Third-order bending magnet optics for Cartesian coordinates

A thorough analysis of the charged particles dynamics on the basis of the matrix formalism up to the 3rd order inclusive for the sector dipole magnets is usually held in a curvilinear coordinate system. In those cases, whenever the dipole is not a sectoral one, transformation to the curvilinear coordinates, associated with the trajectory of the central particle of the beam, doesn´t prove itself, because of the difficulty of the physical interpretation of the obtained results. That makes necessary to analyse the dynamics of the beam in the bending magnet in the cartesian (rectangular) coordinate system