1. Introduction

Lagrangian mechanics employs the least-action principle to derive Newton's equations from a scalar function, the action function L. In classical mechanics, L is the difference of kinetic and potential energies and therefore appears as an artifice. It is somewhat mysterious, then, that the reformulation of Newtonian mechanics in terms of momentum and position, rather than velocity and position as in Lagrangian mechanics, leads immediately to the total energy function H and a plethora of geometric structure that is hidden in the native setting.

Due to the advantages of the Hamiltonian perspective, this chapter studies Lagrangian systems from this dual point of view. The organization of the chapter is this: Section 2 recalls the classic construction of angle-action variables in 1 degree of freedom via several examples, then states and proves the Liouville- Arnol'd theorem; Section 3 discusses the relationship between the topology of the configuration space and the existence of integrable mechanical systems; and it reviews several constructions of integrable systems whose configuration space is the sphere or torus. Section 3 provides a number of open problems that may stimulate interested researchers or students.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and eproduction in any medium, provided the original work is properly cited.
