On Vector Form of Differential Variational Principles of Mechanics

Abstract

We introduce variation of a vector δx which can be interpreted either as a virtual displacement of a system, or as variation of the velocity of a system, or as variation of the acceleration of a system. This vector is used to obtain a unified form of differential variational principles of mechanics from the scalar representative equations of motion. Conversely, this notation implies the original equations of motion, which enables one to consider the obtained scalar products as principles of mechanics. Using the same logical scheme, one constructs a differential principle on the basis of the vector equation of constrained motion of a mechanical system. In this form of notation, it is proposed to conserve the zero scalar products of reactions of ideal constraints and the vector δx. This enables one to obtain also the equations involving generalized constrained forces from this form of notation.