equations:euler_lagrange_equations
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equations:euler_lagrange_equations [2018/03/27 09:08] jakobadmin [Concrete] |
equations:euler_lagrange_equations [2018/04/08 16:13] (current) jakobadmin ↷ Links adapted because of a move operation |
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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 $ ====== | + | <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]]// | ||
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| <tabbox Concrete> | <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]]. | ||
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| + | $$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 . $$ | ||
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| + | 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. | ||
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| + | $$ \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 .$$ | ||
| - | $$ \text{For particles: } \frac{\partial L}{\partial q_i} - \frac{d }{d t}\frac{\partial L}{\partial \dot{q_i}} = 0 \qquad \text{For fields: } \frac{\partial \mathscr{L}}{\partial \Phi^i} - \partial_\mu \left(\frac{\partial \mathscr{L}}{\partial(\partial_\mu\Phi^i)}\right) = 0 $$ | + | The 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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| <tabbox Why is it interesting?> | <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> | </tabbox> | ||
equations/euler_lagrange_equations.1522134493.txt.gz · Last modified: 2018/03/27 07:08 (external edit)
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