + 1,xk) is the classical action.[20]Instead of a single path with stationary action, all possible paths add (the integral overxk), weighted by a complex probability amplitudeeiS⁄ħ. The phase of the amplitude is given by the action divided by thePlanck constant or quantum of action:S/ħ. When the action of a particle is much larger thanħ,S/ħ ≫ 1, the phase changes rapidly along the path: the amplitude averages to a small number.[8]Thus the Planck constant sets the boundary between classical and quantum mechanics.[21]
All of the paths contribute in the quantum action principle. At the end point, where the paths meet, the paths with similar phases add, and those with phases differing byπ subtract. 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. In the classical limit, one path dominates – the path of stationary action.[22]
Schwinger's approach relates variations in the transition amplitudes(qf|qi) to variations in an action matrix element:
where the action operator is
The Schwinger form makes analysis of variation of the Lagrangian itself, for example, variation in potential source strength, especially transparent.[2]: 138
For every path, the action integral builds in value from zero at the starting point to its final value at the end. Any nearby path has similar values at similar distances from the starting point. Lines or surfaces of constant partial action value can be drawn across the paths, creating a wave-like view of the action. Analysis like this connects particle-like rays ofgeometrical optics with the wavefronts ofHuygens–Fresnel principle.
[Maupertuis] … thus pointed to that remarkable analogy between optical and mechanical phenomena which was observed much earlier byJohn Bernoulli and which was later fully developed in Hamilton's ingenious optico-mechanical theory. This analogy played a fundamental role in the development of modern wave-mechanics.
— C. Lanczos[5]: 136
Action principles are applied to derive differential equations like theEuler–Lagrange equations[9]: 44 or as direct applications to physical problems.
Action principles can be directly applied to many problems inclassical mechanics, such as the shape of elastic rods under load,[23]: 9 the shape of a liquid between two vertical plates (acapillary),[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.[24]
Inspired by Einstein's work ongeneral relativity, the renowned mathematicianDavid Hilbert applied the principle of least action to derive the field equations of general relativity.[25]: 186 His action, now known as theEinstein–Hilbert action,
contained a relativistically invariant volume element√−gd4x and the Ricciscalar curvatureR. The scale factor is theEinstein gravitational constant.
The action principle is so central inmodern physics andmathematics that it is widely applied including inthermodynamics,[26][27][28]fluid mechanics,[29] thetheory of relativity,quantum mechanics,[30]particle physics, andstring theory.[31]
The action principle is preceded by earlier ideas inoptics. Inancient Greece,Euclid wrote in hisCatoptrica that, for the path of light reflecting from a mirror, theangle of incidence equals theangle of reflection.[32]Hero of Alexandria later showed that this path has the shortest length and least time.[33]
Building on the early work ofPierre Louis Maupertuis,Leonhard Euler, andJoseph-Louis Lagrange defining versions ofprinciple of least action,[34]: 580 William Rowan Hamilton and in tandemCarl Gustav Jacob Jacobi developed a variational form for classical mechanics known as theHamilton–Jacobi equation.[35]: 201
In 1915,David Hilbert applied the variational principle to deriveAlbert Einstein's equations ofgeneral relativity.[36]
In 1933, the physicistPaul Dirac demonstrated how this principle can be used in quantum calculations by discerning thequantum mechanical underpinning of the principle in thequantum interference of amplitudes.[37] SubsequentlyJulian Schwinger andRichard Feynman independently applied this principle in quantum electrodynamics.[38][39]
InCatoptrics the law of reflection is stated, namely that incoming and outgoing rays form the same angle with the surface normal.