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In [[classical mechanics]], '''Appell's equation of motion''' is an alternative general formulation of [[classical mechanics]] described by [[Paul Émile Appell]] in 1900<ref name="appell_1900a">{{cite journal | last = Appell | first = P | year = 1900 | title = Sur une forme générale des équations de la dynamique.  | journal = Journal für die reine und angewandte Mathematik | volume = 121 | pages = 310&ndash;? }}</ref>

:<math>
\frac{\partial S}{\partial \alpha_{r}} = Q_{r}
</math>

Here, <math>\alpha_r</math> is an arbitrary [[generalized acceleration]] and ''Q<sub>r</sub>'' is its corresponding [[generalized force]]; that is, the work done is given by 

:<math>
dW = \sum_{r=1}^{D} Q_{r} dq_{r} 
</math>

where the index ''r'' runs over the ''D'' generalized coordinates ''q<sub>r</sub>'', which usually correspond to the [[degrees of freedom (physics and chemistry)|degrees of freedom]] of the system.  The function ''S'' is defined as the mass-weighted sum of the particle [[acceleration]]s squared, having the dimension of a generalised force for a generalised acceleration:  

:<math>
S = \frac{1}{2} \sum_{k=1}^{N} m_{k} \mathbf{a}_{k}^{2}
</math>

where the index ''k'' runs over the ''N'' particles.  Although fully equivalent to the other formulations of classical mechanics such as [[Newton's laws of motion|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 system|nonholonomic]] constraints are involved.  Appell’s formulation is an application of [[Gauss' principle of least constraint]].

==Example: Euler's equations==

[[Euler's equations (rigid body dynamics)|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 (geometric)|vector]] <math>\boldsymbol\omega</math>, and the corresponding angular acceleration vector

:<math>
\boldsymbol\alpha = \frac{d\boldsymbol\omega}{dt}
</math>

The generalized force for a rotation is the torque '''N''', since the work done for an infinitesimal rotation <math>\delta \boldsymbol\phi</math> is <math>dW = \mathbf{N} \cdot \delta \boldsymbol\phi</math>.  The velocity of the ''k''<sup>th</sup> particle is given by 

:<math>
\mathbf{v}_{k} = \boldsymbol\omega \times \mathbf{r}_{k}
</math>

where '''r'''<sub>''k''</sub> is the particle's position in Cartesian coordinates; its corresponding acceleration is

:<math>
\mathbf{a}_{k} = \frac{d\mathbf{v}_{k}}{dt} = 
\boldsymbol\alpha \times \mathbf{r}_{k} + \boldsymbol\omega \times \mathbf{v}_{k}
</math>

Therefore, the function ''S'' may be written as 

:<math>
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\}
</math>

Setting the derivative of ''S'' with respect to <math>\boldsymbol\alpha</math> equal to the torque yields Euler's equations

:<math>
I_{xx} \alpha_{x} - \left( I_{yy} - I_{zz} \right)\omega_{y} \omega_{z} = N_{x}
</math>

:<math>
I_{yy} \alpha_{y} - \left( I_{zz} - I_{xx} \right)\omega_{z} \omega_{x} = N_{y}
</math>

:<math>
I_{zz} \alpha_{z} - \left( I_{xx} - I_{yy} \right)\omega_{x} \omega_{y} = N_{z}
</math>

==Derivation==

The change in the particle positions '''r'''<sub>''k''</sub> for an infinitesimal change in the ''D'' generalized coordinates is 

:<math>
d\mathbf{r}_{k} = \sum_{r=1}^{D} dq_{r} \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}
</math>

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

:<math>
\frac{\partial \mathbf{a}_{k}}{\partial \alpha_{r}} = \frac{\partial \mathbf{r}_{k}}{\partial q_{r}}
</math>

The work done by an infinitesimal change ''dq<sub>r</sub>'' in the generalized coordinates is 

:<math>
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}
</math>

Substituting the formula for ''d'''''r'''<sub>''k''</sub> and swapping the order of the two summations yields the formulae

:<math>
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)

</math>

Therefore, the generalized forces are

:<math>
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)
</math>

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

:<math>
\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)
</math>

yielding Appell’s equation of motion

:<math>
\frac{\partial S}{\partial \alpha_{r}} = Q_{r}
</math>

==See also==

* [[Gauss' principle of least constraint]]

==References==

{{reflist|1}}

==Further reading==

* {{cite book | last = Whittaker | first = ET | year = 1937 | title = A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies | edition = 4th | publisher = Dover Publications | location = New York | id = ISBN }}

* {{cite journal | last = Seeger | year = 1930 | title = Unknown title | journal = Journal of the Washington Academy of Science | volume = 20 | pages = 481&ndash;? }}

* {{cite journal | last = Brell | first = H | year = 1913 | title = Unknown title | journal = Wien. Sitz. | volume = 122 | pages = 933&ndash;? }} Connection of Appell's formulation with the [[principle of least action]].
* [http://www.digizeitschriften.de/resolveppn/GDZPPN002164566 PDF copy of Appell's article at Goettingen University] 
* [http://www.digizeitschriften.de/resolveppn/GDZPPN002164760 PDF copy of a second article on Appell's equations and Gauss's principle]

[[Category:Classical mechanics]]