Learning objectives

The class is devoted to a modern study of classical mechanics from a mathematical point of view. The aim of the class is to introduce the tools and techniques of global and numerical analysis, differential geometry and dynamical systems to formalise a model of classical mechanics. At the end of the class a student should be able to construct a model of physical phenomena of mechanical type, write the equations of motion in Lagrangian and Hamiltonian form and analyse the dynamical aspects of the problem.

Prerequisites and basic notions

Qualitative analysis of Ordinary Differential Equations, and stability theory. Theory of differential manifolds, tangent and cotangent bundles, vector fields, Lie derivatives, differential forms and Riemannian metrics.

Program

• Introduction. At the beginning of the course we will quickly review the basic aspects of Newtonian mechanics. The structure of the Galilean space-time and the axioms of mechanics. Systems of particles: cardinal equations. Conservative force fields. Mass particle in a central field force and the problem of two bodies.

• Lagrangian and Hamiltonian mechanics on Rn. Equivalence of Euler-Lagrange, Hamilton and Newton equations in the mechanical case. Hamilton's principle, conservation of generalised energy and invariance of Euler-Lagrange equation with respect to lifted change of coordinates. Legendre transformation. Cyclic variables and reduction in the Hamiltonian contest. Poisson brackets and first integrals.

• Lagrangian mechanics on manifolds. Constrained systems: d’Alembert principle and Lagrange equations. Models of constraints and their equivalence. Invariance of Lagrange equations for change of coordinates. Jacobi integral. Stability theory for Lagrangian systems and small oscillations. Noether’s Theorem, conserved quantities and Routh’s reduction.

Applications: the Foucault pendulum, the magnetic stabilisation and others.

• Rigid bodies. Orthonormal basis, orthogonal and skew-symmetric matrices. Space and body frame: angular velocities. Cardinal equations in different reference frames. A model for rigid bodies. Euler’s equations.

• Introduction to Lie groups and algebras. Group actions, trivializations and Euler-Poincare' theory.

Bibliography

Didactic methods

In-room lectures, team working, homeworks and weekly summary in teams

Learning assessment procedures

The exam is divided in two part: a written and an oral test. Only students who have passed the written exam will be admitted to the oral examination.

Evaluation criteria

The written test is based on the solution of open-form problems and the oral test in which students are required to discuss the written test and to answer some questions proposed in open form. If positive, the mark obtained in the written test will be valid until the last session of the present
academic year (February 2024).

In particular, objective of evaluation will be:
- Knowledge and understanding: a part of the written and the oral tests will be devoted to verify the effective knowledge and understanding of the course's contents (mainly, the third exercise of the written test and the oral test).

- Applying knowledge and understanding: both during the written and the oral tests, the student will be required to solve problems based on the course's contents.

- Making judgements: during the tests, the student can be asked to solve problems requiring a contribution basing on the material of the course assigned for personal study.

- Communication skills: during the written and the oral tests, the solutions expressed in a clear, complete and short way will be preferred.

- Learning skills: part of the course's contents will be based on textbook or scientific articles left to the students for personal study.

Criteria for the composition of the final grade

A student must obtain a mark of at least 18/30 (best) in both the written and oral part to pass the exam,
and the final grade will be given by the average of the marks of the written and of the oral part.

Exam language

Inglese