✎ Euler-Lagrange Equations [The Physics Travel Guide]

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====== Euler-Lagrange Equations: $\quad \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $ ======

//see also [[frameworks: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 $B$.  
  
<tabbox Concrete>

$$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 \qquad \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $$

----
** Euler-Lagrange Expression **

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 $(q, \dot q)\in T\mathcal Q$. Youo can thing $\dot{q}$ as the vector on the point  $q$.

** Euler-Lagrange Equations**

The form to the Euler-Lagrange equations is
$$
[L] = \frac{\partial L}{\partial q} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q}} = 0
$$

<blockquote>
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 $\Theta$.<cite>https://arxiv.org/abs/1601.05956</cite></blockquote>

----

* 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.
  
<tabbox Why is it interesting?>

The Euler-Lagrange equations are used in the [[frameworks:lagrangian_formalism|Lagrange formalism]] to derive from a given Lagrangian the corresponding equations of motion.

</tabbox>

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