Lagrangian and Equation of Motion
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Euler-Lagrangian equation is
.. math::
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\boldsymbol q}} \right) - \frac{\partial L}{\partial \boldsymbol q} = 0.
:label: eqn-euler-lagragian-equation
The component form is
.. math::
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) - \frac{\partial L}{\partial q_i} = 0.
:label: eqn-euler-lagragian-equation-component
.. admonition:: Conserved Quantities
:class: notes
A quantity is conserved through time if :math:`\frac{d}{dt}Q = 0`.
We notice that the second term in :eq:`eqn-euler-lagragian-equation-component` vanishes if the lagragian doesn't depend on :math:`q_i`. That is
.. math::
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q_i}} \right) = 0
for Lagragian that doesn't depend on :math:`q_i`.
We immediately spot that the quantity
.. math::
\frac{\partial L}{\partial \dot{q_i} }
is a conserved quantity.