MATHEMATICAL PHYSICS

Second cycle degree in ASTRONOMY

Language: English

Teaching period: Second Semester

Lecturer: MASSIMILIANO GUZZO

Number of ECTS credits allocated: 6

Syllabus
 Prerequisites: Linear algebra and calculus with functions of several variables. Examination methods: Written examination with open questions and exercises on the topics discussed during lectures. Course unit contents: 1. Ordinary differential equations: Cauchy theorem, phase-space flow, dependence on the initial conditions; linear equations; phase-portraits, first integrals; equilibrium points; linearizations, stable, center and unstable spaces. 2. Integrable systems: elementary examples from population dynamics, from Mechanics and from Astronomy; integrability of mechanical systems, action-angle variables, examples. 3. Non-integrable Systems: discrete dynamical systems, Poincare' sections; bifurcations, elementary examples. Stable and Unstable manifols, homoclinic chaos; Lyapunov exponents, the forced pendulum and other examples; Center manifolds and partial hyperbolicity. The three body-problem, the Lagrange equilibria, Laypunov orbits, the tube manifolds. THE FOLLOWING TOPICS (4) AND (5) ARE ONLY IN THE PART FOR THE STUDENTS OF THE SECOND CYCLE DEGREE IN ASTRONOMY 4. Linear PDEs of first and second order, well-posed problems, the vibrating string, 1-dimensional wave equation, normal modes of vibrations, heat equation, Fourier series, 2-dimensional wave equation, Laplace operator and polar coordinates, separation of variables, Bessel functions, eigenfunctions of the Laplacian operator. 5. Laplace operator and spherical coordinates, separation of variables, Legendre polynomials and associate functions, Spherical harmonics, multipole expansions, L2 operator-eigenvalues and eigenfunctions, complete solution of the wave equation in space, Schrodinger polynomials. THE FOLLOWING TOPICS (6) ARE ONLY IN THE PART FOR THE STUDENTS OF THE SECOND CYCLE DEGREE IN MATHEMATICAL ENGINEERING 6. Examples and Applications: examples of analysis of three and four dimensional systems; limit cycles; the Lorenz system, the three-body problem; examples from fluid dynamics, non autonomous dynamical systems, chaos indicators, Lagrangian Coherent Structures.