Studying at the University of Verona
Here you can find information on the organisational aspects of the Programme, lecture timetables, learning activities and useful contact details for your time at the University, from enrolment to graduation.
Study Plan
The Study Plan includes all modules, teaching and learning activities that each student will need to undertake during their time at the University.
Please select your Study Plan based on your enrollment year.
1° Year
| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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| Modules | Credits | TAF | SSD |
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1 module between the following1 module between the following 3 modules among the followingLegend | Type of training activity (TTA)
TAF (Type of Educational Activity) All courses and activities are classified into different types of educational activities, indicated by a letter.
Analytical mechanics (2020/2021)
Teaching code
4S001102
Teacher
Coordinator
Credits
6
Language
English
Scientific Disciplinary Sector (SSD)
MAT/07 - MATHEMATICAL PHYSICS
Period
II semestre dal Mar 1, 2021 al Jun 11, 2021.
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.
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.
• Review of dynamical systems and differential geometry. 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.
• 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.
| Author | Title | Publishing house | Year | ISBN | Notes |
|---|---|---|---|---|---|
| R. Abraham and J.E. Marsden | Foundations of mechanics. Second Edition. (Edizione 2) | Addison-Wesley | 1987 | 080530102X | Freely available at https://authors.library.caltech.edu/25029/ |
| D.D. Holm, T. Schmah and C. Stoica | Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions (Edizione 1) | Oxford University Press | 2009 | ||
| J.E. Marsden and T.S. Ratiu | Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems (Edizione 2) | Springer-Verlag | 1999 |
Examination Methods
The student is expected to demonstrate the ability to mathematically formalize and solve models used in several scientific discipline, using, adapting and developing the models and advanced methods discussed during the lectures. To that end the final evaluation will consist in a written and oral exam.
Written exam: One question/exercise for each part of the course (Part I and Part II), the solution will possible require the use of computer.
Oral exam: Subject of student choice and discussion of the written exam with questions.
The subject of student choice can be substituted with the development of a small-project to be decided together with the teacher.
