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 1900cite 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–? ]

:$frac\{partial\; S\}\{partial\; alpha\_\{r\; =\; G\_\{r\}$

Here, $alpha\_r$ is an arbitrary generalized acceleration and "G_r" is its corresponding generalized force; that is, the work done is given by

:$dW\; =\; sum\_\{r=1\}^\{D\}\; G\_\{r\}\; dq\_\{r\}$

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

:$S\; =\; frac\{1\}\{2\}\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; left|\; mathbf\{a\}\_\{k\}\; ight|^\{2\}$

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.

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 , and the corresponding angular acceleration vector

:

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation is . The velocity of the "k"^th particle is given by

:

where r_"k" is the particle's position in Cartesian coordinates; its corresponding acceleration is

:

Therefore, the function "S" may be written as

:

Setting the derivative of "S" with respect to equal to the torque yields Euler's equations

:$I\_\{xx\}\; alpha\_\{x\}\; -\; left(\; I\_\{yy\}\; -\; I\_\{zz\}\; ight)omega\_\{y\}\; omega\_\{z\}\; =\; N\_\{x\}$

:$I\_\{yy\}\; alpha\_\{y\}\; -\; left(\; I\_\{zz\}\; -\; I\_\{xx\}\; ight)omega\_\{z\}\; omega\_\{x\}\; =\; N\_\{y\}$

:$I\_\{zz\}\; alpha\_\{z\}\; -\; left(\; I\_\{xx\}\; -\; I\_\{yy\}\; ight)omega\_\{x\}\; omega\_\{y\}\; =\; N\_\{z\}$

Derivation

The change in the particle positions r_"k" for an infinitesimal change in the "D" generalized coordinates is

:$dmathbf\{r\}\_\{k\}\; =\; sum\_\{r=1\}^\{D\}\; dq\_\{r\}\; frac\{partial\; mathbf\{r\}\_\{k\{partial\; q\_\{r$

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

:$dW\; =\; sum\_\{r=1\}^\{D\}\; G\_\{r\}\; dq\_\{r\}\; =\; sum\_\{k=1\}^\{N\}\; mathbf\{F\}\_\{k\}\; cdot\; dmathbf\{r\}\_\{k\}\; =\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; dmathbf\{r\}\_\{k\}$

Substituting the formula for "d"r_"k" and swapping the order of the two summations yields the formulae

:$dW\; =\; sum\_\{r=1\}^\{D\}\; G\_\{r\}\; dq\_\{r\}\; =\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; sum\_\{r=1\}^\{D\}\; dq\_\{r\}\; left(\; frac\{partial\; mathbf\{r\}\_\{k\{partial\; q\_\{r\; ight)\; =\; sum\_\{r=1\}^\{D\}\; dq\_\{r\}\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; left(\; frac\{partial\; mathbf\{r\}\_\{k\{partial\; q\_\{r\; ight)$

Therefore, the generalized forces are

:$G\_\{r\}\; =\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; left(\; frac\{partial\; mathbf\{r\}\_\{k\{partial\; q\_\{r\; ight)\; =sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; left(\; frac\{partial\; mathbf\{a\}\_\{k\{partial\; alpha\_\{r\; ight)$

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

:$frac\{partial\; S\}\{partial\; alpha\_\{r\; =\; frac\{partial\}\{partial\; alpha\_\{r\; frac\{1\}\{2\}\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; left|\; mathbf\{a\}\_\{k\}\; ight|^\{2\}\; =\; sum\_\{k=1\}^\{N\}\; m\_\{k\}\; mathbf\{a\}\_\{k\}\; cdot\; left(\; frac\{partial\; mathbf\{a\}\_\{k\{partial\; alpha\_\{r\; ight)$

yielding Appell’s equation of motion

:$frac\{partial\; S\}\{partial\; alpha\_\{r\; =\; G\_\{r\}$

ee also

* Gauss' principle of least constraint

References

Further reading

*

*

* 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]