Action (physics)

[1] Action and the variational principle are used in Feynman's formulation of quantum mechanics[2] and in general relativity.[3] For systems with small values of action similar to the Planck constant, quantum effects are significant.[4] In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy.More formally, action is a mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result.[5] Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant h).[6] Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages.[1] However, the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students.[7] For a trajectory of a ball moving in the air on Earth the action is defined between two points in time,This makes the action an input to the powerful stationary-action principle for classical and for quantum mechanics.Replace the ball with an electron: classical mechanics fails but stationary action continues to work.It figures in all significant quantum equations, like the uncertainty principle and the de Broglie wavelength.Whenever the value of the action approaches the Planck constant, quantum effects are significant.[4] Pierre Louis Maupertuis and Leonhard Euler working in the 1740s developed early versions of the action principle.Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations.which takes a function of time and (for fields) space as input and returns a scalar.[13][14] In classical mechanics, the input function is the evolution q(t) of the system between two times t1 and t2, where q represents the generalized coordinates.In the abbreviated action, the input function is the path followed by the physical system without regard to its parameterization by time.When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:[11]: 225The physical significance of this function is understood by taking its total time derivative[15]: 434 A variable Jk in the action-angle coordinates, called the "action" of the generalized coordinate qk, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:[15]: 454The corresponding canonical variable conjugate to Jk is its "angle" wk, for reasons described more fully under action-angle coordinates.[15]: 477 When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper timeAction is a part of an alternative approach to finding such equations of motion.Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is stationary.In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length).The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics.The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle.An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely).Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics.
action at a distanceSI unitjoule-secondSI base unitsphysicsscalar quantityphysical systemprinciple of stationary actionFeynman's formulation of quantum mechanicsPlanck constantuniform linear motionmomentumadded upkinetic energypotential energymathematical functionaltrajectoryreal numberdimensionsenergylengthSI unitNewton's laws of motionstationary-action principleclassicalquantum mechanicsuncertainty principlede Broglie wavelengthHistory of variational principles in physicsPierre Louis MaupertuisLeonhard EulerJoseph Louis Lagrangecalculus of variationsWilliam Rowan HamiltonRichard FeynmanJulian Schwingerevolutionstationary point[energy][time]angular momentumintegralfieldsLagrangianfunctionalfunctionscalarclassical mechanicsgeneralized coordinatesHamilton's principlestationarysaddle pointLagrangian mechanicsMaupertuis's principleHamilton–Jacobi equationaction-angle coordinatesphase spacedot productadiabatic invariantsRelativistic Lagrangian mechanicsTheory of relativityworld lineproper timedifferential equationspositioncontinuouslyinitialboundaryfunctionsequations of motionvariationaltheoretical physicsintegral equationclassical fieldselectromagneticgravitationalquantum field theorypath integral formulationSchrödinger equationEuler–Lagrange equationsEinstein–Hilbert actionelectromagnetic fieldgravitational fieldMaxwell's equationsEinstein equationvariational principlespacetimegeodesicConservation lawsNoether's theoremcontinuous symmetryconservation lawpath integralprobability amplitudesNewton's lawsdestructive interferencenonlocal actionsfunctional spacenoncommutative geometryFunctional derivativeFunctional integrationHamiltonian mechanicsPrinciple of least actionPrinciple of maximum entropyNambu–Goto actionPolyakov actionBagger–Lambert–Gustavsson actionBibcodeR.G. LernerSI unitsabsementdistancedisplacementangular displacementsolid anglerad2, srfrequencyvelocitym s−1kinematic viscosityspecific angular momentumrotational speedrotational velocityangular speedangular velocityrad s−1accelerationm s−2rotational accelerationangular accelerationrad s−2weighted positionmoment of inertiakg m2Mass flow ratekg s−1impulsekg m s−1kg m2 s−1kg m rad s−1weighttorquerotatum