Action (physics)

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In physics, the action is a particular quantity in a physical system that can be used to describe its operation. Action is an alternative to differential equations. The action is not necessarily the same for different types of systems.

The action yields the same results as using differential equations. Action 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.

Contents

History of term 'action'

The term "action" was defined in several (now obsolete) ways during its development.

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, 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 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:

Disambiguation of "action" in classical physics

In classical physics, the term "action" has at least eight distinct meanings.

Action (functional)

\mathcal{S}

\mathcal{S}[\mathbf{q}(t)] = \int_{t_1}^{t_2} L[\mathbf{q}(t),\dot{\mathbf{q}}(t),t]\, \mathrm{d}t
\mathbf{q}_{1} = \mathbf{q}(t_{1})

Abbreviated action (functional)

\mathcal{S}_{0}

\mathcal{S}_{0} = \int \mathbf{p} \cdot \mathrm{d}\mathbf{q} = \int p_i \,dq_i
\mathcal{S}_{0}

Hamilton's principal function

Main article: Hamilton's principal function
\mathcal{S}

Hamilton's characteristic function

When the total energy E is conserved, the HJE 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
W(q_{1},\dots,q_{N})
\frac{d W}{d t}= \frac{\partial W}{\partial q_i}\dot q_i=p_i\dot q_i

This can be integrated to give

W(q_{1},\dots,q_{N}) = \int p_i\dot q_i \,dt = \int p_i\,dq_i

which is just the abbreviated action.

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


J_{k} = \oint p_{k} \mathrm{d}q_{k}

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


\mathcal{S} = \int_{t_1}^{t_2}\; L(x,\dot{x})\,\mathrm{d}t
\varepsilon(t)

\varepsilon(t) = x_{\mathrm{per}}(t) - x_{\mathrm{true}}(t)
\varepsilon(t_{1}) = \varepsilon(t_{2}) = 0

Expanded to first order, the difference between the actions integrals for the two evolutions is

\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}
\varepsilon(t_{1}) = \varepsilon(t_{2}) = 0

\delta \mathcal{S} = 
\int_{t_1}^{t_2}\; 
\left(
\varepsilon{\partial L\over \partial x} -
\varepsilon{d\over dt }{\partial L\over\partial \dot x} 
\right)\,\mathrm{d}t.
\mathcal{S}
 
{\partial L\over\partial x} - {\mathrm{d}\over \mathrm{d}t }{\partial L\over\partial
\dot{x}} = 0
   Euler–Lagrange equation

Those familiar with functional analysis will note that the Euler–Lagrange equations simplify to

\frac{\delta \mathcal{S}}{\delta x(t)}=0
\frac{\partial L}{\partial\dot x}
 \frac{\partial L}{\partial x}=0

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

\frac{1}{2} mv^2= \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2 \right)

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


L = \frac{1}{2}m \left( \dot{r}^2 + r^2\dot\varphi^2 \right).

The radial r and φ components of the Euler–Lagrangian equations become, respectively

\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

\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 single relativistic particle

When relativistic effects are significant, the action of a point particle of mass m traveling a world line C parameterized by the proper time τ is

S = - m c^2 \int_{C} \, d \tau

If instead, the particle is parameterized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes

\int_{t1}^{t2} L \, dt

where the Lagrangian is

L = - m c^2 \sqrt {1 - \frac{v^2}{c^2}}

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.