A Jacobi field is a potential force field whose potential is a homogeneous function of degree \(-2\) . The problem of the motion of a particle in such a field admits an additional integralquadratic in velocities. It can be used to reduce the number of degrees of freedom and topass to the study of a reduced system with spherical configuration space. These resultsare extended to the more general case of the motion of a particlein spaces of constant curvature. An analysis is made of particle motionon a cone whose vertex coincides with the singular point of theJacobi potential. A lower estimate of the distance from the moving particle to the vertex of the cone is given. This approach is also applicable to a more general case where the charged particleis additionally located in the magnetic field of a monopole. A billiard inside the conewith a particle bouncing elastically off its boundary isconsidered.
