An action integral is a functional obtained by integrating a Lagrangian function over time. For a curve
,
![S[q]=int_(t_1)^(t_2)L(q,q^.,t)dt,](/images/equations/ActionIntegral/NumberedEquation1.svg) |
where 
depends on the coordinates 
, their velocities 
, and time 
. For a field theory, the analogous
action is a spacetime integral whose integrand depends on the fields and their derivatives.
Under suitable regularity and endpoint conditions, stationary curves of the action
satisfy the Euler-Lagrange differential
equation. In mechanics, these stationary curves describe the equations of motion.
After Wick rotation, a Lorentzian action gives a Euclidean action.
