Appell's Equation of Motion

Derivation

The change in the particle positions r_k 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 dq_r 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 dr_k 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

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

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Appell's Equation of Motion - Derivation