where the index runs over the generalized coordinates , which usually correspond to the degrees of freedom of the system. The function is defined as the mass-weighted sum of the particle accelerations squared,
where the index runs over the particles, and
is the acceleration of the -th particle, the second time derivative of its position vector. Each is expressed in terms of generalized coordinates, and is expressed in terms of the generalized accelerations.
Relations to other formulations of classical mechanics
Appell's formulation does not introduce any new physics to classical mechanics and as such is equivalent to other reformulations of classical mechanics, such as Lagrangian mechanics, and Hamiltonian mechanics. All physics is contained within Newton's laws of motion. In some cases, Appell's equation of motion may be more convenient than the commonly used Lagrangian mechanics, particularly when nonholonomic constraints are involved. In fact, Appell's equation leads directly to Lagrange's equations of motion. Moreover, it can be used to derive Kane's equations, which are particularly suited for describing the motion of complex spacecraft. Appell's formulation is an application of Gauss' principle of least constraint.
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
where Newton's second law for the kth particle
has been used. 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
^Gibbs, JW (1879). "On the Fundamental Formulae of Dynamics". American Journal of Mathematics. 2 (1): 49–64. doi:10.2307/2369196. JSTOR 2369196.
^Appell, P (1900). "Sur une forme générale des équations de la dynamique". Journal für die reine und angewandte Mathematik. 121: 310–?.
^Deslodge, Edward A. (1988). "The Gibbs–Appell equations of motion" (PDF). American Journal of Physics. 56 (9): 841–46. doi:10.1119/1.15463.
^Deslodge, Edward A. (1987). "Relationship between Kane's equations and the Gibbs-Appell equations". Journal of Guidance, Control, and Dynamics. American Institute of Aeronautics and Astronautics. 10 (1): 120–22. doi:10.2514/3.20192.
^Lewis, Andrew D. (August 1996). "The geometry of the Gibbs-Appell equations and Gauss' principle of least constraint" (PDF). Reports on Mathematical Physics. 38 (1): 11–28. doi:10.1016/0034-4877(96)87675-0.
Pars, LA (1965). A Treatise on Analytical Dynamics. Woodbridge, Connecticut: Ox Bow Press. pp. 197–227, 631–632.
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