Lagrange Mechanics: Important Special Systems

In this chapter, we study some important mechanical systems from the viewpoint of the Lagrangian formalism. First we consider time-independent systems with one degree of freedom and show that they can always be solved by quadratures. In the case of bounded motion, we describe the functional relation between the shape of the potential and the period of the motion. Then we consider the two-body problem with a potential which depends only on the distance of the two bodies, and reduce it to the problem of a particle moving in a central potential. We solve the motion in the central potential in terms of quadratures, study the precession of the perihelion, and state the Bertrand theorem. The Kepler problem is studied in detail. Finally we consider the rigid body, discuss its kinematics, and deduce Euler’s equations. After introducing Euler’s angles, we conclude with a study of the Lagrange symmetric top in a constant gravitational field.