Gauss principle for a super-thin elastic rod dynamics

Abstract

The analytical formulation of dynamics of a super-thin elastic rod was studied on the basis of Gauss variation. The definition of virtual displacement in the generalized acceleration space with respect to the arc-coordinate and the time were given with expression of Gauss variation，respectively. The nonholonomic constraint equations of the virtual displacements expressed by Gauss variation were given. The Gauss’s principle of the dynamics of a super-thin elastic rod was established, from which the Kirchhoff equation, the Lagrange equation, the Nielsen equation and the Appell equation of the rod can be derived. The Lagrange equation with indeterminate multipliers was obtained for the case when the rod is subjected to the nonholonomic constraints. The Gauss's principle of least compulsion of a super-thin elastic rod was proved and the compulsion function has a minimum for the actual motion and its physical meaning was indicated.