Hamilton dynamics and $H$-planar curves

TL;DR

This paper investigates Hamilton flows on Kähler manifolds where trajectories are $H$-planar curves, deriving equations for such flows, and analyzes charged particle trajectories in magnetic fields on these manifolds, reducing the problem to ODEs.

Contribution

It introduces the concept of $H$-planar Hamilton flows, derives their governing equations, and applies these to study charged particle trajectories on Kähler manifolds with constant holomorphic sectional curvature.

Findings

Derived the equation for $H$-planar Hamilton flows.

Reduced particle motion equations to second-order ODEs.

Analyzed trajectories of charged particles in specific magnetic fields.

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.