Monogenic system

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.

Abstract: In this paper we consider operators acting on Clifford algebra valued polynomials and, in particular, differential operators with polynomial coefficients. The decomposition of polynomials into homogeneous pieces leads to the classical homogeneous decomposition of operators and the further decomposition of homogeneous polynomials into monogenic polynomials leads to the concept of monogenic operator. Monogenic operators are characterized in terms of commutation relations and the monogenic decomposition of differential operators is studied in detail.

Abstract: Some mechanical aspects of a generalization with s-order derivatives of the Lagrange function are examined. The fundamental equations of a generalized Lagrangian mechanics are established