Explicit equations from orbit reduction: One and two stages

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 three physical examples.