Scientific Disciplinary Sector (SSD)
MAT/05 - MATHEMATICAL ANALYSIS
The teaching is organized as follows:
Esercitazioni di teoria 1
The aim of the course is to introduce the theory and some applications of dynamical systems, which describe the time evolution of quantitative variables. By the end of the course, the students will be able to investigate the stability and the character of an equilibrium, the qualitative analysis of a system of ordinary differential equations, the phase portrait of a (parametric) dynamical system in dimension 1 and 2, and to analyse finite-dimensional Hamiltonian systems. Moreover, the students will be able to study some basic applications of dynamical systems arising from population dynamics, mechanics and traffic flows. Finally, students will be also able to produce proofs using the typical tools of modern dynamical systems and will be able to read and report specific books and articles on dynamical systems and related applications.
Prerequisites and basic notions
The material covered in first-year and second-year, first-semester courses - especially mathematical Analysis 1 and 2, linear algebra.
1. Topics in the theory of ordinary differential equations
Qualitative analysis of ODE: existence and uniqueness of solutions; maximal and global solutions; Gronwall’s Lemma; continuous dependence on the initial data.
2. Vector fields and ordinary differential equations
Vector fields: phase space, integral curves, orbits, equilibria, phase portrait. 1-dimensional examples of phase portraits. Second-order systems of differential equations; phase-space analysis and equilibria.
3. Linear systems
Linearisation of a vector field about an equilibrium. Classification of two-dimensional linear systems (over the real numbers) that are diagonalisable over the complex numbers. (If time permits, we will briefly discuss the nilpotent case as well.) n-dimensional linear systems: invariant subspace decomposition; the stable, unstable and central subspaces. Comparing a vector field with its linearisation about a hyperbolic equilibrium.
4. Flow of a vector field
Flow of a vector field. Change of coordinates: conjugate vector fields; pull-back and push-forward of a vector field by a diffeomorphism. Non-autonomous differential equations: time-dependent change of coordinates; scaling of vector fields and time reparametrisations. The local rectification theorem.
5. First integrals
Invariant sets; first integrals; Lie derivative. Invariant foliations; reduction of the order. First integrals and attractive equilibria.
6. Stability theory
Stability 'à la Lyapunov' of an equilibrium; the method of Lyapunov functions; the spectral method. Applications and examples.
7. 1-dimensional Newton equation.
Phase portraits of the 1-dimensional Newton equation, in the conservative case. Linearisation. Reduction of the order. Systems with friction.
Bifurcatios from equilibria, with 1-dimensional examples; applications.
9. Introduction to the 1-dimensional Calculus of Variations
The indirect method for one-dimensional integral functionals. Necessary conditions for the existence of minimisers: the Euler-Lagrange equations. Jacobi integral; conservation laws. Geodesics on a surface.
10. Hamiltonian systems
Hamiltonian vector fields. Legendre transform. Poisson brackets. Canonical transformations. Lie conditions, generating functions. The Hamilton-Jacobi equations. Integrability. Geometry of the phase space: Liouville's theorem and Poincaré's recurrence theorem.
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Lectures and exercises (often, combined).
Learning assessment procedures
The exam consists of two parts: a written part, and an oral one.
The written part consists of exercises - for instance, qualitative analysis of an ordinary differential equation; explicit solution of an ordinary differential equation; phase portrait of a two-dimensional, non-linear system; stability of equilibria; change of coordinates; first integrals; bifurcations; Hamiltonian systems and canonical transformations...
The written part tests the following learning outcomes:
- To have adequate analytical skills;
- To have adequate computational skills;
- To be able to translate problems from natural language to mathematical formulations;
- To be able to define and develop mathematical models for physics and natural sciences.
The oral part consists of 3 questions. The oral part is compulsory and must be completed within the same session as written part of the exam.
The oral exam tests the following learning outcomes:
- To be able to present precise proofs and recognise them.
According to the pandemic situation the structure of the exam may vary.
Criteria for the composition of the final grade
The written part of the exam is assigned a mark, usually over 30. This serves as basis for the final grade, which is over 30 and accounts for the oral part as well. (The final grade may be greater than, equal to or less than the mark for the written part.)