Abstract
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 Gauss' principle of least constraint in the general case.
Highlights
In this paper we study the so-called Gibbs-Appell equations and relate these equations to Gauss' principle of least constraint
The equations have held their appeal mainly because of their very simple form. These equations have yet to be placed in a general geometric framework since the presentation typically relies on the mechanical system being a collection of point masses and rigid bodies
In this case the Gibbs function is related to the Sasaki metric on the tangent bundle of the configuration manifold
Summary
In this paper we study the so-called Gibbs-Appell equations and relate these equations to Gauss' principle of least constraint. With the Gibbs function in hand, it is an easy matter to give the Gibbs-Appell equations in the absence of constraints and show that these equations are equivalent to Lagrange's equations In this case the Gibbs function is related to the Sasaki metric on the tangent bundle of the configuration manifold.
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