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.
