# Appell's equation of motion

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In classical mechanics, **Appell's equation of motion** (aka Gibbs–Appell equation of motion) is an alternative general formulation of classical mechanics described by Paul Émile Appell in 1900^{[1]} and Josiah Willard Gibbs in 1879^{[2]}

Here, is an arbitrary generalized acceleration, the second time derivative of the generalized coordinates *q _{r}*, and

*Q*is its corresponding generalized force; that is, the work done is given by

_{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,

where the index *k* runs over the *N* particles, and

is the acceleration of the *k*th 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[edit]

The change in the particle positions **r**_{k} for an infinitesimal change in the *D* generalized coordinates is

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

The work done by an infinitesimal change *dq _{r}* in the generalized coordinates is

where Newton's second law for the *k*th particle

has been used. Substituting the formula for *d***r**_{k} and swapping the order of the two summations yields the formulae

Therefore, the generalized forces are

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

yielding Appell’s equation of motion

## Examples[edit]

### Euler's equations[edit]

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

## See also[edit]

## References[edit]

**^**Appell, P (1900). "Sur une forme générale des équations de la dynamique".*Journal für die reine und angewandte Mathematik*.**121**: 310–?.**^**Gibbs, JW (1879). "On the Fundamental Formulae of Dynamics".*American Journal of Mathematics*.**2**: 49–64. doi:10.2307/2369196.

## Further reading[edit]

- Pars, LA (1965).
*A Treatise on Analytical Dynamics*. Woodbridge, Connecticut: Ox Bow Press. pp. 197–227, 631–632. - Whittaker, ET (1937).
*A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies*(4th ed.). New York: Dover Publications. ISBN. - Seeger (1930). "Appell's equations".
*Journal of the Washington Academy of Science*.**20**: 481–484. - Brell, H (1913). "Nachweis der Aquivalenz des verallgemeinerten Prinzipes der kleinsten Aktion mit dem Prinzip des kleinsten Zwanges".
*Wien. Sitz*.**122**: 933–944. 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