Cartesian approach for constrained mechanical systems with three degrees of freedom

In the history of mechanics, there have been two points of view for studying mechanical systems: Newtonian and Cartesian. ing to the Cartesian point of view, the motion of mechanical systems is described by the first-order differential equations in an N-dimensional configuration space Q. s paper we develop the Cartesian approach to mechanical systems with three degrees of freedom and with constraint which is linear with respect to velocity. The obtained results we apply to discuss integrability of geodesic flows on the surface in the three-dimensional Euclidian space and to analyze integrability of a heavy rigid body in the Suslov and the Veselova cases.