More generally, Lagrange equations$^1$ read
$$\begin{align} \frac{d}{dt}\frac{\partial (T-U)}{\partial \dot{q}^j}-\frac{\partial (T-U)}{\partial q^j}~=~&Q_j-\frac{\partial{\cal F}}{\partial\dot{q}^j}+\sum_{\ell=1}^m\lambda^{\ell} a_{\ell j}, \cr j~\in \{1,\ldots, n\},\end{align} \tag{L}$$
where
$q^1,\ldots ,q^n,$ are $n$ generalized position coordinates;
$T$ is the kinetic energy;
$U$ is a generalized potential;
${\cal F}$ is the Rayleigh dissipation function for friction forces;
$Q_1,\ldots ,Q_n,$ are the remaining parts of the generalized forces, which are not described by the generalized potential $U$ or the Rayleigh dissipation function ${\cal F}$;
$\lambda^1,\ldots ,\lambda^m$, are $m$ Lagrange multipliers for $m$ semi-holonomic constraints
$$\begin{align} \sum_{j=1}^n a_{\ell j}(q,t)\dot{q}^j+a_{\ell t}(q,t)~=~0, \cr \ell~\in ~&\{1,\ldots, m\}. \end{align}\tag{SHC}$$
One may think of the last term on the right-hand side of eq. (L) as the generalized constraint forces for the semi-holonomic constraints (SHC). All other constraints are assumed to be holonomic.
For a discussion of conservative & non-conservative forces, see also e.g. this Phys.SE post.
References:
- H. Goldstein, Classical Mechanics; Chapter 1 & 2.
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$^1$ We distinguish between Lagrange equations (L) and Euler-Lagrange equations
$$\begin{align} \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^j}-\frac{\partial L}{\partial q^j}~=~&0, \cr j~\in~&\{1,\ldots, n\}.\end{align}\tag{EL} $$
In contrast to the Lagrange equations (L), the EL equations are by definition always assumed to be derived from a stationary action principle.
We should stress that it is not possible to apply the stationary action principle to derive the Lagrange equations (L) unless all generalized forces have generalized potentials $U$. See also e.g. this and this Phys.SE posts.
