01379 - Relational Mechanics

Learning outcomes

At the end of the module, the student is expected to be familiar with the language of Lagrangian mechanics and the tools and methods required for the analytical analysis of motion in the classical setting.

Course contents

Recalls of Newtonian Mechanics — Review on curves. Galilean spacetime and the laws of mechanics; work, conservative forces, kinetic energy, momentum, torque, and angular momentum. Systems of multiple material points: cardinal equations. Non-inertial reference frames: Euler angles and Poisson's theorem. First and second König’s theorems.

Lagrangian Mechanics — Regular submanifolds and Lagrangian coordinates. d'Alembert-Lagrange principle. Lagrangian and Lagrange’s equations. Generalized forces and potentials. One-dimensional motion: motion on smooth guides under positional forces; harmonic motion, damped and forced oscillations. Equilibrium and phase space: functions and Lyapunov’s theorem, Lagrange-Dirichlet theorem, phase plane and qualitative study in the one-dimensional case. Small oscillations and normal modes. Lagrangian formalism in non-inertial frames: Foucault's pendulum. Variational formulation of Hamilton and Noether’s theorem.

Motion in a Central Field — Properties of orbits and Kepler’s second law, Lagrange stability, first and second forms of the orbit equation. Bertrand’s theorem (statement only). The Kepler problem.

Rigid Body Mechanics — Kinematics of rigid body motion: Mozzi-Chasles theorem, rolling; ruled surfaces, base curve and roulette. Dynamics of rigid body motion: kinetic energy; inertia homography and its fundamental properties, Huygens-Steiner theorem, principal axes of inertia and associated quadratic form; angular momentum of a rigid system; Euler’s equations, polhodes, Poinsot motion and precession, Lagrange top.

Readings/Bibliography

The reference textbooks for the module will be

Antonio Fasano, Stefano Marmi
Analytical Mechanics
Oxford Graduate Texts, 2013

Vladimir I. Arnold
Mathematical Methods of Classical Mechanics
Springer Verlag, 1997

Teaching methods

Blackboard lectures.

Assessment methods

The assessment consists of a written exam and an oral exam, both to be held during the same examination session. The written exam lasts two hours, during which the use of notes, textbooks, or electronic devices is not allowed. Access to the oral exam is conditional upon achieving a minimum score of 16/30 on the written exam.

Teaching tools

Supplementary materials and lecture notes can be found on the personal page of the lecturer.

Office hours

See the website of Gabriele Sicuro