The geometry of the Gibbs-Appell equations and Gaussʹ principle of least constraint

We present a generalisation of the Gibbs-Appell equations which is valid for general Lagrangians. The general form of the Gibbs-Appell equations is shown to be valid in the case when constraints and external forces are present. In the case when the Lagrangian is the kinetic energy with respect to a Riemannian metric, the Gibbs function is shown to be related to the kinetic energy on the tangent bundle of the configuration manifold with respect to the Sasaki metric. We also make a connection with the Gibbs-Appell equations and Gaussʹ principle of least constraint in the general case.