NationMaster - Encyclopedia: Action (physics)

In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. Several different definitions of the action are in common use in physics, as described below. The action is usually an integral over time, but may be integrated over spatial variables as well (for action pertaining to fields); in still other cases, the action is integrated along the path followed by the physical system. The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ...

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. Conversely, an action principle is a method for reformulating differential equations of motion for a physical system as an equivalent integral equation. Although several variants have been defined (see below), the most commonly used action principle is Hamilton's principle. An earlier, less informative action principle is Maupertuis' principle, which is sometimes called by its (less correct) historical name, the principle of least action. Stationary points (red pluses) and inflection points (green circles). ... An illustration of a differential equation. ... In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... In physics, Hamiltons principle is an alternative formulation of the differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. ... In classical mechanics, Maupertuis principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ...

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. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Electromagnetism is the force observed as static electricity, and causes the flow of electric charge (electric current) in electrical conductors. ... Gravity is a force of attraction that acts between bodies that have mass. ... Fig. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ...

Disambiguation of "action" in classical physics

In classical physics, the term action has at least eight distinct meanings. Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ...

Action (functional)

Most commonly, the term is used for a functional $mathcal{S}$ 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 $mathbf{q}(t)$ of the system between two time points

t 1

and

t 2

, where $mathbf{q}$ represent the generalized coordinates. The action $mathcal{S}[mathbf{q}(t)]$ is defined as the integral of the Lagrangian

L

for an input evolution between the two time points In mathematics, the term functional is applied to certain functions. ... Partial plot of a function f. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ... In calculus, the integral of a function is an extension of the concept of a sum. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

where the endpoints of the evolution are fixed and defined as $mathbf{q}_{1} = mathbf{q}(t_{1})$ and $mathbf{q}_{2} = mathbf{q}(t_{2})$ . According to Hamilton's principle, the true evolution $mathbf{q}_{mathrm{true}}(t)$ is an evolution for which the action $mathcal{S}[mathbf{q}(t)]$ is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics. In physics, Hamiltons principle is an alternative formulation of the differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. ... Stationary points (red pluses) and inflection points (green circles). ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ...

Abbreviated action (functional)

Usually denoted as $mathcal{S}_{0}$ , 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 $mathcal{S}_{0}$ is defined as the integral of the generalized momenta along a path in the generalized coordinates Generally, functional refers to something with and able to fulfill its purpose or function. ... In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. ...

According to Maupertuis' principle, the true path is a path for which the abbreviated action $mathcal{S}_{0}$ is stationary. In classical mechanics, Maupertuis principle is an integral equation that determines the path followed by a physical system without specifying the time parameterization of that path. ... Stationary points (red pluses) and inflection points (green circles). ...

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 $mathcal{S}$ 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. In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In an experimental design, the independent variable (also known as predictor or regressor or manipulated variable) is the variable which is manipulated or selected by the experimenter to determine its relationship to an observed phenomenon (the dependent variable). ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...

Hamilton's characteristic function

When total energy

E

is conserved, the HJE can be solved with the time-independent function $W(q_{1},dots,q_{N}) = S(q_{1},dots,q_{N},t) - Ecdot t$ , which is called Hamilton's characteristic function. (See Hamilton-Jacobi equations: Separation of variables.) In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ...

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.,

S k (q k)

, are also called an "action". In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ...

Action of a generalized coordinate

This is a single variable

J k

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 In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ... Phase space of a dynamical system with focal stability. ...

The variable

J k

is called the "action" of the generalized coordinate

q k

; the corresponding canonical variable conjugate to

J k

is its "angle"

w k

, for reasons described more fully under action-angle coordinates. NB! The integration is only over a single variable

q k

and, therefore, unlike the integrated dot product in the abbreviated action integral above. The

J k

variable equals the change in

S k (q k)

q k

is varied around the closed path. For several physical systems of interest,

J k

is either a constant or varies very slowly; hence, the variable

J k

is often used in perturbation calculations and in determining adiabatic invariants. In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems. ... An adiabatic invariant in general is a property of motion which is conserved to exponential accuracy in the small parameter representing the typical rate of change of the gross properties of the body. ...

Action for a Hamiltonian flow

See tautological one-form. In mathematics, the tautological one-form is a special 1-form defined on symplectic manifolds that plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. ...

Historical uses

The term "action" was defined in several 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. This article is 82 kilobytes or more in size. ... Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... Pierre Louis Maupertuis, here wearing lapmudes or a fur coat from his Lapland expedition. ... Hahahahahahahahahahahahhahahahahahahahahahahahahahahahahahahahahahhahaahahhhahhhahaahhhahahahahahahahahahahahahahahahahahahahahahahahahahahahhahahahahahaahhaaahahahahahahahahahahahahhahahahahahahahhahahahahahahahahahahahahahahahahhahahahahahhahahahhahahahhahahahahahhahahahhahahahhahahahhahahahahahahahahahhahahaahhaahahhaahhahaahahahahhahaahahsahahahahahshhsshshshagagaggagagagagagaggagagahahahahhahahahahahahahahahahahahhahahahahahahahahhahahahahahahahahahahyhahahahahhahaahahahahahahahahahahhahahahaah! your moma is a hoe. ... Euler redirects here. ...

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. Stationary points (red pluses) and inflection points (green circles). ... An illustration of a differential equation. ... Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...

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 In physics, Hamiltons principle is an alternative formulation of the differential equations of motion for a physical system as an equivalent integral equation, using the calculus of variations. ...

where the initial and final times (

t 1

and

t 2

) and the final and initial positions are specified in advance as

x 1 = x (t 1)

and

x 2 = x (t 2)

. Let

x true (t)

represent the true evolution that we seek, and let

x per (t)

be a slightly perturbed version of it, albeit with the same endpoints,

x per (t 1) = x 1

and

x per (t 2) = x 2

. The difference between these two evolutions, which we will call $varepsilon(t)$ , is infinitesimally small at all times

At the endpoints, the difference vanishes, i.e., $varepsilon(t_{1}) = varepsilon(t_{2}) = 0$ .

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

Integration by parts of the last term, together with the boundary conditions $varepsilon(t_{1}) = varepsilon(t_{2}) = 0$ , yields the equation In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

The requirement that $mathcal{S}$ be stationary implies that the first-order change $deltamathcal{S}$ must be zero for any possible perturbation $varepsilon(t)$ about the true evolution. This can be true only if Stationary points (red pluses) and inflection points (green circles). ...

Those familiar with functional analysis will note that the Euler-Lagrange equations simplify to Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ...

The quantity $frac{partial L}{partialdot x}$ 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.

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 This article describes some of the common coordinate systems that appear in elementary mathematics. ...

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

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. In advanced physics, equations of motion usually refer to the Euler-Lagrange equations, differential equations derived from the Lagrangian. ... This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen...

The Einstein equation utilizes the Einstein-Hilbert action as constrained by a variational principle. For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ...

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. This article or section is in need of attention from an expert on the subject. ... In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. ...

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. Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... Richard Phillips Feynman (May 11, 1918 in Queens, New York â€“ February 15, 1988 in Los Angeles, California) (surname pronounced FINE-man; in IPA) was an influential American physicist known for expanding greatly on the theory of quantum electrodynamics, particle theory, and the physics of the superfluidity of supercooled liquid helium. ... This article or section is in need of attention from an expert on the subject. ... In electromagnetics, Maxwells equations are a set of four equations, compiled by James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

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. 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. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ... In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to e. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

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. Physical theories are said to exhibit nonlocality if it is not possible to treat widely separated systems as independent. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ...

Contents

Disambiguation of "action" in classical physics

Action (functional)

Abbreviated action (functional)

Hamilton's principal function

Hamilton's characteristic function

Other solutions of Hamilton-Jacobi equations

Action of a generalized coordinate

Action for a Hamiltonian flow

Historical uses

Euler-Lagrange equations for the action integral

Example: Free particle in polar coordinates

Action principle for classical fields

Action principle in quantum mechanics and quantum field theory

Action principle and conservation laws

Modern extensions of the action principle

See also

References

Contents

Disambiguation of "action" in classical physics GA_googleFillSlot("encyclopedia_square");

Action (functional)

Abbreviated action (functional)

Hamilton's principal function

Hamilton's characteristic function

Other solutions of Hamilton-Jacobi equations

Action of a generalized coordinate

Action for a Hamiltonian flow

Historical uses

Euler-Lagrange equations for the action integral

Example: Free particle in polar coordinates

Action principle for classical fields

Action principle in quantum mechanics and quantum field theory

Action principle and conservation laws

Modern extensions of the action principle

See also

References

Disambiguation of "action" in classical physics