Euler-Lagrange Equations

#Physics

$\displaystyle \frac{ \partial L }{ \partial x_{i} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \dot{\partial}x_{i} }=0,,i=1,2,3$

• E.g.: simple harmonic oscillator derivation

$\displaystyle \frac{ \partial L }{ \partial q_{j} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \dot{\partial}q_{j} }=0,,j=1,2,\ldots s$

• Generalized coordinates version
• $\displaystyle s$ is the degrees of freedom of the system

$\displaystyle \frac{ \partial L }{ \partial q_{j} }-\frac{ \mathrm{d} }{ \mathrm{d}t }\frac{ \partial L }{ \partial \dot{q_{j}} }+\sum_{k}\lambda_{k}(t)\frac{ \partial f_{k} }{ \partial q_{j} }$

• With undetermined multipliers