Varying the Cross-Section

The importance of coordinate transformations in the study of dynamical systems cannot be overestimated. For example, in the study of systems of linear constant coefficient ordinary differential equations, coordinate transformations allow one to decouple the system and hence reduce the system to a set of decoupled linear first-order equations which are easily solved. In the study of completely integrable Hamiltonian systems, the transformations to action-angle coordinates results in a trivially solvable system, and these coordinates are also useful in the study of near integrable systems. If we consider general properties of dynamical systems, coordinate transformations provide us with a way of classifying dynamical
systems according to properties which remain unchanged after a coordinate transformation.

(Wiggins, 1990)