# Master's degree in Mathematics

### Analytical mechanics

Course code
4S001102
Name of lecturers
Number of ECTS credits allocated
6
MAT/07 - MATHEMATICAL PHYSICS
Language of instruction
English
Period
II semestre dal Mar 2, 2020 al Jun 12, 2020.

## Learning outcomes

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.

## Syllabus

• Introduction. At the beginning of the course we will quickly review some basic aspects of dynamical systems using the modern tools of differential geometry and global analysis. Vector fields on a manifolds, flow and conjugation of vector fields. First integrals, foliation of the phase space and reduction of order for a ODE. 1-dimensional mechanical systems.

• 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 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.

• 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.

Applications: the Foucault pendulum, the Kepler problem and the magnetic stabilisation.

• Symplectic manfolds and Hamiltonian dynamics. Hamilton equations, Poisson brackets. Noether’s Theorem from the Hamiltonian point of view.

Some qualitative numerical aspects will also been investigated. The course will also include seminars in geometric mechanics, geometric control theory and applications to robotics and surgical robotics.

 Reference books Author Title Publisher Year ISBN Note A. Fasano and S. Marmi Analytical Mechanics: an Introduction. Oxford University Press 2006 D.D. Holm, T. Schmah and C. Stoica Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions (Edizione 1) Oxford University Press 2009 N. Sansonetto Notes for the course of Analytical Mechanics 2019

## Assessment methods and criteria

To success in the exam, students must show that:
- they know and understand the fundamental concepts of Newtonian, Lagrangian and Hamiltonian mechanics;
- they have abilities in solving problems in mechanics, both from the abstract and the computational point of view
- they support their argumentation with mathematical rigor.

The exam is divided in two part: a written test based on the solution of open-form problems and an oral
test in which the student is required to discuss the written test and to answer some questions proposed
in open form. Only students who have passed the written exam will be admitted to the oral examination.
If positive, the mark obtained in the written test will be valid until the last session of the present