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.
The generalized force gives the work done where the index
, which usually correspond to the degrees of freedom of the system.
is defined as the mass-weighted sum of the particle accelerations squared, where the index
-th particle, the second time derivative of its position vector
is expressed in terms of generalized coordinates, and
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.
[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] 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 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
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