Generalized symmetries in mechanics and field theories

Generalized symmetries are introduced in a geometrical and global formalism. Such a framework applies naturally to field theories and specializes to mechanics. Generalized symmetries are characterized in a Lagrangian context by means of the transformation rules of the Poincaré–Cartan form and the (generalized) Nöther theorem is applied to obtain conserved quantities (first integrals in mechanics). In the particular case of mechanics it is shown how to use generalized symmetries to study the separation of variables of Hamilton–Jacobi equations recovering standard results by means of this new method. Supersymmetries (Wess–Zumino model) are considered as an intriguing example in field theory.