In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, action is a
scalar quantity describing how 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 ...
has
changed over time. Action is significant because the
equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
of the system can be derived through 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 ...
.
In the simple case of a single particle moving with a constant velocity (
uniform linear motion), 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 ...
of the particle times the distance it moves,
added up along its path; equivalently, action is twice the particle's
kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
times the duration for which it has that amount of energy. For more complicated systems, all such quantities are combined.
More formally, action is a
mathematical functional which takes the
trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
(also called path or history) of the system as its argument and has a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
as its result. Generally, the action takes different values for different paths.
Action has
dimensions of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
×
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
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 ...
×
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 Inte ...
, and its
SI unit is
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
-second (like the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
''h'').
Introduction
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.
It applies not only to the
classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
of a single particle, but also to
classical field
In physics, a field is a physical quantity, represented by a scalar, vector, or tensor, that has a value for each point in space and time. For example, on a weather map, the surface temperature is described by assigning a number to each poin ...
s 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 str ...
fields. Hamilton's principle has also been extended to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
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
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.
[Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ]
Solution of differential equation
Empirical laws are frequently expressed as
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s, which describe how physical quantities such as
position 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 ...
change
continuously with
time
Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
,
space
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
or a generalization thereof. Given the
initial
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph tha ...
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
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (V ...
''.
Minimization of action integral
''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 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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
, and the classical equations of motion of a system can be derived by minimizing the value of that integral.
This simple principle provides deep insights into physics, and is an important concept in modern
theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experim ...
.
History
''Action'' was defined in several now obsolete ways during the development of the concept.
[Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ]
*
Gottfried Leibniz,
Johann Bernoulli and
Pierre Louis Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
defined the action for light as the integral of its speed or inverse speed along its path length.
*
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
(and, possibly, Leibniz) defined action for a material particle as the integral of the particle's speed along its path through space.
*
Pierre Louis Maupertuis
Pierre Louis Moreau de Maupertuis (; ; 1698 – 27 July 1759) was a French mathematician, philosopher and man of letters. He became the Director of the Académie des Sciences, and the first President of the Prussian Academy of Science, at the ...
introduced several ''ad hoc'' and contradictory definitions of action within a single
article
Article often refers to:
* Article (grammar), a grammatical element used to indicate definiteness or indefiniteness
* Article (publishing), a piece of nonfictional prose that is an independent part of a publication
Article may also refer to:
...
, defining action as potential energy, as virtual kinetic energy, and as a hybrid that ensured conservation of momentum in collisions.
Mathematical definition
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
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
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 action is usually an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over time, taken along the path of the system between the initial time and the final time of the development of the system:
where the integrand ''L'' is called the
Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.
Action has the
dimensions of
">nergynbsp;×
">ime and its
SI unit is
joule
The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
-second, which is identical to the unit of
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
.
Action in classical physics
In
classical physics
Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
, the term "action" has a number of meanings.
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 ...
which takes a
function of time and (for
fields) space as input and returns a
scalar.
[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
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, 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