Preservation of Canonical Structure in Non-planar Curvilinear Coordinates

In a separate paper, the transformation rules to curvilinear coordinates and the form of common differential operators in these coordinates are derived. In this paper we study transformations that preserve an underlying Lagrangian or Hamiltonian structure by providing transformations to suitably constructed canonical variables. Within the canonical framework, various advanced techniques to study the dynamics can be applied. One important such application is the interchange of the independent variable to the arc length along the reference orbit under preservation of existing canonical structure. To illustrate the approach, we derive the canonical curvilinear equations of motion for relativistic dynamics in gravitational and electromagnetic fields, for which the use of perturbative techniques is important and widespread. AMS Subj. Classification: 70G99, 70H09, 70H99, 78A35