28375 - Mathematical Physics 2

Learning outcomes

At the end of the course, the student knows the general methods of mechanics to set-up and possibly solve any problem of free and constrained motion. Autonomy of judgment and critical spirit in relation to the analytical solutions of motion problems will be developed.

Course contents

Newtonian Mechanics — Overview: introduction to the properties of curves, including their geometric and differential properties, and the Frenet theorem; vector fields and integral curves; surfaces, first fundamental form, and geodesics on surfaces. Galilean Spacetime and Laws of Mechanics: the Galilean group; postulates of mechanics and Newton’s principle of determinism; work and conservative fields, the work-energy theorem, and conservation of mechanical energy; linear momentum and angular momentum. Constraints: general concepts of smooth and rough constraints, and the d'Alembert-Lagrange principle. One-dimensional motion: general principles and Weierstrass's approach;
phase space and Lyapunov stability; phase space analysis of the simple pendulum. Small oscillations, damped and forced harmonic motion, and beat phenomena. Motion in a central field: general properties; study of closed orbits, Lagrange stability, and Bertrand's theorem; the Kepler problem. Many-body systems: cardinal equations of motion; the two-body problem. Scaling laws and similarity principles in mechanical systems.

Lagrangian Mechanics — Regular submanifolds and Lagrangian Coordinates. Introduction to Variational Calculus: geodesics. Constrained Systems and holonomic systems, d'Alembert-Lagrange principle; Lagrange equations, generalized potentials, Hamilton's variational principle, Maupertuis principle and Jacobi metric. Noether's Theorem and Conservation Principles: gauge transformations; brachistochrone curve. Equilibrium and Oscillations: Dirichlet stability; small oscillations and Lyapunov functions. Hamiltonian Formalism: Legendre transform, Young's inequality, Hamilton's equations and variational principle; Liouville's theorem and Poincaré recurrence theorem.

Rigid Systems — Reference Changes: Euler angles, angular velocity and Poisson formulas, laws of velocity change, accelerations, and angular momentum, apparent forces and Foucault pendulum; first and second König theorems. Kinematics of Rigid Motion: Mozzi-Chasles theorem, rolling; ruled surfaces, base and roulette, pure rolling disk, and crank-slider mechanism. Dynamics of Rigid 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 equations, trajectories on the inertia ellipsoid, 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.

Links to further information

https://gsicuro.github.io/docs/teaching/fmath2/fismat2/

Office hours

See the website of Gabriele Sicuro