Mechanics, Symmetries and Noether’s Theorem

Abstract

The Lagrangian description of mechanics allows to derive the equations of motion from a variational principle based on conserved quantities of the system. In the first part of this chapter, the Lagrangian formulation of dynamics and the properties of the Lagrangian operator are synthetically reviewed starting from Hamilton’s Principle of First Action. In the second part of the chapter, the important link between continuous symmetries of the Lagrangian operator and conserved quantities of the system is introduced through Noether’s Theorem. The proof of the Theorem is reported both for material particles and for continuous systems such as fluids.