Action | |
---|---|

Common symbols | S |

SI unit | joule-second |

Other units | J⋅Hz^{−1} |

In SI base units | kg⋅m^{2}⋅s^{−1} |

Dimension |

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.^{[1]} Action and the variational principle are used in Feynman's quantum mechanics^{[2]} and in general relativity.^{[3]} For systems with small values of action similar to the Planck constant, quantum effects are significant.^{[4]}

In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy.

More formally, action 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.^{[5]} Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant *h*).^{[6]}

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.^{[1]}

For a trajectory of a baseball 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.^{[4]}

The action balances kinetic against potential energy.^{[4]}
The kinetic energy of a baseball of mass is where is the velocity of the ball; the potential energy is where is the gravitational constant. Then the action between and is

The action value depends upon the trajectory taken by the baseball through and . 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.^{[4]} The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases.

The Planck constant, written as or when including a factor of , is called *the quantum of action*.^{[7]} 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 the Planck constant, quantum effects are significant.^{[4]} The smallest possible action is ; larger action values must be integer multiples of this quantum.^{[8]}

The energy of light quanta, , increases with frequency , but the product of the energy and time for a vibration of a light wave—the action of the quanta—is the constant .^{[9]}

Main article: History of variational principles in physics |

Pierre Louis Maupertuis and Leonhard Euler working in the 1740s developed early versions of the action principle. Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations. William Rowan Hamilton made the next big breakthrough, formulating Hamilton's principle in 1853.^{[10]}^{: 740 } Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles.^{[11]}^{: 127 }

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. 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.^{[12]}^{[13]} 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:^{[12]}

where the integrand

Most commonly, the term is used for a functional which takes a function of time and (for fields) space as input and returns a scalar.^{[14]}^{[15]} In classical mechanics, the input function is the evolution **q**(*t*) of the system between two times *t*_{1} and *t*_{2}, where **q** represents the generalized coordinates. The action is defined as the integral of the Lagrangian *L* for an input evolution between the two times:

where the endpoints of the evolution are fixed and defined as and . According to Hamilton's principle, the true evolution

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 (sometime written as ) is defined as the integral of the generalized momenta,

for a system Lagrangian along a path in the generalized coordinates :

where and 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.

When the total energy *E* is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:^{[12]}^{: 225 }

where the time-independent function

This can be integrated to give

which is just the abbreviated action.^{[16]}^{: 434 }

A variable *J _{k}* in the action-angle coordinates, called the "action" of the generalized coordinate

The corresponding canonical variable conjugate to *J _{k}* is its "angle"

Main articles: Relativistic Lagrangian mechanics and Theory of relativity |

When relativistic effects are significant, the action of a point particle of mass *m* travelling a world line *C* parametrized by the proper time is

If instead, the particle is parametrized by the coordinate time *t* of the particle and the coordinate time ranges from *t*_{1} to *t*_{2}, then the action becomes

where the Lagrangian is