4.13: Generalized Momenta and Forces
- Page ID
- 29995
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For the above orbital Lagrangian, \(\begin{equation}
d L / d \dot{r}=m \dot{r}=p_{r}
\end{equation}\) the momentum in the r -direction, and \(\begin{equation}
d L / d \dot{\theta}=m r^{2} \dot{\theta}=p_{\theta}
\end{equation}\), the angular momentum associated with the variable \(\theta\).
The generalized momenta for a mechanical system are defined by
\begin{equation}
p_{i}=\frac{\partial L}{\partial \dot{q}_{i}}
\end{equation}
Less frequently used are the generalized forces, \(\begin{equation}
F_{i}=\partial L / \partial q_{i}
\end{equation}\), defined to make the Lagrange equations look Newtonian, \(\begin{equation}
F_{i}=\dot{p}_{i}
\end{equation}\).