Lagrangian and Equation of Motion

Euler-Lagrangian equation is

(1)\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\boldsymbol q}} \right) - \frac{\partial L}{\partial \boldsymbol q} = 0.\]

The component form is

(2)\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) - \frac{\partial L}{\partial q_i} = 0.\]

Conserved Quantities

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

We notice that the second term in (2) 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.

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