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[1]

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

Here, $\alpha_r$ 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 nonholonomic constraints are involved. Appell’s formulation is an application of Gauss' principle of least constraint.

Example: Euler's equations

Euler's equations provide an excellent illustration of Appell's formulation.

Consider a rigid body of N particles joined by rigid rods. The rotation of the body may be described by an angular velocity vector $\boldsymbol\omega$, and the corresponding angular acceleration vector

$\boldsymbol\alpha = \frac{d\boldsymbol\omega}{dt}$

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation $\delta \boldsymbol\phi$ is $dW = \mathbf{N} \cdot \delta \boldsymbol\phi$. The velocity of the kth particle is given by

$\mathbf{v}_{k} = \boldsymbol\omega \times \mathbf{r}_{k}$

where rk is the particle's position in Cartesian coordinates; its corresponding acceleration is

$\mathbf{a}_{k} = \frac{d\mathbf{v}_{k}}{dt} = \boldsymbol\alpha \times \mathbf{r}_{k} + \boldsymbol\omega \times \mathbf{v}_{k}$

Therefore, the function S may be written as

$S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \left( \mathbf{a}_{k} \cdot \mathbf{a}_{k} \right) = \frac{1}{2} \sum_{k=1}^{N} m_{k} \left\{ \left(\boldsymbol\alpha \times \mathbf{r}_{k} \right)^{2} + \left( \boldsymbol\omega \times \mathbf{v}_{k} \right)^{2} + 2 \left( \boldsymbol\alpha \times \mathbf{r}_{k} \right) \cdot \left(\boldsymbol\omega \times \mathbf{v}_{k}\right) \right\}$

Setting the derivative of S with respect to $\boldsymbol\alpha$ equal to the torque yields Euler's equations

$I_{xx} \alpha_{x} - \left( I_{yy} - I_{zz} \right)\omega_{y} \omega_{z} = N_{x}$
$I_{yy} \alpha_{y} - \left( I_{zz} - I_{xx} \right)\omega_{z} \omega_{x} = N_{y}$
$I_{zz} \alpha_{z} - \left( I_{xx} - I_{yy} \right)\omega_{x} \omega_{y} = N_{z}$

Derivation

The change in the particle positions rk for an infinitesimal change in the D generalized coordinates is

$d\mathbf{r}_{k} = \sum_{r=1}^{D} dq_{r} \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}$

Taking two derivatives with respect to time yields an equivalent equation for the accelerations

$\frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} = \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}$

The work done by an infinitesimal change dqr in the generalized coordinates is

$dW = \sum_{r=1}^{D} Q_{r} dq_{r} = \sum_{k=1}^{N} \mathbf{F}_{k} \cdot d\mathbf{r}_{k} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot d\mathbf{r}_{k}$

Substituting the formula for drk and swapping the order of the two summations yields the formulae

$dW = \sum_{r=1}^{D} Q_{r} dq_{r} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \sum_{r=1}^{D} dq_{r} \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right) = \sum_{r=1}^{D} dq_{r} \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right)$

Therefore, the generalized forces are

$Q_{r} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{r}_{k}}{\partial q_{r}} \right) = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} \right)$

This equals the derivative of S with respect to the generalized accelerations

$\frac{\partial S}{\partial \alpha_{r}} = \frac{\partial}{\partial \alpha_{r}} \frac{1}{2} \sum_{k=1}^{N} m_{k} \left| \mathbf{a}_{k} \right|^{2} = \sum_{k=1}^{N} m_{k} \mathbf{a}_{k} \cdot \left( \frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} \right)$

yielding Appell’s equation of motion

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