# Lagrangian and Equation of Motion¶

Euler-Lagrangian equation is

(7)$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\boldsymbol q}} \right) - \frac{\partial L}{\partial \boldsymbol q} = 0.$

The component form is

(8)$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) - \frac{\partial L}{\partial q_i} = 0.$

Conserved Quantities

A quantity is conserved through time if $$\frac{d}{dt}Q = 0$$.

We notice that the second term in (8) vanishes if the lagragian doesn’t depend on $$q_i$$. That is

$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) = 0$

for Lagragian that doesn’t depend on $$q_i$$.

We immediately spot that the quantity

$\frac{\partial L}{\partial \dot{q_i} }$

is a conserved quantity.