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[1] and Josiah Willard Gibbs in 1879[2]

Here, is an arbitrary generalized acceleration, the second time derivative of the generalized coordinates qr, and Qr is its corresponding generalized force; that is, the work done is given by

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.

DerivationEdit

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

 

ExamplesEdit

Euler's equationsEdit

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 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

 
 
 

See alsoEdit

ReferencesEdit

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

Further readingEdit

  • 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