# Appell's Equation of Motion

In classical mechanics, Appell's equation of motion is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900

$frac{partial S}{partial alpha_{r}} = Q_{r}$

Here, is an arbitrary generalized acceleration and Qr is its corresponding generalized force; that is, the work done is given by

$dW = sum_{r=1}^{D} Q_{r} dq_{r}$

where the index r runs over the D generalized coordinates qr, which usually correspond to the degrees of freedom of the system. The function S is defined as the mass-weighted sum of the particle accelerations squared, having the dimension of a generalised force for a generalised acceleration:

$S = frac{1}{2} sum_{k=1}^{N} m_{k} mathbf{a}_{k}^{2}$

where the index k runs over the N particles. Although fully equivalent to the other formulations of classical mechanics such as Newton's second law and the principle of least action, Appell's equation of motion may be more convenient in some cases, particularly when constraints are involved. Appell’s formulation can be viewed as a variation of Gauss' principle of least constraint.