Generalizing Lagrange's Equations

In presence of non-conservative forces #

$$ \begin{align} \frac{d}{dt}\left(\frac{\partial T}{\partial q_k’}\right)-\frac{\partial T}{\partial q_k}=G_k=G_{k,conservative}+G_{k,nonconservative}=-\frac{\partial V}{\partial q_k}+G_k’\\ \Leftrightarrow \\ \frac{d}{dt}\left(\frac{\partial L}{\partial q_k’}\right)-\frac{\partial L}{\partial q_k}=G_k' \end{align} $$

$G_k’$: Component of generalized force for non conservative force $\vec{f_i}$

$$ \begin{align} G_k’=\sum_{i=1}^N \vec{f_i}\cdot\frac{\partial \vec{r_i}}{\partial q_k} \end{align} $$

In the case of frictional force #

$$ \begin{align} \vec{f_i}=-k_i\vec{v_i}=-\nabla_{v_i}R\\ where\\ R=\frac{1}{2}\sum_i k_i\vec{v_i}^2\\ \\ G_k’=\sum_{i=1}^N -\nabla_{v_i}R\cdot \frac{\partial \vec{r_i}}{\partial q_k}\\ =\sum_{i=1}^N -\nabla_{v_i}R\cdot \frac{\partial \vec{v_i}}{\partial q_k’}\\ -\frac{\partial R}{\partial q_k’}\\ \So, \frac{d}{dt}\left(\frac{\partial L}{\partial q_k’}\right)-\frac{\partial L}{\partial q_k}=-\frac{\partial R}{\partial q_k'}

\end{align} $$

R: Rayleigh’s dissipation function (it’s half of the rate of dissipation energy against friction)