Action (physics)

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.

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.

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.[7][8] 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.

The abbreviated action is also a functional. It is usually denoted as ${\displaystyle {\mathcal {S}}_{0}}$. 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:

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

Spelled out concretely, this is

$${\displaystyle {\mathcal {S}}_{0}=\int _{t_{1}}^{t_{2}}\mathbf {p} (t)\cdot {\dot {\mathbf {q} }}(t)\,dt=\int _{t_{1}}^{t_{2}}p_{i}(t)\,{\frac {dq_{i}(t)}{dt}}\,dt.}$$

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 ${\displaystyle S=S(q,t;q_{0},t_{0})}$ is obtained from the action functional ${\displaystyle {\mathcal {S}}}$ by fixing the initial time ${\displaystyle t_{0}}$ and the initial endpoint ${\displaystyle q_{0},}$ while allowing the upper time limit ${\displaystyle t}$ and the second endpoint ${\displaystyle q}$ 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.

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.

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".[4]

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.

See tautological one-form.

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.

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.[3]

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, which 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, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.

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

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

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

where the Lagrangian is[9]

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

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.[7]