Generalized Coordinates

Reducing Coordinates

Independent Coordinate and Momenta: : If exist an generalized coordinate that the Lagrangian \(\mathcal{L}\) is independent of, say \(\phi\) in the following coordinate

$$
\begin{gather}
    \boldsymbol{r} = \boldsymbol{r}(r,\phi)\\
    \mathcal{L} = \mathcal{L}(r, \dot r, \dot \phi)
\end{gather}
$$

> Even if $\mathcal{L}$ is dependent of $\dot \phi$, it is still independent of $\phi$.

: Then we are sure that the momenta of the independent coordinate is conserved. For \(\phi\), angular momenta is conserved,

$$ L_\phi = \text{const} $$

$$  $$