Lecturer: M. Fabbrichesi , SISSA

In this lesson the professor solve the problems of homework number 5. Then he continues with the **canonical transformation** and introduces the infinitesimal contact transformations. In the second part of the lecture the professor introduce the **Hamilton–Jacobi** theory by deriving the **Hamilton–Jacobi** equation for the **Hamilton's principal function?**. Then he solves the harmonic oscillator problem as an example of the **Hamilton–Jacobi** method solving this problem he obtain the **Hamilton’s characteristic function**.

In this lesson the profesor focused in the **Hamiltonian **formulation of classical mechanics and its physical consequences, the relation between the **Hamiltonian** and the **Lagrangian.** The functional form of **Hamilton's equations** and the **Poisson bracket** and their properties were introduced.

This lesson continues with the study of **Hamiltonian** and the **Lagrangian** formulation of classical mechanics. The lecture starts with the introduction of the **Liouville's theorem** and was demostrated that the **Hamiltonian** flow in phase space is incompressible. Some concepts were introduced as the **information conservation**, **friction** and **chaos**. It was also shown that the area in phase space is always the same, a result that is related with the **Uncertainty principle.** Finally the motion of a particle in a magnetic field was studied.