Lagrangian Mechanics on Manifolds

In this chapter, we study Lagrangian mechanics on the total space of the tangent bundle of the configuration space. We discuss non-uniqueness of the Lagrangian, its invariances in value and form, and we describe the most general force consistent with a Lagrangian formulation. In this context, we describe the mechanics of a particle moving in a general curved space-time in General Relativity. Most of the chapter is devoted to the relation between Lagrangian symmetries and conserved quantities. We prove a generalized version of the Noether problem and present a large number of examples of conserved Noether charges in different contexts.