The equations of Lagrange written for a non-material volume

Abstract

The Lagrange equations are extended with respect to a non-material volume which instantaneously coincides with some material volume of a continuous body. The surface of the non-material volume is allowed to move at a velocity which is different from the velocity of the material surface. The non-material volume thus represents an arbitrarily moving control volume in the terminology of fluid mechanics. The extension of the Lagrange equations to a control volume is derived by using the method of fictitious particles. Within a continuum mechanics based framework, it is assumed that, the instantaneous positions of both, the original particles included in the material volume, and the fictitious particles included in the control volume, are given as function of their positions in the respective reference configurations, of a set of time-dependent generalized coordinates, and of time. The corresonding spatial formulations are also assumed to be available. Imagining that the fictitious particles do transport the density of kinetic energy of the original particles, the partial derivatives of the total kinetic energy included in the material volume with respect to generalized coordinates and velocities are related to the respective partial derivatives of the total kinetic energy contained in the control volume. Hence follow the Lagrange equations for a control volume by substituting the above relations into the classical formulations for a material volume. In the present paper, holonomic problems are considered. The correction terms in the newly derived version of the Lagrange equations contain the flux of kinetic energy appearing to be transported through the surface of the control volume. This flux comes into the play in the form of properly formulated partial derivatives. Our version of the Lagrange equations is tested using the rocket equation and a folded falling string as illustrative examples.