Explicit equations from orbit reduction: one and two stages

TL;DR

This paper derives explicit equations for orbit reduction in mechanical systems, comparing one and two-stage reduction methods, and illustrates the approach with a rigid body example.

Contribution

It provides explicit formulas for the reduced two-form and equations of motion in both one and two-stage orbit reduction, including cases with normal subgroups.

Findings

Explicit reduced two-form expressions for one-stage reduction

Derived equations of motion for both reduction methods

Validated methods with a rigid body example

Abstract

It is known that orbit reduction can be performed in one or two stages and it has been proven that the two processes are symplectically equivalent. In the context of orbit reduction by one stage we shall write an expression for the reduced two-form in the general case and obtain the equations of motion derived from this theory. Then we shall develop the same process in the case in which the symmetry group has a normal subgroup to get the reduced symplectic form by two stages and the consequent orbit reduced equations. In both cases we shall illustrate the method with the example of a rigid body with rotors and compare the obtained equations with the ones given by other authors in different frameworks.