Euler-Lagrange Equations [The Physics Travel Guide]

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jakobadmin [Concrete]
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jakobadmin [Concrete]
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 <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 [[frameworks:lagrangian_formalism|Lagrangian]].
 $$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 \qquad \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $$
+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.
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