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Action (physics)

From Wikipedia, the free encyclopedia

Physical quantity of dimension energy × time

This article is about a property of a trajectory. For the central force concept, seeaction at a distance.

Action
Common symbols	S
SI unit	joule-second
Other units	J⋅Hz⁻¹
InSI base units	kg⋅m²⋅s⁻¹
Dimension	${\mathsf {M}}\cdot {\mathsf {L}}^{2}\cdot {\mathsf {T}}^{-1}$

Inphysics,action is ascalar quantity that describes how the balance of kinetic versus potential energy of aphysical system changes with trajectory. Action is significant because it is an input to theprinciple of stationary action, an approach to classical mechanics that is simpler for multiple objects.^[1] Action and the variational principle are used inFeynman's formulation of quantum mechanics^[2] and in general relativity.^[3] For systems with small values of action close to thePlanck constant, quantum effects are significant.^[4]

In the simple case of a single particle moving with a constant velocity (thereby undergoinguniform linear motion), the action is themomentum of the particle times the distance it moves,added up along its path; equivalently, action is the difference between the particle'skinetic energy and itspotential energy, times the duration for which it has that amount of energy.

More formally, action is amathematical functional which takes thetrajectory (also called path or history) of the system as its argument and has areal number as its result. Generally, the action takes different values for different paths.^[5] Action hasdimensions ofenergy × time ormomentum × length, and itsSI unit isjoule-second (like thePlanck constanth).^[6]

Introduction

Introductory physics often begins withNewton'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]

Simple example

For a trajectory of a ball moving in the air on Earth theaction is defined between two points in time, $t_{1}$ and $t_{2}$ as the kinetic energy (KE) minus the potential energy (PE), integrated over time.^[4]

S=\int _{t_{1}}^{t_{2}}\left(KE(t)-PE(t)\right)dt

The action balances kinetic against potential energy.^[4] The kinetic energy of a ball of mass $m {\displaystyle m}$ is $(1/2)mv^{2}$ where $v {\displaystyle v}$ is the velocity of the ball; the potential energy is $m g x {\displaystyle mgx}$ where $g {\displaystyle g}$ is the acceleration due to gravity. Then the action between $t_{1}$ and $t_{2}$ is

S=\int _{t_{1}}^{t_{2}}\left({\frac {1}{2}}mv^{2}(t)-mgx(t)\right)dt

The action value depends upon the trajectory taken by the ball through $x(t)$ and $v(t)$ . This makes the action an input to the powerfulstationary-action principle forclassical and forquantum mechanics. Newton's equations of motion for the ball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work.^[4] The energy difference in the simple action definition, kinetic minus potential energy, is generalized and calledthe Lagrangian for more complex cases.

Planck's quantum of action

ThePlanck constant, written as $h {\displaystyle h}$ is the quantum of action.^[8] The quantum ofangular momentum is $\hbar ={\frac {h}{2\pi }}$ . These constants have units of energy times time. They appear in all significant quantum equations, like theuncertainty principle and thede Broglie wavelength. Whenever the value of the action approaches the Planck constant, quantum effects are significant.^[4]

History

Main article:History of variational principles in physics

Pierre Louis Maupertuis andLeonhard Euler working in the 1740s developed early versions of the action principle.Joseph Louis Lagrange clarified the mathematics when he invented thecalculus of variations.William Rowan Hamilton made the next big breakthrough, formulating Hamilton's principle in 1853.^[9]^: 740 Hamilton's principle became the cornerstone for classical work with different forms of action untilRichard Feynman andJulian Schwinger developed quantum action principles.^[10]^: 127

Definitions

Expressed in mathematical language, using thecalculus of variations, theevolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to astationary point (usually, a minimum) of the action. Action has thedimensions of[energy] × [time], and itsSI unit isjoule-second, which is identical to the unit ofangular momentum.

Several different definitions of "the action" are in common use in physics.^[11]^[12] The action is usually anintegral over time. However, when the action pertains tofields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.

The action is typically represented as anintegral over time, taken along the path of the system between the initial time and the final time of the development of the system:^[11] ${\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,dt,$ where the integrandL is called theLagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.

Action (functional)

Most commonly, the term is used for afunctional ${\mathcal {S}}$ which takes afunction of time and (forfields) space as input and returns ascalar.^[13]^[14] Inclassical mechanics, the input function is the evolutionq(t) of the system between two timest₁ andt₂, whereq represents thegeneralized coordinates. The action ${\mathcal {S}}[\mathbf {q} (t)]$ is defined as theintegral of theLagrangianL for an input evolution between the two times: ${\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt,$ where the endpoints of the evolution are fixed and defined as $\mathbf {q} _{1}=\mathbf {q} (t_{1})$ and $\mathbf {q} _{2}=\mathbf {q} (t_{2})$ . According toHamilton's principle, the true evolutionq_true(t) is an evolution for which the action ${\mathcal {S}}[\mathbf {q} (t)]$ isstationary (a minimum, maximum, or asaddle point). This principle results in the equations of motion inLagrangian mechanics.

Abbreviated action (functional)

In addition to the action functional, there is another functional called theabbreviated action. In the abbreviated action, the input function is thepath followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path.

The abbreviated action ${\mathcal {S}}_{0}$ (sometime written as $W {\displaystyle W}$ ) is defined as the integral of the generalized momenta, $p_{i}={\frac {\partial L(q,t)}{\partial {\dot {q}}_{i}}},$ for a system Lagrangian $L {\displaystyle L}$ along a path in thegeneralized coordinates $q_{i}$ : ${\mathcal {S}}_{0}=\int _{q_{1}}^{q_{2}}\mathbf {p} \cdot d\mathbf {q} =\int _{q_{1}}^{q_{2}}\Sigma _{i}p_{i}\,dq_{i}.$ where $q_{1}$ and $q_{2}$ are the starting and ending coordinates.According toMaupertuis's principle, the true path of the system is a path for which the abbreviated action isstationary.

Hamilton's characteristic function

When the total energyE is conserved, theHamilton–Jacobi equation can be solved with theadditive separation of variables:^[11]^: 225 $S(q_{1},\dots ,q_{N},t)=W(q_{1},\dots ,q_{N})-E\cdot t,$ where the time-independent functionW(q₁,q₂, ...,q_N) is calledHamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative

${\frac {dW}{dt}}={\frac {\partial W}{\partial q_{i}}}{\dot {q}}_{i}=p_{i}{\dot {q}}_{i}.$

This can be integrated to give

$W(q_{1},\dots ,q_{N})=\int p_{i}{\dot {q}}_{i}\,dt=\int p_{i}\,dq_{i},$

which is just theabbreviated action.^[15]^: 434

Action of a generalized coordinate

A variableJ_k in theaction-angle coordinates, called the "action" of the generalized coordinateq_k, is defined by integrating a single generalized momentum around a closed path inphase space, corresponding to rotating or oscillating motion:^[15]^: 454

$J_{k}=\oint p_{k}\,dq_{k}$

The corresponding canonical variable conjugate toJ_k is its "angle"w_k, for reasons described more fully underaction-angle coordinates. The integration is only over a single variableq_k and, therefore, unlike the integrateddot product in the abbreviated action integral above. TheJ_k variable equals the change inS_k(q_k) asq_k is varied around the closed path. For several physical systems of interest, J_k is either a constant or varies very slowly; hence, the variableJ_k is often used in perturbation calculations and in determiningadiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.^[15]^: 477

Single relativistic particle

Main articles:Relativistic Lagrangian mechanics andTheory of relativity

When relativistic effects are significant, the action of a point particle of massm travelling aworld lineC parametrized by theproper time $\tau$ is $S=-mc^{2}\int _{C}\,d\tau .$

If instead, the particle is parametrized by the coordinate timet of the particle and the coordinate time ranges fromt₁ tot₂, then the action becomes $S=\int _{t1}^{t2}L\,dt,$ where theLagrangian is^[16] $L=-mc^{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.$

Action principles and related ideas

Main article:Principle of stationary action

Physical laws are frequently expressed asdifferential equations, which describe how physical quantities such asposition andmomentum changecontinuously withtime,space or a generalization thereof. Given theinitial andboundary conditions for the situation, the "solution" to these empirical equations is one or morefunctions that describe the behavior of the system and are calledequations of motion.

Action 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 theaction is minimized, or more generally, isstationary. In other words, the action satisfies avariational principle: theprinciple of stationary action (see also below). The action is defined by anintegral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.

The action principle provides deep insights into physics, and is an important concept in moderntheoretical physics. Various action principles and related concepts are summarized below.

Maupertuis's principle

Main article:Maupertuis's principle

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). Maupertuis's principle uses theabbreviated action between two generalized points on a path.

Hamilton's principal function

Main article:Hamilton's principle

Hamilton's principle states that the differential equations of motion forany physical system can be re-formulated as an equivalentintegral equation. Thus, there are two distinct approaches for formulating dynamical models.

Hamilton's principle applies not only to theclassical mechanics of a single particle, but also toclassical fields such as theelectromagnetic andgravitational fields. Hamilton's principle has also been extended toquantum mechanics andquantum field theory—in particular thepath integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase.^[17]

Hamilton–Jacobi equation

Main article:Hamilton–Jacobi equation

Hamilton's principal function $S=S(q,t;q_{0},t_{0})$ is obtained from the action functional ${\mathcal {S}}$ by fixing the initial time $t_{0}$ and the initial endpoint $q_{0},$ while allowing the upper time limit $t {\displaystyle t}$ and the second endpoint $q {\displaystyle q}$ to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation ofclassical mechanics. Due to a similarity with theSchrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link withquantum mechanics.

Euler–Lagrange equations

Main article:Euler–Lagrange equations

In Lagrangian mechanics, the requirement that the action integral bestationary under small perturbations is equivalent to a set ofdifferential equations (called the Euler–Lagrange equations) that may be obtained using thecalculus of variations.

Classical fields

See also:Einstein–Hilbert action

Theaction principle can be extended to obtain theequations of motion for fields, such as theelectromagnetic field orgravitational field.Maxwell's equations canbe derived as conditions of stationary action.

TheEinstein equation utilizes theEinstein–Hilbert action as constrained by avariational principle. Thetrajectory (path inspacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is ageodesic.

Conservation laws

Main article:Conservation laws

Implications of symmetries in a physical situation can be found with the action principle, together with theEuler–Lagrange equations, which are derived from the action principle. An example isNoether's theorem, which states that to everycontinuous symmetry in a physical situation there corresponds aconservation law (and conversely). This deep connection requires that the action principle be assumed.^[17]

Path integral formulation of quantum field theory

Main article:Path integral formulation

Inquantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate thepath integral, which gives theprobability amplitudes of the various outcomes.

Although equivalent in classical mechanics withNewton's laws, theaction principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly inRichard Feynman'spath integral formulation, where it arises out ofdestructive interference of quantum amplitudes.

Modern extensions

The action principle can be generalized still further. For example, the action need not be an integral, becausenonlocal actions are possible. The configuration space need not even be afunctional space, given certain features such asnoncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.^[13]

See also

References

^^a ^bNeuenschwander, Dwight E.; Taylor, Edwin F.; Tuleja, Slavomir (2006-03-01)."Action: Forcing Energy to Predict Motion".The Physics Teacher.44 (3):146–152.Bibcode:2006PhTea..44..146N.doi:10.1119/1.2173320.ISSN 0031-921X.
^Ogborn, Jon; Taylor, Edwin F (2005-01-01)."Quantum physics explains Newtons laws of motion"(PDF).Physics Education.40 (1):26–34.Bibcode:2005PhyEd..40...26O.doi:10.1088/0031-9120/40/1/001.ISSN 0031-9120.S2CID 250809103.
^Taylor, Edwin F. (2003-05-01)."A call to action".American Journal of Physics.71 (5):423–425.Bibcode:2003AmJPh..71..423T.doi:10.1119/1.1555874.ISSN 0002-9505.
^^a ^b ^c ^d ^e"The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action".www.feynmanlectures.caltech.edu. Retrieved2023-11-03.
^Goodman, Bernard (1993)."Action". In Parker, S. P. (ed.).McGraw-Hill Encyclopaedia of Physics (2nd ed.). New York: McGraw-Hill. p. 22.ISBN 0-07-051400-3.
^Stehle, Philip M. (1993)."Least-action principle". In Parker, S. P. (ed.).McGraw-Hill Encyclopaedia of Physics (2nd ed.). New York: McGraw-Hill. p. 670.ISBN 0-07-051400-3.
^Fee, Jerome (1942)."Maupertuis and the Principle of Least Action".American Scientist.30 (2):149–158.ISSN 0003-0996.JSTOR 27825934.
^"Max Planck Nobel Lecture".Archived from the original on 2023-07-14. Retrieved2023-07-14.
^Kline, Morris (1972).Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press. pp. 167–168.ISBN 0-19-501496-0.
^Yourgrau, Wolfgang; Mandelstam, Stanley (1979).Variational principles in dynamics and quantum theory. Dover books on physics and chemistry (Republ. of the 3rd ed., publ. in 1968 ed.). New York, NY: Dover Publ.ISBN 978-0-486-63773-0.
^^a ^b ^cAnalytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008,ISBN 978-0-521-57572-0
^Encyclopaedia of Physics (2nd Edition),R.G. Lerner, G.L. Trigg, VHC publishers, 1991,ISBN 3-527-26954-1 (Verlagsgesellschaft),ISBN 0-89573-752-3 (VHC Inc.)
^^a ^bThe Road to Reality, Roger Penrose, Vintage books, 2007,ISBN 0-679-77631-1
^T. W. B. Kibble,Classical Mechanics, European Physics Series, McGraw-Hill (UK), 1973,ISBN 0-07-084018-0
^^a ^b ^cGoldstein, Herbert; Poole, Charles P.; Safko, John L. (2008).Classical mechanics (3, [Nachdr.] ed.). San Francisco Munich: Addison Wesley.ISBN 978-0-201-65702-9.
^L. D. Landau and E. M. Lifshitz (1971).The Classical Theory of Fields. Addison-Wesley. Sec. 8. p. 24–25.
^^a ^bQuantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,ISBN 978-0-13-146100-0

Further reading

The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,ISBN 978-0-521-57507-2.
Dare A. Wells, Lagrangian Dynamics, Schaum's Outline Series (McGraw-Hill, 1967)ISBN 0-07-069258-0, A 350-page comprehensive "outline" of the subject.

External links

Principle of least action interactive Interactive explanation/webpage

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Classical mechanics SI units

Dimensions	1	L	L²		Dimensions	1	θ	θ²
Linear/translational quantities					Angular/rotational quantities
T	time:t s	absement:A m s			T	time:t s
1		distance:d,position:r,s,x,displacement m	area:A m²		1		angle:θ,angular displacement:θ rad	solid angle:Ω rad², sr
T⁻¹	frequency:f s⁻¹,Hz	speed:v,velocity:v m s⁻¹	kinematic viscosity:ν, specific angular momentum: h m² s⁻¹		T⁻¹	frequency:f,rotational speed:n,rotational velocity:n s⁻¹,Hz	angular speed:ω,angular velocity:ω rad s⁻¹
T⁻²		acceleration:a m s⁻²			T⁻²	rotational acceleration s⁻²	angular acceleration:α rad s⁻²
T⁻³		jerk:j m s⁻³			T⁻³		angular jerk:ζ rad s⁻³
M	mass:m kg	weighted position:M ⟨x⟩ = ∑mx	moment of inertia: I kg m²		ML
MT⁻¹	Mass flow rate: ${\dot {m}}$ kg s⁻¹	momentum:p,impulse:J kg m s⁻¹,N s	action:𝒮,actergy:ℵ kg m² s⁻¹,J s		MLT⁻¹		angular momentum:L,angular impulse:ΔL kg m rad s⁻¹
MT⁻²		force:F,weight:F_g kg m s⁻²,N	energy:E,work:W,Lagrangian:L kg m² s⁻²,J		MLT⁻²		torque:τ,moment:M kg m rad s⁻²,N m
MT⁻³		yank:Y kg m s⁻³, N s⁻¹	power:P kg m² s⁻³, W		MLT⁻³		rotatum:P kg m rad s⁻³, N m s⁻¹

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