In physics, action is a scalar quantity assigned to a possible history of a physical system. Usually denoted (S), it is constructed by integrating a Lagrangian over time, or a Lagrangian density over spacetime. Its importance lies in the principle of stationary action: classical motions make its first-order variation vanish under appropriate boundary conditions. In quantum theory, action instead determines the phase associated with each history in a sum over alternatives. It therefore connects classical mechanics, field theory, and quantum mechanics. (mitp-content-server.mit.edu)
Definition and dimensions
For a mechanical system described by coordinates (q_i(t)), the action is
[ S[q]=\int_{t_1}^{t_2}L(q_i,\dot q_i,t),dt. ]
Here (\dot q_i=dq_i/dt), and the coordinates describe the system’s degrees of freedom. The brackets emphasize that action is a functional: its input is an entire trajectory, rather than simply a coordinate value at one instant. Two trajectories connecting the same endpoints can have different actions. The integral accumulates the Lagrangian along each trajectory. (mitp-content-server.mit.edu)
For many nonrelativistic systems with conservative forces, (L=T-V), where (T) is kinetic energy and (V) is potential energy. Action consequently has dimensions of energy multiplied by time. In the International System of Units, its unit is the joule-second, (\mathrm{J,s}=\mathrm{kg,m^2,s^{-1}}). This differs from work, which has units of energy alone. The Planck constant has the same dimensions as action. (feynmanlectures.caltech.edu)
Stationary action and equations of motion
Hamilton’s principle states that a physical trajectory satisfies
[ \delta S=0 ]
for small admissible changes of the path that leave the endpoint coordinates and endpoint times fixed. Writing a varied path as (q_i(t)+\epsilon\eta_i(t)), with (\eta_i(t_1)=\eta_i(t_2)=0), stationarity means that the derivative of its action with respect to (\epsilon) vanishes at (\epsilon=0). (mitp-content-server.mit.edu)
The calculus of variations converts this condition into the Euler–Lagrange equations:
[ \frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right) -\frac{\partial L}{\partial q_i}=0. ]
These local differential equations determine motion once suitable initial conditions are supplied. Although the action principle compares complete histories, its equivalence to local equations does not require a system to anticipate its future endpoint. (mitp-content-server.mit.edu)
The traditional expression “principle of least action” can be misleading. The required condition is stationarity, not an absolute minimum among all possible paths. A stationary trajectory may be a local minimum or a saddle, depending on the system and the interval considered. Establishing a minimum requires examining higher-order variations, not merely setting the first variation to zero. (mitp-content-server.mit.edu)
Mechanical examples and alternative forms
For a particle of mass (m) moving in one dimension,
[ L=\frac12m\dot x^2-V(x). ]
The Euler–Lagrange equation becomes
[ m\ddot x=-\frac{dV}{dx}, ]
recovering Newton’s second law for a conservative force. For a free particle, (V=0), so (\ddot x=0): the stationary trajectory between fixed spacetime endpoints has constant velocity. The same procedure applies to many-particle systems and to coordinates adapted to mechanical constraints. (damtp.cam.ac.uk)
In Hamiltonian mechanics, action can instead be written on phase space:
[ S[q,p]=\int_{t_1}^{t_2} \left(\sum_i p_i\dot q_i-H(q,p,t)\right)dt. ]
The (p_i) are canonical momenta, and (H) is the Hamiltonian. Independent variations of coordinates and momenta yield Hamilton’s equations. Related action variables, constructed from integrals such as (\oint p,dq), describe periodic motion and play an important role in adiabatic invariance. (damtp.cam.ac.uk)
Fields, symmetries, and boundaries
For fields (\phi_a(x)), a common action is
[ S[\phi]=\int d^4x, \mathcal L(\phi_a,\partial_\mu\phi_a), ]
using relativistic units with (c=1). The Lagrangian density (\mathcal L) encodes field dynamics and interactions. Varying the fields gives field Euler–Lagrange equations, extending the mechanical construction to continuously distributed degrees of freedom. This framework underlies quantum field theory. (damtp.cam.ac.uk)
Noether’s theorem relates continuous symmetries of the action to conservation laws. Time-translation symmetry yields conserved energy, spatial-translation symmetry yields conserved momentum, and rotational symmetry yields conserved angular momentum. In field theory, these quantities arise from conserved currents. The theorem applies to transformations that preserve the action, including cases where the Lagrangian changes by an appropriate total derivative. (damtp.cam.ac.uk)
An action is not uniquely determined by its bulk equations. Adding (dF(q,t)/dt) to a mechanical Lagrangian changes the action by (F(t_2)-F(t_1)), but leaves the equations unchanged for fixed endpoints. Boundary conditions therefore form part of an action principle, and boundary terms cannot always be discarded. (damtp.cam.ac.uk)
In general relativity, the Einstein–Hilbert action integrates scalar curvature with the invariant volume element of spacetime. Varying the metric in the gravitational action, together with a matter action and suitable boundary treatment, produces the Einstein field equations. (damtp.cam.ac.uk)
Quantum significance
In the path-integral formulation, a transition amplitude is represented schematically by
[ K=\int\mathcal Dq,e^{iS[q]/\hbar}, ]
where (\hbar=h/(2\pi)). Histories contribute complex amplitudes, not ordinary probabilities; their relative phases depend on action differences. Thus quantum dynamics does not simply select one classical path. (feynmanlectures.caltech.edu)
In a semiclassical regime, phases often vary rapidly between neighboring nonstationary histories, causing substantial cancellation through interference. Near stationary histories, the first-order change in action vanishes, allowing nearby contributions to reinforce one another. This stationary-phase behavior explains how classical equations emerge as an approximation to quantum dynamics; the relevant scale is variation of action relative to (\hbar), rather than an arbitrary absolute value of action. (feynmanlectures.caltech.edu)