Descriptif

This course introduces students to the Lagrangian and Hamiltonian mechanics.

Starting from the concepts of Newtonian mechanics, the course extends these

concepts to a more systematic description of the mechanics, adapted to complex

systems. The course will mostly use examples from the dynamics and vibrations of

mechanical systems, with progressively increasing complexity. Examples from

other fields of physics will be also proposed (electromagnetism, astrophysics,

chaos,...)

After a reminder of the classical concepts of point mechanics, the course

extends these concepts to the Lagrangian formalism and to the least action

principle. The Lagrangian formalism will be used to describe the mechanics of

rigid bodies. Lagrangian formalism will then be extended to the Hamiltonian

mechanics which is at the core of quantum physics and other modern theories in

physics. We will also present some extensions of Lagrangian and Hamiltonian

mechanics to other fields of physics.

Upon completion of this course, students master equations and principles in

analytical mechanics. They will be able to discuss the relevance of the chosen

model, as well as derive and solve simple models taken from their environment.

Main concepts covered: Fundamental law of dynamics; kinetic and potential

energy. Linearized equations of motion, dynamics of linear coupled oscillators.

Constraints and generalized coordinates, D'Alembert principle, Hamilton

principle, Euler-Lagrange equations of motion, conservation of energy and

momentum. Rigid body, center of mass, Euler angles, Moment of inertia and

inertia tensor, Euler equation of motion. Equations of Hamilton, conservation

theorem, canonical transformation, Poisson brackets.