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MATH-F-204 Analytical mechanics: Lagrangian and Hamiltonian formalism

Glenn Barnich

Physique Théorique et Mathématique
Université libre de Bruxelles and International Solvay Institutes
Campus Plaine C.P. 231, B-1050 Bruxelles, Belgique
Bureau 2.O6. 217, E-mail: Glenn.Barnich@ulb.be, Tel: 02 650 58 01

March 29, 2024

Abstract

The aim of the course is an introduction to Lagrangian and Hamiltonian mechanics. The fundamental equations as well as standard applications of Newtonian mechanics are derived from variational principles.

From a mathematical perspective, the course and exercises (i) constitute an application of differential and integral calculus (primitives, graphs and functions); (ii) provide examples and explicit solutions of first and second order systems of differential equations; (iii) give an implicit introduction to differential manifolds and to realizations of Lie groups and algebras (change of coordinates, parametrization of a rotation, Galilean group, canonical transformations, one-parameter subgroups, infinitesimal transformations).

From a physical viewpoint, explicit and elegant solutions to idealized, but nevertheless concrete problems are provided. The more formal aspects of the course, such as (i) the covariance of the Lagrangian formalism under changes of Lagrange coordinates, (ii) symmetries and conservation laws in the form of Noether’s theorem as applied to Galilean invariance, (iii) canonical transformations in the Hamiltonian formalism, pave the way for a thorough understanding of the structure of quantum mechanics, both in its operatorial and path integral formulations, and of field theories with Poincaré invariance or general relativity.

The course of 30 hours goes hand in hand with 30 hours of exercises. The focus of the course is the general theoretical setting and abstract developments. Simple examples and concrete applications of these general concepts are treated during the exercise sessions. A full understanding of the material requires both of these complementary parts.

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