In this study, the dynamics of a relativistic particle that does not have an electric charge and is under the action of an external force has been analyzed on the basis of the special theory of relativity. The accelerated motion of such a particle was also investigated in the absence of an external electromagnetic field specified by the scalar and vector potentials. An analytical method for efficient writing of classical equations of relativistic dynamics was developed, and an estimate of integrals of motion was carried out. The motion integral was established to be valid both for charged particles and uncharged ones. The use of motion integrals made it possible to describe the relationship between dynamic parameters. The dependence of the x space-time coordinate on the integral of motion was also obtained.