Lagrange's equations from D'Alembert's principle

Consider $N$ particles with positions $\mathbf{r}_i(q_1,\dots,q_n,t)$ in terms of $n$ independent generalised coordinates $q_j$ (holonomic constraints). Let $\mathbf{F}_i$ be the applied force on particle $i$ (constraint forces do no virtual work on smooth constraints).

Step 1: D'Alembert-Lagrange principle

D'Alembert's principle with virtual displacements: the virtual work of the real forces plus the inertia forces is zero:

$$\sum_i (\mathbf{F}_i - m_i\ddot{\mathbf{r}}_i)\cdot\delta\mathbf{r}_i = 0$$

Step 2: Express in generalised coordinates

Since $\delta\mathbf{r}_i = \sum_j \dfrac{\partial\mathbf{r}_i}{\partial q_j}\delta q_j$, the applied-force term becomes $\sum_j Q_j\,\delta q_j$, where the generalised force is

$$Q_j = \sum_i \mathbf{F}_i\cdot\frac{\partial\mathbf{r}_i}{\partial q_j}$$

The inertia term is $\sum_j\left[\sum_i m_i\ddot{\mathbf{r}}_i\cdot\dfrac{\partial\mathbf{r}_i}{\partial q_j}\right]\delta q_j$.

Step 3: Convert the inertia term to kinetic energy

The velocity is $\dot{\mathbf{r}}_i = \sum_j \dfrac{\partial\mathbf{r}_i}{\partial q_j}\dot q_j + \dfrac{\partial\mathbf{r}_i}{\partial t}$. Two identities follow:

$$\frac{\partial\dot{\mathbf{r}}_i}{\partial\dot q_j} = \frac{\partial\mathbf{r}_i}{\partial q_j} \qquad \frac{d}{dt}\left(\frac{\partial\mathbf{r}_i}{\partial q_j}\right) = \frac{\partial\dot{\mathbf{r}}_i}{\partial q_j}$$

Then

$$\sum_i m_i\ddot{\mathbf{r}}_i\cdot\frac{\partial\mathbf{r}_i}{\partial q_j} = \frac{d}{dt}\left(\sum_i m_i\dot{\mathbf{r}}_i\cdot\frac{\partial\dot{\mathbf{r}}_i}{\partial\dot q_j}\right) - \sum_i m_i\dot{\mathbf{r}}_i\cdot\frac{\partial\dot{\mathbf{r}}_i}{\partial q_j}$$

With kinetic energy $T = \sum_i\tfrac12 m_i\dot{\mathbf{r}}_i^2$, this equals $\dfrac{d}{dt}\left(\dfrac{\partial T}{\partial\dot q_j}\right) - \dfrac{\partial T}{\partial q_j}$.

Step 4: Lagrange's equations

Because the $\delta q_j$ are independent, each coefficient must vanish:

$$\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j}\right) - \frac{\partial T}{\partial q_j} = Q_j \qquad (j = 1, \dots, n)$$

Conservative systems

For conservative forces $Q_j = -\dfrac{\partial V}{\partial q_j}$, where $V = V(q)$ does not depend on velocities. Define the Lagrangian $L = T - V$. Since $\dfrac{\partial V}{\partial\dot q_j} = 0$:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_j}\right) - \frac{\partial L}{\partial q_j} = 0$$

If some forces are non-conservative, they appear as $Q_j^{nc}$ on the right-hand side. The number of equations equals the number of DOF, and constraint forces never appear.