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 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.
Prerequisites: PHY101, PHY102, PHY201, PHY202
The course describes waves in fluids, with a preference for illustrations coming from the Earth system, in particular the atmosphere and ocean. Waves are one essential type of motion present in many fluids. One goal of the course is to demonstrate how one proceeds to obtain wave solutions starting from a physical description of a system and its equations of motion. Acoustic waves will be considered as a first example, surface water waves at different scales (from ripples in the pond to tsunamis) will be derived as further examples. Basics of fluid mechanics (Euler equations, kinematics) will be introduced in order to make these developments possible. Similarities in the behavior of fluid waves and optical waves seen in PHY202 will be discussed.
At the end of the course, the students will understand how one characterizes a family of waves (dispersion relation, polarisation relations), and how to proceed to obtain, in a given system, wave solutions if they exist. Some preliminary considerations for exploring behaviors beyond linearity will have been introduced, as an opening. Finally, some elements of the study of the Earth, and of the atmosphere in particular, will have been introduced.
