In physics, the action is a particular quantity in a physical system that can be used
to describe its operation in an alternative manner to the usual differential equation approach. The action is not necessarily the
same for different types of system.
The contemporary action approach for physical systems yields the same results as those found using differential
equations to describe the system, but only requires the states of the physical variable
to be specified at two points, called the initial and final states. The values of the physical variable at all intermediate
points may then be determined by 'minimizing' the action.
History of term 'action'
The term "action" was defined in several (now obsolete) ways during its development.
- Gottfried Leibniz, Johann Bernoulli and
Pierre Louis Maupertuis defined the "action" for light as the integral of its speed (or inverse speed) along its path length.
- Leonhard Euler (and, possibly, Leibniz) defined it for a material particle as the
integral of the particle speed along its path through space.
- Maupertuis 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 strange hybrid that ensured
conservation of momentum in collisions.
Concepts
Physical laws are most often expressed as differential equations, which specify
how a physical variable changes from its present value with infinitesimally small changes in time or position or some
other variable. By adding up these small changes, a differential equation provides a recipe for determining the value of the
physical variable at any point, given only its starting value at one point and possibly some initial derivatives.
The action takes a different but equivalent approach that yields the same results as the differential equation but only
requires the states of the physical variable to be specified at two points, called the initial and final states. The values of
the physical variable at all intermediate points may then be determined by 'minimizing' the action.
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.
Mathematical definition
Expressed in mathematical language,using the calculus of variations, the
evolution of a physical system (i.e. how the system actually progresses from one state to
another) corresponds to an extremum (usually, a minimum) of the action.
Several different definitions of 'the action' are in common use in physics:
- The action is usually an integral over time. But for action pertaining 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 evolution of a physical system between two states is determined by requiring the action be minimized or, more
generally, be stationary for small perturbations about the true evolution. This
requirement leads to differential equations that describe the true evolution.
Disambiguation of "action" in classical physics
In classical physics, the term action has at least eight
distinct meanings.
Action (functional)
Most commonly, the term is used for a functional 
which takes a function of time and (for fields) space as input and
returns a scalar. Specifically, in classical
mechanics, the input function is the evolution 
of the system between two time points
t1 and t2, where 
represent the generalized coordinates. The action ![\mathcal{S}[\mathbf{q}(t)]](http://content.answers.com/main/content/wp/en/math/b/3/c/b3c291e94128405fd44791176583e4d1.png)
is defined as the
integral of the Lagrangian L for an input evolution between the two time points
![\mathcal{S}[\mathbf{q}(t)] = \int_{t_1}^{t_2} L[\mathbf{q}(t),\dot{\mathbf{q}}(t),t]\, \mathrm{d}t](http://content.answers.com/main/content/wp/en/math/e/d/1/ed1d2492f39299a56130de7a20819be1.png)
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.
Abbreviated action (functional)
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.
Hamilton's principal function
Hamilton's principal function is defined by the Hamilton–Jacobi equations
(HJE), another alternative formulation of classical mechanics. This function
S is related to the functional by fixing the initial time t1 and endpoint 
and allowing the upper limits
t2 and the second endpoint 
to vary; these variables are the
arguments of the function S. In
other words, the action function S is the indefinite
integral of the Lagrangian with respect to time.
Hamilton's characteristic function
When total energy E is conserved, the HJE can be solved with the time-independent function
, which is called Hamilton's characteristic function. (See
Hamilton–Jacobi equations: Separation of variables.)
Other solutions of Hamilton–Jacobi equations
The Hamilton–Jacobi equations are often solved by additive separability; in
some cases, the individual terms of the solution, e.g., Sk(qk), are also called an "action".
Action of a generalized coordinate
This is a single variable Jk 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 Jk is called the "action" of the generalized coordinate
qk; the corresponding canonical variable conjugate to Jk is its "angle" wk, for reasons
described more fully under action-angle coordinates. 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.
Action for a Hamiltonian flow
See tautological one-form.
Euler-Lagrange equations for the action integral
As noted above, the requirement that the action integral be stationary under small
perturbations of the evolution is equivalent to a set of differential equations
(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 does not depend on time explicitly. In that case, the action integral can be written
where the initial and final times (t1 and t2) and the final and initial positions are specified in advance as x1 = x(t1) and x2 =
x(t2). Let xtrue(t) represent the true
evolution that we seek, and let xper(t) be a slightly perturbed version of
it, albeit with the same endpoints, xper(t1) =
x1 and xper(t2) = x2.
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
- Failed to parse (unknown function\begin): \begin{align} \delta \mathcal{S} &= \int_{t_1}^{t_2}\;
\left[ L(x_{\mathrm{true}}+\varepsilon,\dot x_{\mathrm{true}} +\dot\varepsilon)- L(x_{\mathrm{true}},\dot x_{\mathrm{true}})
\right]dt \\ &= \int_{t_1}^{t_2}\; \left(\varepsilon{\partial L\over\partial x} + \dot\varepsilon{\partial L\over\partial
\dot x} \right)\,\mathrm{d}t \end{align}
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
Euler-Lagrange
equation
Those familiar with functional analysis will note that the Euler-Lagrange
equations simplify to
.
The quantity 
is called the conjugate momentum for the coordinate x. An important
consequence of the Euler-Lagrange eqations 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 energy
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
- Failed to parse (unknown function\begin): \begin{align} \frac{\mathrm{d}}{\mathrm{d}t} \left(
\frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} &= 0 \qquad \Rightarrow \qquad \ddot{r} -
r\dot{\varphi}^2 &= 0 \\ \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{\partial L}{\partial \dot{\varphi}} \right) -
\frac{\partial L}{\partial \varphi} &= 0 \qquad \Rightarrow \qquad \ddot{\varphi} + \frac{2}{r}\dot{r}\dot{\varphi} &= 0
\end{align}
The solution of these two equations is given by
- Failed to parse (unknown function\begin): \begin{align} r\cos\varphi &= a t + b \\ r\sin\varphi
&= c t + d \end{align}
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 classical fields
The action principle can be extended to obtain the equations of motion for
fields, such as the electromagnetic field 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.
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 imaginable paths and the value of their action. The action corresponding to the various paths is used to calculate
the path integral, that 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. 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.
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 theorem, 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.
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 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
References
For an annotated bibliography, see Edwin F. Taylor [1] who lists, among other things, the following books
- Cornelius Lanczos, The Variational Principles of Mechanics (Dover Publications,
New York, 1986). ISBN 0-486-65067-7. The reference most quoted by all those who explore this field.
- L. D. Landau and E. M. Lifshitz, Mechanics, Course of Theoretical Physics
(Butterworth-Heinenann, 1976), 3rd ed., Vol. 1. ISBN 0-7506-2896-0. Begins with the principle of least action.
- Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia of Physics (Simon & Schuster Macmillan, 1996), Volume
2, ISBN 0-02-897359-3, OCLC 35269891, pages 840 – 842.
- David Morin introduces Lagrange's equations in Chapter 5 of his honors introductory physics text. Concludes with a wonderful
set of 27 problems with solutions. A draft of is available at [2]
- Gerald Jay Sussman and Jack Wisdom, Structure and Interpretation of Classical Mechanics (MIT Press, 2001). Begins with the
principle of least action, uses modern mathematical notation, and checks the clarity and consistency of procedures by programming
them in computer language.
- 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.
- Robert Weinstock, Calculus of Variations, with Applications to Physics and Engineering (Dover Publications, 1974). ISBN
0-486-63069-2. An oldie but goodie, with the formalism carefully defined before use in physics and engineering.
- Wolfgang Yourgrau and Stanley Mandelstam, Variational Principles in Dynamics and Quantum Theory (Dover Publications, 1979). A
nice treatment that does not avoid the philosophical implications of the theory and lauds the Feynman treatment of quantum
mechanics that reduces to the principle of least action in the limit of large mass.
- Edwin F. Taylor's page [3]
- Principle of
least action interactive Excellent interactive explanation/webpage
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