On vector form of differential variational principles of mechanics

The paper introduces the variation of a vector x, which can be interpreteed either as a virtual displacement of a system, the variation of the velocity of a system or the variation of the acceleration of a system. This vector is used to put forward, from scalar representative motion equations, a uniform notation of all differential variational principles of mechanics. Conversely, this notation involves all original motion equations, which enables one to consider the previously obtained scalar products as principles of mechanics. The same approach leads us to a differential principle on the basis of the vector equation of constrained motion of a mechanical system. In this form, it is proposed to retain the nonzero scalar product of ideal constraints by the vector x. This enables one from this notation to derive equations involving generalized constrained forces