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

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

Here, $\alpha_r=\ddot{q_r}$ is an arbitrary generalized acceleration, the second time derivative of the generalized coordinates q_r and Q_r 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 q_r, 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,

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

where the index k runs over the N particles, and

$\mathbf{a}_k = \ddot{\mathbf{r}}_k = \frac{d^2 \mathbf{r}_k}{dt^2}$

is the acceleration of the kth particle, the second time derivative of its position vector r_k. Each r_k is expressed in terms of generalized coordinates, and a_k is expressed in terms of the generalized accelerations.

Appells formulation does not introduce any new physics to classical mechanics. It is fully equivalent to the other formulations of classical mechanics such as Newton's second law, Lagrangian mechanics, Hamiltonian mechanics, 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.

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

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

where Newton's second law for the kth particle

$\mathbf{F}_k = m_k\mathbf{a}_k$

has been used. 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

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

Examples

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 r_k 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

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

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

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

See also

• Gauss' principle of least constraint
• Analytical mechanics

Further reading

• Connection of Appell's formulation with the principle of least action.
• PDF copy of Appell's article at Goettingen University
• PDF copy of a second article on Appell's equations and Gauss's principle