Action (physics)
Encyclopedia
In physics
, action is an attribute of the dynamics of a physical system
. It 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. Action has the dimension of energy
× time
, and its unit is joule-seconds in the International System of Units
(SI). Generally, the action takes different values for different paths. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more strictly, is stationary
. The classical equations of motion of a system can be derived from this principle of least action
. The stationary action formulation of classical mechanics extends to quantum mechanics
in the Feynman path integral formulation, where a physical system follows simultaneously all possible paths with amplitudes determined by the action.
If the action is 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,
the integrand,
, is called the Lagrangian
. For the action integral to be well defined the trajectory has to be bounded in time and space.
s, which specify how a physical quantity varies over infinitesimally small changes in time, position, or other independent variables in its domain. A solution to a differential equation provides the value of the physical variable at any point in its domain (e.g.,
), given both initial conditions and boundary conditions.
In analytical dynamics
, the action represents the final form obtained by working backwards from classical Newtonian mechanics to achieve an integral minimization expression in the form of a variational statement. The statement is profound, simple, and elegant but comes at the cost of several simplifying assumptions. The integral form espoused here can only be applied to conservative holonomic
mechanical systems and to do otherwise can yield incorrect results.
The equivalence of these two approaches is contained in Hamilton's principle
, which states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation
. It applies not only to the classical mechanics
of a single particle, but also to classical fields
such as the electromagnetic
and gravitational fields.
Hamilton's principle has also been extended to quantum mechanics
and quantum field theory
.
, 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.
Several different definitions of 'the action' are in common use in physics:
, the term "action" has at least eight distinct meanings.
which takes a function
of time and (for fields
) space as input and returns a scalar
. In classical mechanics
, the input function is the evolution
of the system between two times 
and 
, where 
represent the generalized coordinates. The action 
is defined as the integral
of the Lagrangian
for an input evolution between the two times
where the endpoints of the evolution are fixed and defined as
and 
. According to Hamilton's principle
, the true evolution
is an evolution for which the action 
is stationary
(a minimum, maximum, or a saddle point
). This principle results in the equations of motion in Lagrangian mechanics
.
Usually denoted as
, this is also a functional
. Here 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
is defined as the integral of the generalized momenta along a path in the generalized coordinates
According to Maupertuis' principle
, the true path is a path for which the abbreviated action
is stationary
.
s (HJE), another alternative formulation of classical mechanics
. This function
is related to the functional 
by fixing the initial time 
and endpoint 
and allowing the upper limits 
and the second endpoint 
to vary; these variables are the arguments
of the function
. In other words, the action function 
is the indefinite integral
of the Lagrangian with respect to time.
is conserved, the HJE
can be solved with the additive separation of variables
,
where the time independent function
is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative
.
This can be integrated to give
,
which is just the abbreviated action.
s are often solved by additive separability; in some cases, the individual terms of the solution, e.g.,
, are also called an "action".
in the action-angle coordinates
, defined by integrating a single generalized momentum around a closed path in phase space
, corresponding to rotating or oscillating motion
The variable
is called the "action" of the generalized coordinate 
; the corresponding canonical variable conjugate to 
is its "angle" 
, for reasons described more fully under action-angle coordinates
. The integration is only over a single variable
and, therefore, unlike the integrated dot product in the abbreviated action integral above. The 
variable equals the change in 
as 
is varied around the closed path. For several physical systems of interest, 
is either a constant or varies very slowly; hence, the variable 
is often used in perturbation calculations and in determining adiabatic invariant
s.
under small perturbations of the evolution is equivalent to a set of differential equation
s (called the Euler–Lagrange equations) that may be determined using the calculus of variations
. We illustrate this derivation here using only one coordinate, x; the extension to multiple coordinates is straightforward.
Adopting Hamilton's principle
, we assume that the Lagrangian L (the integrand of the action integral) depends only on the coordinate x(t) and its time derivative dx(t)/dt, and may also depend explicitly on time. In that case, the action integral can be written
where the initial and final times (
and 
) and the final and initial positions are specified in advance as 
and 
. Let 
represent the true evolution that we seek, and let 
be a slightly perturbed version of it, albeit with the same endpoints, 
and 
. The difference between these two evolutions, which we will call 
, is infinitesimally small at all times
At the endpoints, the difference vanishes, i.e.,
.
Expanded to first order, the difference between the actions integrals for the two evolutions is
Integration by parts
of the last term, together with the boundary conditions
, yields the equation
The requirement that
be stationary
implies that the first-order change
must be zero for any possible perturbation 
about the true evolution. This can be true only if
Those familiar with the concept and notation of a functional derivative
will note that the Euler–Lagrange equation is simply the functional derivative of the action integral:
.
The quantity
is called the
conjugate momentum for the coordinate x. An important consequence of the Euler–Lagrange equations is that if L does not explicitly contain coordinate x, i.e.
In such cases, the coordinate x is called a cyclic coordinate,
and its conjugate momentum is conserved.
in orthonormal (x,y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time, t).
In polar coordinates (r, φ) the kinetic energy and hence the Lagrangian becomes
The radial r and φ components of the Euler–Lagrangian equations become, respectively
The solution of these two equations is given by
for a set of constants a, b, c, d determined by initial conditions.
Thus, indeed, the solution is a straight line given in polar coordinates.
C parametrized by the proper time
is
.
If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes
where the Lagrangian
is
.
or gravity
.
The Einstein equation utilizes the Einstein-Hilbert action
as constrained by a variational principle
.
The path of a body in a gravitational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
, that gives the probability amplitude
s 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. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman
's path integral formulation
of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations
can be derived as conditions of stationary action.
, which states that to every continuous symmetry
in a physical situation there corresponds a conservation law
(and conversely). This deep connection requires that the action principle be assumed.
s 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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, action is an attribute of the dynamics of a physical system
Physical system
In physics, the word system has a technical meaning, namely, it is the portion of the physical universe chosen for analysis. Everything outside the system is known as the environment, which in analysis is ignored except for its effects on the system. The cut between system and the world is a free...
. It is a mathematical functional
Functional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
which takes the trajectory
Trajectory
A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...
, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
× time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, and its unit is joule-seconds in the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...
(SI). Generally, the action takes different values for different paths. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or, more strictly, is stationary
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
. The classical equations of motion of a system can be derived from this principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
. The stationary action formulation of classical mechanics extends to quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
in the Feynman path integral formulation, where a physical system follows simultaneously all possible paths with amplitudes determined by the action.
If the action is 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,
the integrand,
, is called the LagrangianLagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
. For the action integral to be well defined the trajectory has to be bounded in time and space.
History
Action was defined in several, now obsolete, ways during the development of the concept.- Gottfried LeibnizGottfried LeibnizGottfried Wilhelm Leibniz was a German philosopher and mathematician. He wrote in different languages, primarily in Latin , French and German ....
, Johann BernoulliJohann BernoulliJohann Bernoulli was a Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family...
and Pierre Louis MaupertuisPierre Louis MaupertuisPierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great....
defined the action for lightLightLight or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
as the integral of its speed or inverse speed along its path length. - Leonhard EulerLeonhard EulerLeonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...
(and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space. - Pierre Louis MaupertuisPierre Louis MaupertuisPierre-Louis Moreau de Maupertuis was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Berlin Academy of Science, at the invitation of Frederick the Great....
introduced several ad hoc and contradictory definitions of action within a single article, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.
Concepts
Physical laws are most often expressed as differential equationDifferential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s, which specify how a physical quantity varies over infinitesimally small changes in time, position, or other independent variables in its domain. A solution to a differential equation provides the value of the physical variable at any point in its domain (e.g.,
), given both initial conditions and boundary conditions.In analytical dynamics
Analytical dynamics
In classical mechanics, analytical dynamics, or more briefly dynamics, is concerned about the relationship between motion of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies...
, the action represents the final form obtained by working backwards from classical Newtonian mechanics to achieve an integral minimization expression in the form of a variational statement. The statement is profound, simple, and elegant but comes at the cost of several simplifying assumptions. The integral form espoused here can only be applied to conservative holonomic
Holonomic
In mathematics and physics, the term holonomic may occur with several different meanings.-Holonomic basis:A holonomic basis for a manifold is a set of basis vectors ek for which all Lie derivatives vanish:[e_j,e_k]=0 \,...
mechanical systems and to do otherwise can yield incorrect results.
The equivalence of these two approaches is contained in Hamilton's principle
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, which states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
. It applies not only to the classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
of a single particle, but also to classical fields
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
such as the electromagnetic
Electromagnetism
Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...
and gravitational fields.
Hamilton's principle has also been extended to quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
.
Mathematical definition
Expressed in mathematical language, using the calculus of variationsCalculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
, the evolution
Time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state . In this formulation, time is not required to be a continuous parameter, but may be discrete or even finite. In classical physics, time evolution of a collection of rigid bodies...
of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point
Stationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
(usually, a minimum) of the action.
Several different definitions of 'the action' are in common use in physics:
- The action is usually an integralIntegralIntegration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
over time. But for action pertaining to fieldsField (physics)In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
, 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 evolution of a physical system between two states is determined by requiring the action be minimized or, more generally, be stationaryStationary pointIn mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
for small perturbations about the true evolution. This requirement leads to differential equations that describe the true evolution.
- Conversely, an action principle is a method for reformulating differentialDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
equations of motion for a physical system as an equivalent integral equationIntegral equationIn mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
. Although several variants have been defined (see below), the most commonly used action principle is Hamilton's principleHamilton's principleIn physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
.
- An earlier, less informative action principle is Maupertuis' principleMaupertuis' principleIn classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action...
, which is sometimes called by its (less correct) historical name, the principle of least actionPrinciple of least actionIn physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
.
Disambiguation of action in classical physics
In classical physicsClassical physics
What "classical physics" refers to depends on the context. When discussing special relativity, it refers to the Newtonian physics which preceded relativity, i.e. the branches of physics based on principles developed before the rise of relativity and quantum mechanics...
, the term "action" has at least eight distinct meanings.
Action (functional)
Most commonly, the term is used for a functionalFunctional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
which takes a functionFunction (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of time and (for fields
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
) space as input and returns a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
. In classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, the input function is the evolution
of the system between two times and , where represent the generalized coordinates. The action is defined as the integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
of the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
for an input evolution between the two timeswhere the endpoints of the evolution are fixed and defined as
and . According to Hamilton's principleHamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, the true evolution
is an evolution for which the action is stationaryStationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
(a minimum, maximum, or a saddle point
Saddle point
In mathematics, a saddle point is a point in the domain of a function that is a stationary point but not a local extremum. The name derives from the fact that in two dimensions the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
). This principle results in the equations of motion in Lagrangian mechanics
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
.
Abbreviated action (functional)
Usually denoted as
, this is also a functionalFunctional (mathematics)
In mathematics, and particularly in functional analysis, a functional is a map from a vector space into its underlying scalar field. In other words, it is a function that takes a vector as its input argument, and returns a scalar...
. Here 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
is defined as the integral of the generalized momenta along a path in the generalized coordinatesGeneralized coordinates
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....
According to Maupertuis' principle
Maupertuis' principle
In classical mechanics, Maupertuis' principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. It is a special case of the more generally stated principle of least action...
, the true path is a path for which the abbreviated action
is stationaryStationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
.
Hamilton's principal function
Hamilton's principal function is defined by the Hamilton–Jacobi equationHamilton–Jacobi equation
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
s (HJE), another alternative formulation of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. This function
is related to the functional by fixing the initial time and endpoint and allowing the upper limits and the second endpoint to vary; these variables are the argumentsIndependent variable
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...
of the function
. In other words, the action function is the indefinite integralAntiderivative
In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite integralof a function f is a function F whose derivative is equal to f, i.e., F ′ = f...
of the Lagrangian with respect to time.
Hamilton's characteristic function
When the total energy is conserved, the HJEHamilton–Jacobi equation
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
can be solved with the additive separation of variables
,where the time independent function
is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative.This can be integrated to give
,which is just the abbreviated action.
Other solutions of Hamilton–Jacobi equations
The Hamilton–Jacobi equationHamilton–Jacobi equation
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
s are often solved by additive separability; in some cases, the individual terms of the solution, e.g.,
, are also called an "action".Action of a generalized coordinate
This is a single variable in the action-angle coordinatesAction-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 frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates...
, defined by integrating a single generalized momentum around a closed path in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, corresponding to rotating or oscillating motion
The variable
is called the "action" of the generalized coordinate ; the corresponding canonical variable conjugate to is its "angle" , for reasons described more fully under action-angle coordinatesAction-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 frequencies of oscillatory or rotational motion without solving the equations of motion. Action-angle coordinates...
. The integration is only over a single variable
and, therefore, unlike the integrated dot product in the abbreviated action integral above. The variable equals the change in as is varied around the closed path. For several physical systems of interest, is either a constant or varies very slowly; hence, the variable is often used in perturbation calculations and in determining adiabatic invariantAdiabatic invariant
An adiabatic invariant is a property of a physical system that stays constant when changes occur slowly.In thermodynamics, an adiabatic process is a change that occurs without heat flow, and slowly compared to the time to reach equilibrium. In an adiabatic process, the system is in equilibrium at...
s.
Euler–Lagrange equations for the action integral
As noted above, the requirement that the action integral be stationaryStationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
under small perturbations of the evolution is equivalent to a set of differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s (called the Euler–Lagrange equations) that may be determined using the calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
. We illustrate this derivation here using only one coordinate, x; the extension to multiple coordinates is straightforward.
Adopting Hamilton's principle
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action...
, we assume that the Lagrangian L (the integrand of the action integral) depends only on the coordinate x(t) and its time derivative dx(t)/dt, and may also depend explicitly on time. In that case, the action integral can be written
where the initial and final times (
and ) and the final and initial positions are specified in advance as and . Let represent the true evolution that we seek, and let be a slightly perturbed version of it, albeit with the same endpoints, and . The difference between these two evolutions, which we will call , is infinitesimally small at all timesAt the endpoints, the difference vanishes, i.e.,
.Expanded to first order, the difference between the actions integrals for the two evolutions is
Integration by parts
Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...
of the last term, together with the boundary conditions
, yields the equationThe requirement that
be stationaryStationary point
In mathematics, particularly in calculus, a stationary point is an input to a function where the derivative is zero : where the function "stops" increasing or decreasing ....
implies that the first-order change
must be zero for any possible perturbation about the true evolution. This can be true only ifThose familiar with the concept and notation of a functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
will note that the Euler–Lagrange equation is simply the functional derivative of the action integral:
.The quantity
is called theconjugate momentum for the coordinate x. An important consequence of the Euler–Lagrange equations is that if L does not explicitly contain coordinate x, i.e.
- if
, then 
is constant.
In such cases, the coordinate x is called a cyclic coordinate,
and its conjugate momentum is conserved.
Example: Free particle in polar coordinates
Simple examples help to appreciate the use of the action principle via the Euler–Lagrangian equations. A free particle (mass m and velocity v) in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energyin orthonormal (x,y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time, t).
In polar coordinates (r, φ) the kinetic energy and hence the Lagrangian becomes
The radial r and φ components of the Euler–Lagrangian equations become, respectively
The solution of these two equations is given by
for a set of constants a, b, c, d determined by initial conditions.
Thus, indeed, the solution is a straight line given in polar coordinates.
Action principle for single relativistic particle
When relativistic effects are significant, the action of a point particle of mass m travelling a world lineWorld line
In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. The concept of "world line" is distinguished from the concept of "orbit" or "trajectory" by the time dimension, and typically encompasses a large area of spacetime wherein...
C parametrized by the proper time
Proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
is.If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes
where the Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
is
.Action principle for classical fields
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic fieldElectromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
or gravity
Einstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
.
The Einstein equation utilizes the Einstein-Hilbert action
Einstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
as constrained by a variational principle
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
.
The path of a body in a gravitational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.
Action principle in quantum mechanics and 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 integralPath integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
, that gives the probability amplitude
Probability amplitude
In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density.For example, if the probability amplitude of a quantum state is \alpha, the probability of measuring that state is |\alpha|^2...
s 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. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
's path integral formulation
Path integral formulation
The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics...
of quantum mechanics is based on a stationary-action principle, using path integrals. Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
can be derived as conditions of stationary action.
Action principle and conservation laws
Symmetries in a physical situation can better be treated with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theoremNoether's theorem
Noether's theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918...
, which states that to every continuous symmetry
Continuous symmetry
In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e.g. reflection symmetry, which is invariance under a kind of flip from one state to another. It has largely and successfully been formalised in the...
in a physical situation there corresponds a conservation law
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....
(and conversely). This deep connection requires that the action principle be assumed.
Modern extensions of the action principle
The action principle can be generalized still further. For example, the action need not be an integral because nonlocal actionNonlocality
In Classical physics, nonlocality is the direct influence of one object on another, distant object. In Quantum mechanics, nonlocality refers to the absence of a local, realist model in agreement with quantum mechanical predictions.Nonlocality may refer to:...
s are possible. The configuration space need not even be a functional space given certain features such as noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
. However, a physical basis for these mathematical extensions remains to be established experimentally.