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Non-conservative forces

I guess next I should explain how we account for non-conservative forces. Essentially we replace the $0$ in Equations (??) with the generalized force $Q_i$. It is defined as:

Where $\textbf{F}_j$ represents the vector of the $j$th non-conservative force we wish to account for in our equations of motion. As you can see by the summation bounds, we have $m$ non-conservative forces we wish to account for. $\textbf{r}_j$ is the position vector for force $j$. $\textbf{F}_j\cdot\dfrac{\partial \textbf{r}_j}{\partial q_i}$ denotes the dot product between $\textbf{F}_j$ and $\dfrac{\partial \textbf{r}_j}{\partial q_i}$. Alternatively, to calculate $Q_i$ we can use velocities instead of positions:

$\dot{\textbf{r}}_j$ is the time derivative of the position vector (that is, the velocity vector) for force $j$. Hence the more generalized form of Equations (??) is:

The velocity-based definition of $Q_i$ can also be used, which yields: