In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action close to the Planck constant, quantum effects are significant. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant ).
Introduction
Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages.
Simple example
For a trajectory of a ball moving in the air on Earth the action is defined between two points in time, and as the kinetic energy () minus the potential energy (), integrated over time.
<math display="block">S = \int_{t_1}^{t_2} \bigl( \mathrm{KE}(t) - \mathrm{PE}(t)\bigr) dt</math>
The action balances kinetic against potential energy. It is related to the quantum of angular momentum, , by the relation . These constants have units of energy times time. They appear in all significant quantum equations, such as the uncertainty principle and the de Broglie wavelength. Whenever the value of the action approaches the Planck constant, quantum effects are significant. Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles.
Definitions
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (that is, how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action. Action has the dimensions of [energy] × [time], and its SI unit is joule-second, which is identical to the unit of angular momentum.
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: In classical mechanics, the input function is the evolution of the system between times and , where represents the generalized coordinates. The action is defined as the integral of the Lagrangian for an input evolution between the two times:
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\mathcal{S}[\mathbf{q}(t)] = \int_{t_1}^{t_2} L\bigl(\mathbf{q}(t),\dot{\mathbf{q(t),t\bigr)\, dt,
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where the endpoints of the evolution are fixed and defined as and . According to Hamilton's principle, the true evolution is an evolution for which the action is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.
Abbreviated action (functional)
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In addition to the action functional, there is another functional called the abbreviated action. In the abbreviated action, 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 (sometimes written as ) is defined as the integral of the generalized momenta,
<math display="block">p_i = \frac{\partial L(q,t)}{\partial \dot{q}_i},</math>
for a system Lagrangian along a path in the generalized coordinates :
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\mathcal{S}_0 = \int_{q_1}^{q_2} \mathbf{p} \cdot d\mathbf{q} = \int_{q_1}^{q_2} \sum_i p_i \,dq_i.
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where and are the starting and ending coordinates.
According to Maupertuis's principle, the true path of the system is a path for which the abbreviated action is stationary.
Hamilton's characteristic function
When the total energy is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:
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J_k = \oint p_k \,dq_k
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The corresponding canonical variable conjugate to is its "angle" , for reasons described more fully under action-angle coordinates. The integration is only over a single variable and, therefore, unlike the integrated dot product in the abbreviated action integral above. The variable equals the change in as is varied around the closed path. For several physical systems of interest, is either a constant or varies very slowly; hence, the variable is often used in perturbation calculations and in determining adiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.
<math display="block">L = - m c^2 \sqrt {- c^{-2} g_{\mu\nu} \frac{d x^{\mu{d t} \frac{d x^{\nu{d t \approx -mc^2 \sqrt{1 - \frac{v^2}{c^2,</math>
where is the metric tensor .
Action principles and related ideas
Physical 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.
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.
The action principle provides deep insights into physics, and is an important concept in modern theoretical physics. Various action principles and related concepts are summarized below.
Maupertuis's principle
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the abbreviated action between two generalized points on a path.
Hamilton's principal function
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.
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—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase.
Hamilton–Jacobi equation
Hamilton's principal function is obtained from the action functional by fixing the initial time and the initial endpoint , while allowing the upper time limit and the second endpoint to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.
Euler–Lagrange equations
In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.
Classical fields
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.
Maxwell's equations can be derived as conditions of stationary action.
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.