Action (physics)

In physics, the action principle is an assertion about the nature of motion from which the trajectory of an object subject to forces can be determined. The path of an object is the one that yields a stationary value for a quantity called the action. Thus, instead of thinking about an object accelerating in response to applied forces, one might think of them picking out the path with a stationary action.

The principle is also called the principle of stationary action and also Hamilton's principle or (less general and in fact incorrect) the principle of least action and the principle of minimal action. The action is a scalar (a number) with the dimension of energy <math>\times<math> time. The principle is a simple, general, and powerful theory for predicting motion in classical mechanics. Extensions of the action principle describe relativistic mechanics, quantum mechanics, electricity and magnetism.

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. Feynman's 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.

Many problems in physics can be represented and solved in the form of an action principle, such as finding the quickest way to run down the beach for reaching a drowning person. Water running downhill seeks the steepest descent, the quickest way down, and water running into a basin distributes itself so that its surface is as low as possible. Light finds the quickest trajectory through an optical system (Fermat's principle of least time). The path of a body in a graviational field (i.e. free fall in space time, a so called geodesic) can be found using the action principle.

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. For example, Noether's theorem which states that with every continuous symmetry in a physical situation there corresponds a conservation law. This deep connection, however, requires that the action principle is assumed.

In classical mechanics (non-relativistic, non-quantum mechanics), the correct choice of the action can be proven from Newton's laws of motion. Conversely, the action principle proves Newton's equation of motion given the correct choice of action. So in classical mechanics the action principle is equivalent to Newton's equation of motion. The use of the action principle often is simpler than the direct application of Newton's equation of motion. The action principle is a scalar theory, with derivations and applications that employ elementary calculus.

History

The principle of least action was first formulated by Maupertuis in 1746 and further developed (from 1748 onwards) by the mathematicians Euler, Lagrange, and Hamilton. Maupertuis arrived at this principle from a feeling that the very perfection of the universe demands a certain economy in nature and is opposed to any needless expenditure of energy. Natural motions must be such as to make some quantity a minimum. It was only necessary to find that quantity, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" or twice what we now call the kinetic energy of the system.

Euler (in "Reflexions sur quelques loix generales de la nature", 1748) adopts the least-action principle, calling the quantity "effort". His expression corresponds to what we would now call potential energy, so that his statement of least action in statics is equivalent to the principle that a system of bodies at rest will adopt a configuration that minimizes total potential energy.

Action principle in classical mechanics

Newton's laws of motion can be stated in various ways. One of them is the Lagrangian formalism, also called Lagrangian mechanics. If we denote the trajectory of a particle as a function of time <math>t<math> as <math>x(t)<math>, with a velocity <math>\dot{x}(t)<math>, then the Lagrangian is a function dependent on these quantities and possibly also explicitly on time:

<math>L(x(t),\dot{x}(t),t)<math>

The action integral <math>S<math> is the integral of the Lagrangian over time between a given starting point <math>x(t_1)<math> at time <math>t_1<math> and a given end point <math>x(t_2)<math> at time <math>t_2<math>

<math> S=\int_{t_1}^{t_2} L(x(t),\dot{x}(t),t)\, dt. <math>

In Lagrangian mechanics, the trajectory of an object is derived by finding the path for which the action integral <math>S<math> is stationary (a minimum or a saddle point). The action integral is a functional (a function depending on a function, in this case <math>x(t)<math>). For a system with conservative forces (forces that can be described in terms of a potential, like the gravitational force and not like friction forces), the choice of a Lagrangian as the kinetic energy minus the potential energy results in the correct laws of Newtionain mechanics (Note that the sum of kinetic and potential energy is the total energy of the system).

Euler-Lagrange equations for the action integral

The stationary point of an integral along a path is equivalent to a set of differental-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 <math>S<math> of an integrand <math>L<math> which depends on coordinates <math>x(t)<math> and <math>\dot{x}(t)<math>, its derivatives with respect to <math>t<math>:

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

Consider a second curve <math>x_1(t)<math> which starts and ends at the same points as the first curve, and assume that the distance between the two curves is small everywhere: <math>\varepsilon(t) = x_1(t)-x(t)<math> is small. At the begin and endpoint we have <math>\varepsilon(t_1)=\varepsilon(t_2) =0<math>.

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

<math> \delta S = \int_{t_1}^{t_2}\; (L(x+\varepsilon,\dot x+\dot\varepsilon)

- L(x,\dot x))dt = \int_{t_1}^{t_2}\; \left( \varepsilon{\partial L\over\partial x} + \dot\varepsilon{\partial L\over\partial \dot x} \right)\,dt <math>

where we have used the first order expansion of <math>L<math> in <math>\varepsilon<math> and <math>\dot\varepsilon<math>. Now use partial integration on the last term and use the conditions <math>\varepsilon (t_1)=\varepsilon (t_2) =0<math> to find:

<math>

<math>

<math>S<math> reaches a stationary point (an extremum), i.e. <math>\delta S =0<math> for each <math>\varepsilon<math>. Note that this is the only requirement: the extremum could either be a minimum, saddle-point or formally even a maximum. <math>\delta S =0<math> for each <math>\varepsilon<math> if and only if

Where we have replaced <math>x^a,\; a=0,1,2,3<math> for <math>x<math>, since this must hold for every coordinate. This set of equations is called the Euler-Lagrange equations for the variational problem. An important simple consequence of these equations is that if <math>L<math> does not explicitly contain coordinate <math>x<math>, i.e.

if <math> {\partial L/\partial x}=0<math> then <math> {\partial L/\partial\dot x}<math> is constant

Such a coordinate <math>x<math> is called a cyclic coordinate, and <math>{\partial L}/{\partial\dot x}<math> is called the conjugate momentum, which is conserved. For example if <math>L<math> does not depend on time, the associated constant of motion (the conjugate momentum) is called the energy. If we use spherical coordinates <math>t,r,\varphi,\theta<math> and <math>L<math> does not depend on <math>\varphi<math>, the conjugate momentum is the conserved angular momentum.

(Those familiar with functional analysis will note that the Euler-Lagrange equations simplify to <math>\frac{\delta}{\delta x^i(t)}S=0<math>.)

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 <math>m<math> and velocity <math>v<math>) 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 <math>\frac{1}{2} mv^2= \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2 \right)<math> in orthonormal <math>(x,y)<math> coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time <math>t<math>). In polar coordinates <math>(r,\varphi)<math> the kinetic energy and hence the Lagrangian becomes

<math>

<math>

The radial <math>r<math> and <math>\phi<math> components of the Euler-Lagrangian equations become, respectively

<math>

<math>

The solution of these two equations is given by

<math> r\cos\varphi = a t + b <math>

<math> r\sin\varphi = c t + d <math>

for a set of constants <math>a, b, c, d<math> determined by initial conditions. Thus, indeed, the solution is a straight line given in polar coordinates.

The formalisms above are valid in classical mechanics in a very restrictive sense of the term. More generally, an action is a functional from the configuration space to the real numbers and in general, it needn't even necessarily be an integral because nonlocal actions are possible. The configuration space needn't even necessarily be a functional space because we could have things like noncommutative geometry.

See also

• Lagrangian
• Lagrangian mechanics
• Noether's theorem
• Hamiltonian mechanics
• Functional derivative
• Functional integral
• path integral formulation
• Quantum physics

Literature

For an annotated bibliography, see Edwin F. Taylor