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 $$oldsymbolomega, and the corresponding angular acceleration vector

:$$oldsymbolalpha = frac{doldsymbolomega}{dt}

The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation $$delta oldsymbolphi is $$dW = mathbf{N} cdot delta oldsymbolphi. The velocity of the "k"^th particle is given by

:$$mathbf{v}_{k} = oldsymbolomega imes mathbf{r}_{k}

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

:$$mathbf{a}_{k} = frac{dmathbf{v}_{k{dt} = oldsymbolalpha imes mathbf{r}_{k} + oldsymbolomega imes 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} ight)= frac{1}{2} sum_{k=1}^{N} m_{k} left{ left(oldsymbolalpha imes mathbf{r}_{k} ight)^{2} + left( oldsymbolomega imes mathbf{v}_{k} ight)^{2} + 2 left( oldsymbolalpha imes mathbf{r}_{k} ight) cdot left(oldsymbolomega imes mathbf{v}_{k} ight) ight}

Setting the derivative of "S" with respect to $$oldsymbolalpha 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]