During the days 9, 10 and 11 of July 2018 I taught three lessons on Symplectic Geometry in Classical Mechanics at the SEMF Summer School, organized in the Burjassot Campus at Universitat de València with the following **contents**:

**Lesson 1**: *Classical Mechanics on manifolds*. We start studying the Newtonian formalism and finish at the Symplectic Geometry formulation, passing through the Lagrangian Formalism. We study a practical example by constructing the configuration space on a Riemannian manifold and the phase space on its tangent bundle.

**Lesson 2**: *Foundations of Symplectic Geometry*. In this class we introduce the basic concepts of symplectic geometry: the Darboux theorem, symplectic and Hamiltonian fields and the Poisson bracket. We see how the concepts of Hamiltonian mechanics appear in a natural way in the symplectic context. Finally, we introduce the Noether theorem in its symplectic version and we study concrete examples.

**Lesson 3**: *The theorem of Arnold-Liouville.* In this class we give a proof of the theorem of Arnold-Liouville and construct the action-angle variables for general systems. After that, we study the case of harmonic oscillators and introduce conditionally periodic motion.

Here are some **references**:

- V.I. Arnold.
*Mathematical methods of Classical Mechanics*. Springer-Verlag, 1989. - H. Goldstein, C.P. Poole, and J.L. Safko.
*Classical Mechanichs*. Pearson, 2011. - L.D. Landau and E.M. Lifshitz.
*Curso de física teórica (vol. 1): Mecánica*. Reverté, 1985. - M. Spivak.
*Physics for Mathematicians: Mechanics I*. Publish or Perish, 2010.

These are the **lecture notes** that I developed for the course (in Spanish):