Families of Canonical Transformations by Hamilton-Jacobi-Poincar\'e equation. Application to Rotational and Orbital Motion

TL;DR

This paper revisits the Hamilton-Jacobi equation in extended phase space, constructing canonical transformations for Hamiltonian reduction, with applications to orbital and attitude dynamics, including new transformations.

Contribution

It develops a method to generate families of canonical transformations using Hamilton-Jacobi-Poincaré equations, including novel transformations for orbital and attitude dynamics.

Findings

Constructed families of canonical transformations for orbital and attitude dynamics.

Demonstrated the use of Whittaker and Andoyer charts in generating transformations.

Identified new canonical transformations applicable to rotational and orbital motion.

Abstract

The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in the extended phase space and including regularization, is revisited building canonical transformations with the purpose of Hamiltonian reduction. We illustrate our approach dealing with orbital and attitude dynamics. Based on the use of Whittaker and Andoyer symplectic charts, for which all but one coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole families of canonical transformations, among which one recognizes the familiar ones used in orbital and attitude dynamics. In addition, new canonical transformations are demonstrated.