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--- equations:euler_lagrange_equations [2018/03/27 09:07]
jakobadmin [Concrete]
+++ equations:euler_lagrange_equations [2018/04/08 16:13] (current)
jakobadmin ↷ Links adapted because of a move operation
@@ Line 1: / Line 1: @@
-====== 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 $ ======
+<WRAP lag>$ \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $</WRAP>
+====== Euler-Lagrange Equations ======
-//see also [[frameworks:lagrangian_formalism|]]//
+//see also [[formalisms:lagrangian_formalism]]//
@@ Line 12: / Line 14: @@
 <tabbox Concrete>
+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]].
-$$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 \qquad \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $$
+$$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 . $$
+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.
 ----
@@ Line 44: / Line 54: @@
 \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}
 <tabbox Abstract>
+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  $q$.
+----
 <blockquote>
@@ Line 95: / Line 119: @@
 <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.
+The Euler-Lagrange equations are used in the [[formalisms:lagrangian_formalism|Lagrange formalism]] to derive from a given Lagrangian the corresponding equations of motion.
 </tabbox>

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