Appell's equation of motion

Remarks

Differentiating the velocity formula gives $\ddot x_k = \sum_r e_{kr}\, \dot u_r + (\text{terms without } \dot u)$, so $\partial \ddot x_k / \partial \dot u_r = e_{kr}$ and $\partial S / \partial \dot u_r = \sum_k m_k \ddot x_k \cdot e_{kr}$. The equations therefore say $\sum_k (m_k \ddot x_k - F_k) \cdot e_{kr} = 0$, which is d'Alembert's principle, because the vectors $(e_{1r}, \dots, e_{Nr})$ span the virtual displacements and ideal constraint forces do no virtual work. For a holonomic system one may take $u_r = \dot q_r$, and then Appell's equations are equivalent to Lagrange's equations. Their advantage is for nonholonomic systems: they give exactly $m$ equations in the independent quasi-velocities, with no Lagrange multipliers. Example: a particle of mass $m$ in the plane under a force $F$, with polar coordinates and $u = (\dot r, \dot\theta)$ has $S = \frac m2\big((\ddot r - r\dot\theta^2)^2 + (r\ddot\theta + 2\dot r\dot\theta)^2\big)$, and $\partial S / \partial \ddot\theta = m r (r\ddot\theta + 2\dot r \dot\theta) = Q_\theta$ is the angular momentum balance.

Going deeper

For a student

Appell's equations are the analytic form of Gauss's principle of least constraint: minimizing $S - \sum_r Q_r \dot u_r$ over $\dot u$ gives exactly these equations. They extend Lagrange's equations to nonholonomic systems written in quasi-velocities, and they are used for rolling bodies and in multibody dynamics. The price is that one must compute accelerations, which is longer than computing the kinetic energy.