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Euler Lagrange Equations

Euler-Lagrange Equations

Oct 06, 20231 min read

Physics

$\frac{\partial L}{\partial x _{i}} - \frac{d}{d t} \frac{\partial L}{\partial ˙ x _{i}} = 0, i = 1, 2, 3$

E.g.: simple harmonic oscillator derivation

$\frac{\partial L}{\partial q _{j}} - \frac{d}{d t} \frac{\partial L}{\partial ˙ q _{j}} = 0, j = 1, 2, \dots s$

Generalized coordinates version
$s$ is the degrees of freedom of the system

$\frac{\partial L}{\partial q _{j}} - \frac{d}{d t} \frac{\partial L}{\partial q _{j} ˙} + k \sum λ_{k} (t) \frac{\partial f _{k}}{\partial q _{j}}$

With undetermined multipliers

Graph View

$\displaystyle \frac{ \partial L }{ \partial x_{i} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \dot{\partial}x_{i} }=0,\,i=1,2,3$
$\displaystyle \frac{ \partial L }{ \partial q_{j} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \dot{\partial}q_{j} }=0,\,j=1,2,\ldots s$
$\displaystyle \frac{ \partial L }{ \partial q_{j} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \partial \dot{q_{j}} }+\sum_{k}\lambda_{k}(t)\frac{ \partial f_{k} }{ \partial q_{j} }$

Backlinks

Lagrangian and Hamiltonian Mechanics

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