Quantum redactiones paginae "Aequationes Lagrangi" differant
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:::<math>\ddot\theta + \frac{\ddot x}{L} \cos\theta + \frac{g}{L} \sin\theta = 0 </math>.
Hae solutiones videntur complexae; sed sine aequationibus Lagrangianis, solum legibus Newtonianis utendo, illas solutiones obtinere difficilior fuerit, quod tunc
===Functio Lagrangiana contextu relativistica simplice===
Methodus Lagrangiana nos sinit ad contextum relativisticum discriptiones mechanicas facilius generalizare. Exempli gratia particulam onerus electricum habentem consideremus, quae in campo electromagnetico gyrat. Functio Lagrangiana huius particulae est:
:::<math> L[t] = - m c^2 \sqrt {1 - \frac{v^2 [t]}{c^2}} - q \phi [\vec{x}[t],t] + q \dot{\vec{x}}[t] \cdot \vec{A} [\vec{x}[t],t]</math>
Derivando respecto <math>\vec{x}</math>, obtinemus
:::<math>0 = - \frac{d}{d t}\left(\frac{m \dot{\vec{x}}[t]} {\sqrt {1 - \frac{v^2 [t]}{c^2}}}\right) - q \nabla\phi [\vec{x}[t],t] - q \partial_t{\vec{A}} [\vec{x}[t],t]
- q \dot{\vec{x}}[t] \cdot \nabla\vec{A} [\vec{x}[t],t]
+ q \nabla{\vec{A}} [\vec{x}[t],t] \cdot \dot{\vec{x}}[t]
</math>
quod est aequationem virium Lorentz
:::<math>\frac{d}{d t}\left(\frac{m \dot{\vec{x}}[t]} {\sqrt {1 - \frac{v^2 [t]}{c^2}}}\right) = q \vec{E}[\vec{x}[t],t]
+ q \dot{\vec{x}}[t] \times \vec{B} [\vec{x}[t],t] </math>
ubi identificamus
:::<math>\vec{E}[\vec{x},t] = - \nabla\phi [\vec{x},t] - \partial_t{\vec{A}} [\vec{x},t] </math>
:::<math>\vec{B}[\vec{x},t] = \nabla \times \vec{A} [\vec{x},t] </math>
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