Appell's equation of motion
where the index r runs over the D generalized coordinates qr, 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, having the dimension of a generalised force for a generalised acceleration:
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 nonholonomic constraints are involved. Appell’s formulation is an application of Gauss' principle of least constraint.
Example: Euler's equations
Euler's equations provide an excellent illustration of Appell's formulation.
The generalized force for a rotation is the torque N, since the work done for an infinitesimal rotation is . The velocity of the kth particle is given by
where rk 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
The change in the particle positions rk 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 dqr in the generalized coordinates is
Substituting the formula for drk 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
- Appell, P (1900). "Sur une forme générale des équations de la dynamique.". Journal für die reine und angewandte Mathematik 121: 310–?.
- 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). "Unknown title". Journal of the Washington Academy of Science 20: 481–?.
- Brell, H (1913). "Unknown title". Wien. Sitz. 122: 933–?. 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