$ \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $
====== Euler-Lagrange Equations ======
//see also [[formalisms:lagrangian_formalism]]//
The basic idea behind the Lagrangian formalism is that nature is guided by a principle of "minimal action". The Euler-Lagrange equations give the path with a minimal amount of "action" that a system follows.
In principle, there are many possible paths how some given particle or multiple particles could get from some point $A$ to another point $B$. The Euler-Lagrange equations are used to calculate the correct path that a particle follows between $A$ and $B$.
The Euler-Lagrange equation tells us which path is the path with minimal action $S = \int_{t_i}^{t_f} dt L(q,\dot{q})$, where $L(q,\dot{q})$ denotes the [[formalisms:lagrangian_formalism|Lagrangian]].
$$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 . $$
The Euler-Lagrange equation can also be used in a field theory and there it tells us which sequence of field configurations has minimal action.
$$ \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 .$$
The general procedure is that we start with a Lagrangian. The Lagrangian is an object that has to be guessed by making use of symmetry considerations and characterizes the system in question. Then we put the Lagrangian into the Euler-Lagrange equation and this gives us the equations of motion of the system.
----
**Derivation**
We consider an arbitrary path $(q,\dot{q})$. If it is the
path that minimizes the action, we have
\begin{eqnarray}
0 &=& \delta S = \delta \int_{t_i}^{t_f} dt L(q,\dot{q}) =
\int_{t_i}^{t_f} dt L(q+\delta q,\dot{q}+\delta \dot{q})-S \\
&=& \int_{t_i}^{t_f} dtL(q,\dot{q}) + \int_{t_i}^{t_f} dt\bigg(\delta q
{\partial L \over \partial q} + \delta \dot{q} {\partial L \over
\partial \dot{q}} \bigg) - S \\
&=& \int_{t_i}^{t_f} dt \bigg(\delta q {\partial L \over \partial q} +
{\partial L \over \partial \dot{q}} {d \over dt} \delta q\bigg)
\end{eqnarray}
If we now integrate the second term by parts, and take the variation of
$\delta q$ to be $0$ at $t_i$ and $t_f$,
\begin{eqnarray}
\delta S = \int_{t_i}^{t_f} dt \bigg(\delta q{\partial L \over \partial
q} - \delta q {d \over dt} {\partial L \over \partial \dot{q}} \bigg) =
\int_{t_i}^{t_f} dt \delta q \bigg({\partial L \over \partial q} - {d
\over dt} {\partial L \over \partial \dot{q}} \bigg) = 0
\end{eqnarray}
Now, the only way this holds for an arbitrary variation $\delta q$
of the path $(q,\dot{q})$ is when
\begin{eqnarray}
{d \over dt}{\partial L \over \partial \dot{q}} - {\partial L \over
\partial q} = 0
\end{eqnarray}
This is the Euler-Lagrange equation.
Generalized to multiple coordinates $q_i$ ($i=1,\ldots,n$) it reads
\begin{eqnarray}
{d \over dt} {\partial L \over \partial \dot{q}_i} - {\partial L \over
\partial q_i} = 0
\end{eqnarray}
Given a Lagrangian function $L:T\mathcal{Q} \longrightarrow \mathbb{R}$, the //Lagrange expression //, denoted as $[]$ is given by:
$$
[L] = \frac{\partial L}{\partial q} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q^i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q}^i}
$$
where $q\in\mathcal{Q}$ and $\dot q$ represents the lift of $q$ to the tangent bundle, i.e $(q, \dot q)\in T\mathcal Q$. You can think of $\dot{q}$ as the vector on the point $q$.
----
With this notation, a globally defined local Lagrangian for fields that are sections of some bundle $E$
over spacetime/worldvolume $\Sigma$ is simply a morphism of the form
$$
L : E \longrightarrow \wedge^{p+1}T^\ast \Sigma
\,.
$$
Unwinding what this means, this is a function that at each point of $\Sigma$ sends the value of field configurations and all their spacetime/worldvolume
derivatives at that point to a $(p+1)$-form on $\Sigma$ at that point. It is this pointwise local (in fact: infinitesimally local) dependence
that the term local in local Lagrangian refers to. [...]
Regarding such $L$ for a moment as just a differential form on $J^\infty_\Sigma(E)$ (= the [[advanced_tools:jet_bundles|jet bundle]]), we may apply the
de Rham differential to it. One finds that this uniquely decomposes as a sum of the form
\begin{equation}
\label{differentialofLagrangian}
d L = \mathrm{EL} - d_H (\Theta + d_H(\cdots))
\,,
\end{equation}
for some $\Theta$ and for $\mathrm{EL}$ pointwise the
pullback of a vertical 1-form on $E$; such a differential form is called a source form:
$\mathrm{EL} \in \Omega^{p+1,1}_S(E)$.
This particular source form is of paramount importance: the equation
$$
\underset{v\in \Gamma(V E)}{\forall} j^\infty(\phi)^\ast \iota_v\mathrm{EL} = 0
$$
on sections $\phi \in \Gamma_\Sigma(E)$
is a partial differential equation, and
this is called the Euler-Lagrange equation of motion induced by $L$. Differential equations arising this way
from a local Lagrangian are called variational.
A little reflection reveals that this is indeed a re-statement of the traditional prescription of obtaining the Euler-Lagrange equations by
locally varying the integral over the Lagrangian and then applying partial integration to turn all variation of derivatives (i.e. of [[advanced_tools:jet_bundles|jets]]) of fields
into variation of the fields themselves. Here we do not consider this under the integral, and hence the boundary terms arising from the
would-be partial integration show up as the contribution $\Theta$.https://arxiv.org/abs/1601.05956
----
* http://www.project-tartarus.com/2017/07/the-euler-lagrange-equation-and-the-principle-of-least-action/
* Bleecker, D., Gauge Theory and Variational Principles, Addison-Wesley, Reading, MA, 1981.
The Euler-Lagrange equations are used in the [[formalisms:lagrangian_formalism|Lagrange formalism]] to derive from a given Lagrangian the corresponding equations of motion.