Action (physics)

In physics, the action principle is a method for reformulating differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. The principle is also called the principle of stationary action and Hamilton's principle. Other names for the principle include the principle of least action and the principle of minimal action. The history of the development of this principle is discussed in the article on the principle of least action.

The differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Hence, 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 fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory.

Action principle in classical physics

In classical physics, the term action can have no less than three distinct (but related) meanings.

The first and foremost meaning of action in classical physics is a functional $\mathcal{S}$ of the system under consideration, i.e., something that takes as its input a function that completely describes the system and its history and returns a single number, a scalar. For example, in classical mechanics, the input function is the trajectory $\mathbf{q}(t)$ of the system between two time points $t_{1}$ and $t_{2}$ , where $\mathbf{q}$ represent the generalized coordinates. In that case, the action is defined as the integral of the Lagrangian for the input trajectory between the two time points

\mathcal{S} \equiv \int_{t_1}^{t_2} L[\mathbf{q}(t),\dot{\mathbf{q}}(t),t \, dt

where the endpoints $\mathbf{q}_{1}\equiv\mathbf{q}(t_{1})$ and $\mathbf{q}_{2}\equiv\mathbf{q}(t_{2})$ of the trajectory are fixed. In Lagrangian mechanics, the physical trajectory is a path $\mathbf{q}_{\mathrm{extremal}}(t)$ for which the action $\mathcal{S}$ ^* is stationary (a minimum, maximum, or a saddle point).

A second meaning of action in classical physics is the function (not functional) $S$ that is used in the Hamilton-Jacobi equations, another alternative to Newton's laws of motion, Lagrangian mechanics and even Hamiltonian mechanics. This function $S$ is related to the functional $\mathcal{S}$ by fixing the initial time $t_{1}$ and endpoint $\mathbf{q}_{1}$ and allowing the upper limits $t_{2}$ and the second endpoint $\mathbf{q}_{2}$ 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. More details on this function and its usefulness can be found at the article about the Hamilton-Jacobi equations.

A third meaning of action in classical physics is a generalized momentum $J_{k}$ in the action-angle coordinates, defined by integrating the action integral around a closed path in phase space, corresponding to rotating or oscillating motion

J_{k} \equiv \oint p_{k} dq_{k}

This variable $J_{k}$ is called the action of variable $q_{k}$ , and the corresponding generalized coordinate $w_{k}$ is called its "angle", for reasons that are described more fully under action-angle coordinates. The integral here is over a single dimension and need not correspond to any physical motion.

Euler-Lagrange equations for the action integral

The stationary point of an integral along a path is equivalent to a set of differential-equations, called the Euler-Lagrange equations. This can be seen as follows where we restrict ourselves to one coordinate only. The extension to more coordinates is straightforward.

Suppose we have an action integral S of an integrand L which depends on coordinates x(t) and dx(t)/dt, its derivative with respect to t:

S = \int_{t_1}^{t_2}\; L(x,\dot{x})\,dt.

Consider a second curve x₁(t) which starts and ends at the same points as the first curve, and assume that the distance between the two curves is small everywhere: ε(t) = x₁(t) - x(t) is small. At the beginning and endpoint we have ε(t₁) = ε(t₂) = 0.

The difference between the integrals along curve one and along curve two is:

dt = \int_{t_1}^{t_2}\; \left(

\varepsilon{\partial L\over\partial x} + \dot\varepsilon{\partial L\over\partial \dot x} \right)\,dt

where we have used the first order expansion of L in ε and ε′. Now use integration by parts on the last term and use the conditions ε(t₁) = ε(t₂) = 0 to find:

\delta S = \int_{t_1}^{t_2}\; \left( \varepsilon{\partial L\over \partial x} - \varepsilon{d\over dt }{\partial L\over\partial \dot x} \right)\,dt.

S reaches a stationary point, i.e. δ S = 0 for each ε. Note that the point could either be a minimum, saddle-point or formally even a maximum. δ S = 0 for each ε if and only if

{\partial L\over\partial x_{a}} - {d\over dt }{\partial L\over\partial \dot{x}_{a}} = 0 Euler-Lagrange equations

Where we have replaced x_a, a = 0,1,2,3 for x, because this must hold for every coordinate. These equations are called the Euler-Lagrange equations for the variational problem. An important simple consequence of these equations is that if L does not explicitly contain coordinate x, i.e.

\frac{\partial L}{\partial x}=0

, then

\frac{\partial L}{\partial\dot x}

is constant.

Then the coordinate x is called a cyclic coordinate, and $\frac{\partial L}{\partial\dot x}$ is called the conjugate momentum, which is conserved. For example, if L does not depend on time, the associated constant of motion (the conjugate momentum) is called the energy. If we use spherical coordinates t, r, φ, θ and L does not depend on φ, the conjugate momentum is the conserved angular momentum.

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

\frac{\delta S}{\delta x_{i}(t)}=0

Example: Free particle in polar coordinates

Trivial 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

\frac{d}{dt} \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{d}{dt} \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.

The solution of these two equations is given by

r\cos\varphi = a t + b

r\sin\varphi = c t + d

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.

References

For an annotated bibliography, see Edwin F. Taylor ^* 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 0-0286457-1, 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 ^*
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 007-069258-0, 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 ^*
Principle of least action interactive Excellent interactive explanation/webpage

Lagrangian mechanics | Hamiltonian mechanics | Calculus of variations

Gourt Home::Gourt :: Action (physics)