Three-dimensional motion of a material point

Abstract

The two-dimensional motion of a material point over the active portion of its trajectory can be generalized in a natural way to three dimensions. Corresponding to the traditional flight plane, to which the trajectory of motion is confined in three dimensions, we have a set of flight surfaces obtained from it by bending. The three-dimensional system of differential equations governing the motion of a material point splits into a two-dimensional system, which describes the motion in the flight surface, and a system of ordinary differential equations, which describes the bending of the surface. By solving this system of equations one can determine by analytical means how the velocity and coordinate vectors over the active portion of the trajectory depend on its three-dimensional distortion. The results obtained may be used to analyse the three-dimensional motion of a material point, to select trajectories in space and to control the three-dimensional motion of the centre of mass over the active portions. In some cases one can actually derive analytical expressions for solutions to boundary-value and extremal problems associated with the three-dimensional motion of a material point.