Lagrange’s equations for open systems, derived via the method of fictitious particles, and written in the Lagrange description of continuum mechanics

Abstract

In the present paper, a formulation of Lagrange’s equations, written in the framework of the Lagrange (material) description of Continuum Mechanics, is provided for open systems. An open system is constituted by (the portion of) a continuous material body that is enclosed by a nonmaterial surface. Such a surface, through which a flow of mass takes place, is also denoted as a control surface, and the corresponding volume is called a control volume. In order to apply Lagrange’s equations of analytical mechanics, the motion and the deformation of the body is modeled in the framework of the Ritz approximation technique by means of a finite number of generalized coordinates. However, since mass may not be conserved in an open system due to the flow of mass through the control surface, the original form of Lagrange’s equations must be accomplished by proper flux terms to be considered at the control surface. In order to derive this extended form, a local version of Lagrange’s equations is derived first, using a proper mathematical manipulation of the local relation of balance of linear momentum written in the Lagrange description of Continuum Mechanics. This local form is integrated over the volume that instantaneously is enclosed by the image of the control surface in a properly chosen reference configuration. In the integrated form, the Truesdell–Toupin method of fictitious particles and generalized Reynolds transport theorems are utilized in order to exchange the integrals and the partial derivatives with respect to time and generalized coordinates and velocities. This yields the desired form of Lagrange’s equations for open systems, written in the Lagrange description of Continuum Mechanics. Illustrative examples demonstrate the consistence of this novel form with the Euler (spatial) version of Lagrange’s equations for open systems, which was derived earlier by the present authors.