Appell's equation of motion
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 and Josiah Willard Gibbs in 1879
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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,
where the index k runs over the N particles, and
is the acceleration of the kth particle, the second time derivative of its position vector rk. Each rk is expressed in terms of generalized coordinates, and ak 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.
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
yielding Appell’s equation of motion
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
- 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