This course introduces the Hamiltonian formulation of classical systems with constraints, emphasizing its role in understanding gauge symmetries and its applications in modern theoretical physics. Beginning with a review of regular Hamiltonian dynamics, the course develops the Dirac–Bergmann formalism for singular systems, introducing key concepts such as constraint classification, Dirac brackets, and reduced phase space. A central focus is the analysis of local symmetries in the Hamiltonian framework, including the construction of symmetry generators, their algebraic structure, and the relationship with Noether’s theorem. These ideas are illustrated through classical gauge theories such as electromagnetism, Yang–Mills theory, and Chern–Simons theory. The course also explores the geometric side of the formalism, introducing symplectic geometry and describing the structure of phase space in geometric terms. In the final part, the formalism is applied to General Relativity, concluding with the interpretation of conserved quantities, such as the mass of a black hole, within the Hamiltonian framework. This course is designed for graduate students in physics or mathematics with prior knowledge of classical mechanics. It provides a solid foundation for further study in gauge theories, general relativity, and quantum field theory.

Guest Lecture: Olivera Miskovic, Pontificia Università Cattolica di Valparaiso, Cile Research lines - Gauge/Gravity duality - Black hole physics - (Super)gravity in diverse dimensions - Asymptotic symmetries in AdS and ¿at spaces - Hamiltonian dynamics of gauge theories Her research activity focuses on theoretical high-energy physics and gravitational theories, with particular emphasis on gauge/gravity duality, black hole physics, and (super)gravity in diverse spacetime dimensions. Her core interests include the study of asymptotic symmetries in anti–de Sitter and flat spacetimes, Hamiltonian dynamics of gauge theories, and holographic approaches to quantum field theories. Over the past years, her research has been supported by multiple competitive national and international grants, including several FONDECYT Regular and Postdoctoral projects. These projects address topics such as boundary dynamics in AdS gravity, holographic descriptions of non-conformal gauge theories, black holes and their asymptotic symmetries, holographic complexity, and the stability of rotating black holes. Her work also encompasses applications of holography to quantum gravity, high-energy physics, and condensed matter systems, notably within the ANID–ANILLO international collaboration project. Her research agenda has been strengthened by international collaboration, including a CNRS-funded research visit to the Centre de Physique Théorique at École Polytechnique, fostering long-term cooperation with leading groups in holography and gravitational physics. Part 1: Hamiltonian Dynamics of Singular Systems 1. Review of Hamiltonian Dynamics of Regular Systems 2. Singular systems 3. Dirac–Bergmann algorithm, consistency conditions 4. Total and extended Hamiltonian 5. Reduced phase space, Dirac brackets 6. Gauge conditions 7. Number of degrees of freedom 8. Examples of classical mechanics Part 2: Local Symmetries in the Hamiltonian Formalism 1. Generators of local symmetries 2. Castellani’s method 3. Conserved charges 4. Lie algebra 5. Connection with Noether’s theorem 6. Examples of gauge theories (electromagnetism, Yang–Mills theory, Chern-Simons theory) Part 3: Symplectic geometry and canonical transformations 1. Symplectic form and canonical structure 2. Canonical transformations as symplectomorphisms 3. Symmetry generators revisited geometrically 4. Examples Part 4: Gravitational Hamiltonian 1. Review of General Relativity and Einstein-Hilbert action 2. ADM decomposition 3. Constraints in General Relativity 4. Gravitational Hamiltonian structure 5. Dirac algebra 6. Example – black hole mass as Hamiltonian conserved charge

On site

Oral presentation

P.D.2-2 - May