action (physics)
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In physics, action is a
scalar quantity Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
that describes how the balance of kinetic versus potential energy of a
physical system A physical system is a collection of physical objects. In physics, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the environment. The environment is ignored except for its effects on the ...
changes with trajectory. Action is significant because it is an input to the
principle of stationary action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action similar to the Planck constant, quantum effects are significant. In the simple case of a single particle moving with a constant velocity (thereby undergoing
uniform linear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...
), the action is the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
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 In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
, 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. Generally, the action takes different values for different paths. Action has dimensions of energy ×  time or
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
 × 
length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
, and its SI unit is joule-second (like the Planck constant ''h'').


Introduction

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. However the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students.


Simple example

For a trajectory of a baseball moving in the air on Earth the action 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. :S = \int_^ \left( KE(t) - PE(t)\right) dt The action balances kinetic against potential energy. The kinetic energy of a baseball of mass m is (1/2)mv^2 where v is the velocity of the ball; the potential energy is mgx where g is the gravitational constant. Then the action between t_1 and t_2 is :S = \int_^ \left(\fracm v^2(t) - mg x(t) \right) dt The action value depends upon the trajectory taken by the baseball through x(t) and v(t). This makes the action an input to the powerful
stationary-action principle The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
for classical and for quantum mechanics. Newton's equations of motion for the baseball 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 baseball with an electron: classical mechanics fails but stationary action continues to work. The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases.


Planck's quantum of action

The Planck constant, written as h or \hbar when including a factor of 1/2\pi, is called ''the quantum of action''. Like action, this constant has unit of energy times time. 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.


History

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 The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
. William Rowan Hamilton made the next big breakthrough, formulating Hamilton's principle in 1853. Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles.


Definitions

Expressed in mathematical language, using the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action. Action has the dimensions of nergynbsp;×  ime and its SI unit is joule-second, which is identical to the unit of angular momentum. Several different definitions of "the action" are in common use in physics.Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, The action is usually an integral over time. However, when the action pertains to fields, 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 an integral over time, taken along the path of the system between the initial time and the final time of the development of the system: \mathcal = \int_^ L \, dt, where the integrand ''L'' is called the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
. 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 a
functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...
\mathcal which takes a function of time and (for fields) space as input and returns a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
.The Road to Reality, Roger Penrose, Vintage books, 2007, T. W. B. Kibble, ''Classical Mechanics'', European Physics Series, McGraw-Hill (UK), 1973, In classical mechanics, the input function is the evolution q(''t'') of the system between two times ''t''1 and ''t''2, where q represents the generalized coordinates. The action \mathcal mathbf(t)/math> is defined as the integral of the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
''L'' for an input evolution between the two times: \mathcal mathbf(t)= \int_^ L(\mathbf(t),\dot(t),t)\, dt, where the endpoints of the evolution are fixed and defined as \mathbf_ = \mathbf(t_) and \mathbf_ = \mathbf(t_). According to Hamilton's principle, the true evolution qtrue(''t'') is an evolution for which the action \mathcal mathbf(t)/math> is
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
(a minimum, maximum, or a
saddle point In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function ...
). This principle results in the equations of motion in Lagrangian mechanics.


Abbreviated action (functional)

In addition to the action functional, there is another functional called the ''abbreviated action''. In the abbreviated action, the input function is the ''path'' 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_ (sometime written as W) is defined as the integral of the generalized momenta, p_i = \frac, for a system Lagrangian L along a path in the generalized coordinates q_i: \mathcal_0 = \int_^ \mathbf \cdot d\mathbf = \int_^ \Sigma_i p_i \,dq_i. where q_1 and q_2 are the starting and ending coordinates. According to
Maupertuis' 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''). It is a special case of ...
, the true path of the system is a path for which the abbreviated action is
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
.


Hamilton's characteristic function

When the total energy ''E'' is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables: S(q_1, \dots, q_N, t) = W(q_1, \dots, q_N) - E \cdot t, where the time-independent function ''W''(''q''1, ''q''2, ..., ''qN'') is called ''Hamilton's characteristic function''. The physical significance of this function is understood by taking its total time derivative \frac = \frac \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 the abbreviated action.


Action of a generalized coordinate

A variable ''Jk'' in the
action-angle coordinates In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion witho ...
, called the "action" of the generalized coordinate ''qk'', is defined by integrating a single generalized momentum around a closed path in
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually ...
, corresponding to rotating or oscillating motion: J_k = \oint p_k \,dq_k The corresponding canonical variable conjugate to ''Jk'' is its "angle" ''wk'', for reasons described more fully under
action-angle coordinates In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. The method of action-angles is useful for obtaining the frequency, frequencies of oscillatory or rotational motion witho ...
. The integration is only over a single variable ''qk'' and, therefore, unlike the integrated dot product in the abbreviated action integral above. The ''Jk'' variable equals the change in ''Sk''(''qk'') as ''qk'' is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable ''Jk'' is often used in perturbation calculations and in determining adiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.


Single relativistic particle

When relativistic effects are significant, the action of a point particle of mass ''m'' travelling a world line ''C'' parametrized by the proper time \tau is S = - m c^2 \int_ \, d \tau. If instead, the particle is parametrized by the coordinate time ''t'' of the particle and the coordinate time ranges from ''t''1 to ''t''2, then the action becomes S = \int_^ L \, dt, where the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
isL. D. Landau and E. M. Lifshitz (1971). ''The Classical Theory of Fields''. Addison-Wesley. Sec. 8. p. 24–25. L = -mc^2 \sqrt.


Action principles and related ideas

Physical laws are frequently expressed as differential equations, which describe how physical quantities such as
position Position often refers to: * Position (geometry), the spatial location (rather than orientation) of an entity * Position, a job or occupation Position may also refer to: Games and recreation * Position (poker), location relative to the dealer * ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass an ...
change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called '' equations 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 the ''action is minimized'', or more generally, is
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
. In other words, the action satisfies a variational principle: the
principle of stationary action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
(see also below). The action is defined by an integral, 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 modern theoretical physics. Various action principles and related concepts are summarized below.


Maupertuis's principle

In classical mechanics,
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''). It is a special case of ...
(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 the abbreviated action between two generalized points on a path.


Hamilton's principal function

Hamilton's principle states that the differential equations of motion for ''any'' physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models. Hamilton's principle applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and
gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
fields. Hamilton's principle has also been extended to quantum mechanics and
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
—in particular the path 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.Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,


Hamilton–Jacobi equation

Hamilton's principal function S=S(q,t;q_0,t_0) is obtained from the action functional \mathcal by fixing the initial time t_0 and the initial endpoint q_0, while allowing the upper time limit t and the second endpoint q to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.


Euler–Lagrange equations

In Lagrangian mechanics, the requirement that the action integral be
stationary In addition to its common meaning, stationary may have the following specialized scientific meanings: Mathematics * Stationary point * Stationary process * Stationary state Meteorology * A stationary front is a weather front that is not moving ...
under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.


Classical fields

The action principle can be extended to obtain the equations of motion for fields, such as the
electromagnetic field An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
or
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
. Maxwell's equations can be derived as conditions of stationary action. The
Einstein equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...
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. For a free falling body, this trajectory is a
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
.


Conservation laws

Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from 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 In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...
(and conversely). This deep connection requires that the action principle be assumed.


Path integral formulation of quantum field theory

In quantum 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 the path integral, which gives the probability amplitudes of the various outcomes. 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. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.


Modern extensions

The action principle can be generalized still further. For example, the action need not be an integral, because nonlocal actions are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally.


See also

*
Calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
*
Functional derivative In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on ...
*
Functional integral Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential ...
* Hamiltonian mechanics *
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
* Lagrangian mechanics *
Measure (physics) The measure in quantum physics is the integration measure used for performing a path integral. In quantum field theory, one must sum over all possible histories of a system. When summing over possible histories, which may be very similar to each ...
* Noether's theorem * Path integral formulation * Principle of least action *
Principle of maximum entropy The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition ...
* Some actions: **
Nambu–Goto action The Nambu–Goto action is the simplest invariant action (physics), action in bosonic string theory, and is also used in other theories that investigate string-like objects (for example, cosmic strings). It is the starting point of the analysis of ...
** Polyakov action **
Bagger–Lambert–Gustavsson action In theoretical physics, in the context of M-theory, the action for the '' N''=8 M2 branes in full is (with some indices hidden): : S = \intd\sigma^3 where is a generalisation of a Lie bracket which gives the group constants. The only known com ...
** Einstein–Hilbert action


References


Further reading

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


External links


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