Appell's equation of motion

Summary

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

Statement edit

The Gibbs-Appell equation reads

 

where   is an arbitrary generalized acceleration, or the second time derivative of the generalized coordinates  , and   is its corresponding generalized force. The generalized force gives the work done

 

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 edit

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 classical mechanics 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.[3] Moreover, it can be used to derive Kane's equations, which are particularly suited for describing the motion of complex spacecraft.[4] Appell's formulation is an application of Gauss' principle of least constraint.[5]

Derivation edit

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

 

Examples edit

Euler's equations of rigid body dynamics 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  , since the work done for an infinitesimal rotation   is  . The velocity of the  -th particle is given by

 

where   is the particle's position in Cartesian coordinates; its corresponding acceleration is

 

Therefore, the function   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

  1. ^ Gibbs, JW (1879). "On the Fundamental Formulae of Dynamics". American Journal of Mathematics. 2 (1): 49–64. doi:10.2307/2369196. JSTOR 2369196.
  2. ^ Appell, P (1900). "Sur une forme générale des équations de la dynamique". Journal für die reine und angewandte Mathematik. 121: 310–?.
  3. ^ Deslodge, Edward A. (1988). "The Gibbs–Appell equations of motion" (PDF). American Journal of Physics. 56 (9): 841–46. Bibcode:1988AmJPh..56..841D. doi:10.1119/1.15463. S2CID 123074999.
  4. ^ 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. Bibcode:1987JGCD...10..120D. doi:10.2514/3.20192.
  5. ^ 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. Bibcode:1996RpMP...38...11L. doi:10.1016/0034-4877(96)87675-0.

Further reading edit