Hamilton dynamics and $H$-planar curves

Abstract

Hamilton flows on K\"ahler manifold for which all trajectories are $H$-planar curves (complex analog of geodesics) are considered. These flows are called $H$-planar. The equation which has to obey the Hamiltonian of $H$-planar Hamilton flow is received and the method of finding general solution of this equation is proposed. Trajectories of charged particles in magnetic fields of special form on K\"ahler manifolds of constant holomorphic sectional curvature are studied. Using the fact that K\"ahler manifolds of constant holomorphic sectional curvature admit $H$-projective mapping on flat space the equation of particle motion is reduced to an ordinary differential equation of second order.