Differential geometric methods in physics 1.

Course description:
Differential geometric methods are applied in many branches of theoretical physics, for example in general relativity, in Yang-Mills theory and in the modern formulation of analytical mechanics. At the beginning of the course, the basic notions of differential geometry (manifolds, exterior forms, tensor fields, Lie groups, Riemann manifolds etc) are surveyed. Then phase spaces (symplectic and Poisson manifolds), Hamiltonian systems, symmetries and the methods of Hamiltonian symmetry reduction are treated in some detail. Finally an introduction is given to fibre bundles and connexion theory, with field theoretical (gravity and Yang-Mills) applications.

Recommended Texts:

• V.I. Arnold: Mathematical Methods of Classical Mechanics, second edition, Springer-Verlag, 1989.
• J.E. Marsden, T.S. Ratiu: Introduction to Mechanics and Symmetry, second edition, Springer-Verlag, 1999.
• B.A. Dubrovin, A.T. Fomenko, S.P. Novikov: Modern Geometry - Methods and Applications, Vols. I and II, Springer-Verlag, 1985.