Lagrangian Dynamics

Abstract

In this chapter, Lagrangian dynamics is described. After introducing the configuration space for a system of constrained rigid bodies, the principle of virtual power and its equivalence to Lagrange’s equations is shown. Then, these equations are formulated in the case of conservative forces and forces deriving from a generalized potential energy. The fundamental relation between conservation laws (first integrals) and the symmetries of the Lagrangian function is proved (Noether’s theorem). Further, Lagrange’s equations for linear nonholonomic constraints are formulated. The chapter contains also the analysis of small oscillations about a stable equilibrium configuration (normal modes) , and an introduction to variational calculus including Hamilton’s principle. Finally, two geometric formulations of Lagrangian’s dynamics are presented together with Legendre’s transformation that allows to transform Lagrange’s equations into Hamilton’s equations.