Lagrange's equations in the form discussed in this chapter

QUESTION:

Lagrange's equations in the form discussed in this chapter hold only if the forces (at least the non constraint forces) are derivable from a potential energy. To get an idea how they can be modified to include forces like friction, consider the following: A single particle in one dimension is subject to various conservative forces (net conservative force \(=F=-\partial U / \partial x\)) and a nonconservative force (let's call it \(F_{\text {fric }}\)) Define the Lagrange as \(\mathcal{L}=T-U\) and show that the appropriate modification is \(\frac{\partial \mathcal{L}}{\partial x}+F_{\mathrm{fric}}=\frac{d}{d t} \frac{\partial \mathcal{L}}{\partial \dot{x}}\).