The _Euler-Lagrange Equation_ is a [2nd order ODE](2nd%20order%20ODE.md) that describes the dynamics of a system in terms of a given [[Lagrangian]] given as $\frac{\partial \mathcal{L}}{\partial q}-\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}}=0$ # Field Euler-Lagrange equation To describe a field in [[Spacetime]] the Euler Lagrange equation is given in terms of a [[Lagrangian density]], $\mathscr{L}$ as $\frac{\partial\mathscr{L}}{\partial \varphi}-\partial_t \frac{\partial \mathscr{L}}{\partial \dot{\varphi}}-\partial_\mu\frac{\partial \mathscr{L}}{\partial (\partial_\mu\varphi)}=0$ # Derivation from the [least action principle](Least%20action%20principle.md) ## The Euler-Lagrange equation for discrete systems ## The Euler-Lagrange equation for [[scalar field]]s We evaluate the [least action principle](Least%20action%20principle.md#Formulation%20on%20a%20continuous%20field) as $\delta S[\varphi,\delta \varphi] = \int d^4x\bigg(\frac{\partial \mathscr{L}}{\partial \varphi}\delta\varphi + \frac{\partial \mathscr{L}}{\partial \dot{\varphi}}\delta\dot{\varphi}+\frac{\partial\mathscr{L}}{\partial(\partial_\mu \varphi)}\delta(\partial_\mu \varphi)\bigg)$ Next we [integrate by parts](Analysis%20(index).md#Integration%20by%20parts) in order to obtain $=\int d^4 x\, \delta \varphi\bigg(\frac{\partial\mathscr{L}}{\partial \varphi}-\partial_t \frac{\partial \mathscr{L}}{\partial \dot{\varphi}}-\partial_\mu\frac{\partial \mathscr{L}}{\partial(\partial_\mu \varphi)}\bigg)+\int_{\delta\Omega}dx^4\,\delta \varphi\bigg(\frac{\partial\mathscr{L}}{\partial\dot{\varphi}} + \frac{\partial \mathscr{L}}{\partial(\partial_\mu \varphi)}\bigg)$ where here we separate out the boundary terms given by the integral over the edge of spacetime, $\delta \Omega.$ The boundary terms go to zero since for a [field](Field.md) we demand that it is $0$ at $\infty.$ From here on, we extract the Euler-Lagrange equation from the first integral term giving $\frac{\partial\mathscr{L}}{\partial \varphi}-\partial_t \frac{\partial \mathscr{L}}{\partial \dot{\varphi}}-\partial_\mu\frac{\partial \mathscr{L}}{\partial (\partial_\mu\varphi)}=0$ # Coordinate invariance of the Euler-Lagrange equations #Mechanics #Mechanics/ClassicalFields #Mechanics/SpecialRelativity