Action (physics)

From Wikipedia, the free encyclopedia

In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has the dimensions of [energy]⋅[time] or [momentum]⋅[length], and its SI unit is joule-second.

Introduction

Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.

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—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system randomly follows one of the possible paths, with the phase of the probability amplitude for each path being determined by the action for the path.

Solution of differential equation

Empirical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion.

Minimization of action integral

Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is stationary. In other words, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.
This simple principle provides deep insights into physics, and is an important concept in modern theoretical physics.

History

Action was defined in several now obsolete ways during the development of the concept.

• 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 action for a material particle as the integral of the particle's speed along its path through space.
• Pierre Louis 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 hybrid that ensured conservation of momentum in collisions.

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 a stationary point (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. However, when the action pertains 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 action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:

${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L\,dt,}$

where the integrand L is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.

Action has the dimensions of [energy]⋅[time], and its SI unit is joule-second, which is identical to the unit of angular momentum.

Action in classical physics

In classical physics, the term "action" has a number of meanings.

Action (functional)

Most commonly, the term is used for a functional ${\displaystyle {\mathcal {S}}}$ which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution q(t) of the system between two times t₁ and t₂, where q represents the generalized coordinates. The action ${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}$ is defined as the integral of the Lagrangian L for an input evolution between the two times:

${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L[\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t]\,dt,}$

where the endpoints of the evolution are fixed and defined as ${\displaystyle \mathbf {q} _{1}=\mathbf {q} (t_{1})}$ and ${\displaystyle \mathbf {q} _{2}=\mathbf {q} (t_{2})}$. According to Hamilton's principle, the true evolution q_true(t) is an evolution for which the action ${\displaystyle {\mathcal {S}}[\mathbf {q} (t)]}$ is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.

Abbreviated action (functional)

Usually denoted as ${\displaystyle {\mathcal {S}}_{0}}$, 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 ${\displaystyle {\mathcal {S}}_{0}}$ is defined as the integral of the generalized momenta along a path in the generalized coordinates:

${\displaystyle {\mathcal {S}}_{0}=\int \mathbf {p} \cdot d\mathbf {q} =\int p_{i}\,dq_{i}.}$

According to Maupertuis' principle, the true path is a path for which the abbreviated action ${\displaystyle {\mathcal {S}}_{0}}$ 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 ${\displaystyle {\mathcal {S}}}$ by fixing the initial time t₁ and the initial endpoint q₁ and allowing the upper limits t₂ and the second endpoint q₂ 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 the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:

${\displaystyle S(q_{1},\dots ,q_{N},t)=W(q_{1},\dots ,q_{N})-E\cdot t,}$

where the time-independent function W(q₁, q₂, … q_N) is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative

${\displaystyle {\frac {dW}{dt}}={\frac {\partial W}{\partial q_{i}}}{\dot {q}}_{i}=p_{i}{\dot {q}}_{i}.}$

This can be integrated to give

${\displaystyle W(q_{1},\dots ,q_{N})=\int p_{i}{\dot {q}}_{i}\,dt=\int p_{i}\,dq_{i},}$

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., S_k(q_k), are also called an "action".

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:

${\displaystyle J_{k}=\oint p_{k}\,dq_{k}}$

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. 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) as 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.

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 may also depend explicitly on time. In that case, the action integral can be written as

${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(x,{\dot {x}},t)\,dt,}$

where the initial and final times (t₁ and t₂) and the final and initial positions are specified in advance as ${\displaystyle x_{1}=x(t_{1})}$ and ${\displaystyle x_{2}=x(t_{2})}$. Let x_true(t) represent the true evolution that we seek, and let ${\displaystyle x_{\text{per}}(t)}$ be a slightly perturbed version of it, albeit with the same endpoints, ${\displaystyle x_{\text{per}}(t_{1})=x_{1}}$ and ${\displaystyle x_{\text{per}}(t_{2})=x_{2}}$. The difference between these two evolutions, which we will call ${\displaystyle \varepsilon (t)}$, is infinitesimally small at all times:

${\displaystyle \varepsilon (t)=x_{\text{per}}(t)-x_{\text{true}}(t).}$

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

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

${\displaystyle {\begin{aligned}\delta {\mathcal {S}}&=\int _{t_{1}}^{t_{2}}\left[L(x_{\text{true}}+\varepsilon ,{\dot {x}}_{\text{true}}+{\dot {\varepsilon }},t)-L(x_{\text{true}},{\dot {x}}_{\text{true}},t)\right]\,dt\\&=\int _{t_{1}}^{t_{2}}\left(\varepsilon {\frac {\partial L}{\partial x}}+{\dot {\varepsilon }}{\frac {\partial L}{\partial {\dot {x}}}}\right)\,dt.\end{aligned}}}$

Integration by parts of the last term, together with the boundary conditions ${\displaystyle \varepsilon (t_{1})=\varepsilon (t_{2})=0}$, yields the equation

${\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\left(\varepsilon {\frac {\partial L}{\partial x}}-\varepsilon {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {x}}}}.\right)\,dt.}$

The requirement that ${\displaystyle {\mathcal {S}}}$ be stationary implies that the first-order change must be zero for any possible perturbation ε(t) about the true evolution:

Principle of stationary action ${\displaystyle \delta {\mathcal {S}}=0}$

This can be true only if

Euler–Lagrange equation ${\displaystyle {\partial L \over \partial x}-{d \over dt}{\partial L \over \partial {\dot {x}}}=0}$

The Euler–Lagrange equation is obeyed provided the functional derivative of the action integral is identically zero:

${\displaystyle {\frac {\delta {\mathcal {S}}}{\delta x(t)}}=0.}$

The quantity ${\displaystyle {\frac {\partial L}{\partial {\dot {x}}}}}$ is called the conjugate momentum for the coordinate x. An important consequence of the Euler–Lagrange equations is that if L does not explicitly contain coordinate x, i.e.

if ${\displaystyle {\frac {\partial L}{\partial x}}=0}$, then ${\displaystyle {\frac {\partial L}{\partial {\dot {x}}}}}$ is constant in time.

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

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

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

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

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

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {r}}}}\right)-{\frac {\partial L}{\partial r}}&=0&&\Rightarrow &{\ddot {r}}-r{\dot {\varphi }}^{2}&=0,\\{\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\varphi }}}}\right)-{\frac {\partial L}{\partial \varphi }}&=0&&\Rightarrow &{\ddot {\varphi }}+{\frac {2}{r}}{\dot {r}}{\dot {\varphi }}&=0.\end{aligned}}}$

The solution of these two equations is given by

${\displaystyle {\begin{aligned}r\cos \varphi &=at+b,\\r\sin \varphi &=ct+d\end{aligned}}}$

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.

The action principle

Classical fields

The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.

The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle.
The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic.

Conservation laws

Implications of symmetries in a physical situation can be found 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.

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 permitted 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. It is best understood within quantum mechanics. In particular, in Richard Feynman's path integral formulation of quantum mechanics, where it arises out of destructive interference of quantum amplitudes.

Maxwell's equations can also be derived as conditions of stationary action.

Single relativistic particle

When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time ${\displaystyle \tau }$ is

${\displaystyle S=-mc^{2}\int _{C}\,d\tau .}$

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

${\displaystyle \int _{t1}^{t2}L\,dt,}$

where the Lagrangian is

${\displaystyle L=-mc^{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}.}$

Modern extensions

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.

Principle of least action

From Wikipedia, the free encyclopedia

The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). The physicist Paul Dirac, and after him Julian Schwinger and Richard Feynman demonstrated how this principle can also be used in quantum calculations. It was historically called "least" because its solution requires finding the path that has the least value. Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.

The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory, and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.

In 1933, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes.

General statement

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).^[18]

The starting point is the action, denoted ${\displaystyle {\mathcal {S}}}$ (calligraphic S), of a physical system. It is defined as the integral of the Lagrangian L between two instants of time t₁ and t₂ - technically a functional of the N generalized coordinates q = (q₁, q₂, ... , q_N) which define the configuration of the system:

${\displaystyle {\mathcal {S}}[\mathbf {q_{1}} (t_{1}),\mathbf {q_{2}} (t_{2}),\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),\mathbf {\dot {q}} (t),t)dt}$

where the dot denotes the time derivative, and t is time.

Mathematically the principle is

${\displaystyle \delta {\mathcal {S}}=0,}$

where δ (lowercase Greek delta) means a small change. In words this reads:

The path taken by the system between times t₁ and t₂ and configurations q₁ and q₂ is the one for which the action is stationary (no change) to first order.

In applications the statement and definition of action are taken together:

${\displaystyle \delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0.}$

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).

Origins, statements, and controversy

Fermat

In the 1600s, Pierre de Fermat postulated that "light travels between two given points along the path of shortest time," which is known as the principle of least time or Fermat's principle.

Maupertuis

Credit for the formulation of the principle of least action is commonly given to Pierre Louis Maupertuis, who felt that "Nature is thrifty in all its actions", and applied the principle broadly:

The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.

— Pierre Louis Maupertuis

This notion of Maupertuis, although somewhat deterministic today, does capture much of the essence of mechanics.

In application to physics, Maupertuis suggested that the quantity to be minimized was the product of the duration (time) of movement within a system by the "vis viva",

Maupertuis' principle ${\displaystyle \delta \int 2T(t)dt=0}$

which is the integral of twice what we now call the kinetic energy T of the system.

Euler

Leonhard Euler gave a formulation of the action principle in 1744, in very recognizable terms, in the Additamentum 2 to his Methodus Inveniendi Lineas Curvas Maximi Minive Proprietate Gaudentes. Beginning with the second paragraph:

“	Let the mass of the projectile be M, and let its speed be v while being moved over an infinitesimal distance ds. The body will have a momentum Mv that, when multiplied by the distance ds, will give Mv ds, the momentum of the body integrated over the distance ds. Now I assert that the curve thus described by the body to be the curve (from among all other curves connecting the same endpoints) that minimizes ${\displaystyle \int Mv\,ds}$ or, provided that M is constant along the path, ${\displaystyle M\int v\,ds}$.	”
— Leonhard Euler

As Euler states, ∫Mvds is the integral of the momentum over distance travelled, which, in modern notation, equals the abbreviated or reduced action

Euler's principle ${\displaystyle \delta \int p\,dq=0}$

Thus, Euler made an equivalent and (apparently) independent statement of the variational principle in the same year as Maupertuis, albeit slightly later. Curiously, Euler did not claim any priority, as the following episode shows.

Disputed priority

Maupertuis' priority was disputed in 1751 by the mathematician Samuel König, who claimed that it had been invented by Gottfried Leibniz in 1707. Although similar to many of Leibniz's arguments, the principle itself has not been documented in Leibniz's works. König himself showed a copy of a 1707 letter from Leibniz to Jacob Hermann with the principle, but the original letter has been lost. In contentious proceedings, König was accused of forgery, and even the King of Prussia entered the debate, defending Maupertuis (the head of his Academy), while Voltaire defended König.

Euler, rather than claiming priority, was a staunch defender of Maupertuis, and Euler himself prosecuted König for forgery before the Berlin Academy on 13 April 1752. The claims of forgery were re-examined 150 years later, and archival work by C.I. Gerhardt in 1898 and W. Kabitz in 1913 uncovered other copies of the letter, and three others cited by König, in the Bernoulli archives.

Further development

Euler continued to write on the topic; in his Reflexions sur quelques loix generales de la nature (1748), he called 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.

Lagrange and Hamilton

Much of the calculus of variations was stated by Joseph-Louis Lagrange in 1760 and he proceeded to apply this to problems in dynamics. In Méchanique Analytique (1788) Lagrange derived the general equations of motion of a mechanical body. William Rowan Hamilton in 1834 and 1835 applied the variational principle to the classical Lagrangian function

${\displaystyle L=T-V}$

to obtain the Euler–Lagrange equations in their present form.

Jacobi and Morse

In 1842, Carl Gustav Jacobi tackled the problem of whether the variational principle always found minima as opposed to other stationary points (maxima or stationary saddle points); most of his work focused on geodesics on two-dimensional surfaces. The first clear general statements were given by Marston Morse in the 1920s and 1930s, leading to what is now known as Morse theory. For example, Morse showed that the number of conjugate points in a trajectory equalled the number of negative eigenvalues in the second variation of the Lagrangian.

Gauss and Hertz

Other extremal principles of classical mechanics have been formulated, such as Gauss's principle of least constraint and its corollary, Hertz's principle of least curvature.

Disputes about possible teleological aspects

The mathematical equivalence of the differential equations of motion and their integral counterpart has important philosophical implications. The differential equations are statements about quantities localized to a single point in space or single moment of time. For example, Newton's second law

${\displaystyle \mathbf {F} =m\mathbf {a} }$

states that the instantaneous force F applied to a mass m produces an acceleration a at the same instant. By contrast, the action principle is not localized to a point; rather, it involves integrals over an interval of time and (for fields) an extended region of space. Moreover, in the usual formulation of classical action principles, the initial and final states of the system are fixed, e.g.,

Given that the particle begins at position x₁ at time t₁ and ends at position x₂ at time t₂, the physical trajectory that connects these two endpoints is an extremum of the action integral.

In particular, the fixing of the final state has been interpreted as giving the action principle a teleological character which has been controversial historically. However, according to W. Yourgrau and S. Mandelstam, the teleological approach... presupposes that the variational principles themselves have mathematical characteristics which they de facto do not possess. In addition, some critics maintain this apparent teleology occurs because of the way in which the question was asked. By specifying some but not all aspects of both the initial and final conditions (the positions but not the velocities) we are making some inferences about the initial conditions from the final conditions, and it is this "backward" inference that can be seen as a teleological explanation. Teleology can also be overcome if we consider the classical description as a limiting case of the quantum formalism of path integration, in which stationary paths are obtained as a result of interference of amplitudes along all possible paths.

The short story Story of Your Life by the speculative fiction writer Ted Chiang contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension. Keith Devlin's The Math Instinct contains a chapter, "Elvis the Welsh Corgi Who Can Do Calculus" that discusses the calculus "embedded" in some animals as they solve the "least time" problem in actual situations.