Simple example

For a trajectory of a baseball moving in the air on Earth the action is defined between two points in time, ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ as the kinetic energy minus the potential energy, integrated over time.^[7]

${\displaystyle S=\int _{t_{1}}^{t_{2}}\left(KE(t)-PE(t)\right)dt}$

The action balances kinetic against potential energy.^[7] The kinetic energy of a baseball of mass ${\displaystyle m}$ is ${\displaystyle (1/2)mv^{2}}$ where ${\displaystyle v}$ is the velocity of the ball; the potential energy is ${\displaystyle mgx}$ where ${\displaystyle g}$ is the gravitational constant. Then the action between ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ is

${\displaystyle S=\int _{t_{1}}^{t_{2}}\left({\frac {1}{2}}mv^{2}(t)-mgx(t)\right)dt}$

The action value depends upon the trajectory taken by the baseball through ${\displaystyle x(t)}$ and ${\displaystyle v(t)}$. This makes the action an input to the powerful stationary-action principle for classical and for quantum mechanics. Newton's equations of motion for the baseball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where the Newton's laws are difficult to apply. Replace the baseball by an electron: classical mechanics fails but stationary action continues to work.^[7] The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases.

Planck's quantum of action

Planck's constant, written as ${\displaystyle h}$ or ${\displaystyle \hbar }$ when including a factor of ${\displaystyle 1/2\pi }$, is also called the quantum of action. Like action, this constant has unit of energy times time. It figures in all significant quantum equations, like the uncertainty principle and the de Broglie wavelength. Whenever the value of the action approaches Planck's constant, quantum effects are significant.^[7] The smallest possible action is ${\displaystyle \hbar /2}$; larger action values must be integer multiples of this quantum.^[1]

The energy of light quanta, ${\displaystyle E=\hbar \omega }$, increases with frequency ${\displaystyle \omega }$, but the product of the energy and time for a vibration of a light wave—the action of the quanta—is the constant ${\displaystyle \hbar }$.^[8]

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.^[13][14] 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)

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 ${\displaystyle {\mathcal {S}}_{0}}$ (sometime written as ${\displaystyle W}$) is defined as the integral of the generalized momenta,

$${\displaystyle p_{i}={\frac {\partial L(q,t)}{\partial {\dot {q}}_{i}}},}$$

for a system Lagrangian ${\displaystyle L}$ along a path in the generalized coordinates ${\displaystyle q_{i}}$:

$${\displaystyle {\mathcal {S}}_{0}=\int _{q_{1}}^{q_{2}}\mathbf {p} \cdot d\mathbf {q} =\int _{q_{1}}^{q_{2}}\Sigma _{i}p_{i}\,dq_{i}.}$$

where ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$ are the starting and ending coordinates. According to Maupertuis' 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 E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:^[11]: 225

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

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.^[15]: 434

Action of a generalized coordinate

A variable J_k in the action-angle coordinates, called the "action" of the generalized coordinate q_k, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:^[15]: 454

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

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. For example, they are used in the calculation of planetary and satellite orbits.^[15]: 477

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

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

where the Lagrangian is^[16]

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

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.^[17]

Hamilton–Jacobi equation

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.

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.^[17]

Path integral formulation of 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, 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.

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.^[13]