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
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
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
Euler's equations of rigid body dynamics
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
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
Seeger (1930). "Appell's equations". Journal of the Washington Academy of Sciences. 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