The _action_ is a [[Functional]], $S$ that takes the _path_ of a system as an argument from which we can obtain the [Euler-Lagrange equation](Euler-Lagrange%20equation.md) or [Hamilton equations of motion](Hamilton%20equations%20of%20motion) via the [Least action principle](Least%20action%20principle.md). In the case of Action integrals we refer to these as _[local functionals](Functional.md#local%20functional)_ # Formulation for discrete systems The action integral over a trajectory from $q(t_i)$ to $q(t_f)$ where $q(t)$ is [generalized coordinate](Generalized%20coordinates.md) in space is as follows: $S[q(t)]=\int_{t_i}^{t_f} dt \mathcal{L}(q(t),\dot{q}(t))$ # Formulation on a continuous field Here we integrate over both [time and volume](Spacetime.md#Integral%20measure) of the [[Lagrangian density]] to obtain $S[\varphi] = \int d^4 x \mathscr{L}(\varphi, \dot{\varphi}, \partial_\mu \phi)$ Here we are generally concerned with [oscillator field](Oscillator%20Field%20in%20Spacetime.md)s. # Least Action Principle In real physical systems the path between the initial and final point is minimized by the [[Least action principle]], also known as _Hamilton's Principle_. #Mechanics