Action (physics)

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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.