Mathematical Physics 2023/24
Laurea magistrale in Fisica, I year, II semester, 6 CFU (60 hours)
Teachers: Alessandro Teta and Guido Cavallaro
NOTICE:
The course will start on February 28, 2024, with the following time-table:
Wednesday 2.00 p.m. - 4.00 p.m. in aula Conversi
Thursday 8.00 a.m. - 10.00 a.m. in aula Conversi
Syllabus
Dynamical systems: basic elements on ordinary differential equations, Liouville transport theorem, equilibrium and stability, Lyapunov theorem.
Hamiltonian systems: basic elements on Hamilton's equations, action-angle variables, first order canonical perturbation theory.
Continuous systems: Euler equation; Navier-Stokes equation; examples and applications.
Detailed program (notice: w.p. means without proof)
1. Ordinary differential equations
Existence and uniqueness theorem (w.p.), Gronwall lemma, continuity w.r. to initial conditions.
Equilibrium position, stable and asymptotically stable equilibrium position, Lyapunov theorem, stability and instability recognized from the linearized system (w.p.), examples and applications.
[HS] ch. 8, 9, [C]
(see also [E] ch. 2 and sect. 10.5, or [B] ch. 2).
2. Hamiltonian systems
Derivation of Hamilton's equations, variational principle, constant of motions, Poisson brackets.
Definition of canonical and completely canonical transformations, examples.
Criterion via Poisson brackets, extended phase space, sufficient condition via differential form, generating functions, examples.
Hamilton-Jacobi equation for the time-dependent and time-independent case, separation of variables, action-angle variables.
Liouville theorem (local integrability)
Arnold theorem (w.p.).
First order perturbation theory.
Applications: action-angle variables for the planar Kepler problem, restricted planar three body problem, precession of the perihelion of Mercury.
The classical limit of Quantum Mechanics.
[T] sect. 6.1, 6.2, 6.4, 6.5, 6.7, 6.8, 6.9, 6.10, 6.11, 6.12 and [T1] Appendix A.
(See also [T1] sect. 1.1,...,1.5; [FM] sect. 11.4, 11.5; [E] ch. 12 and 13, sect. 14.1, 14.2; [Ce] ch. 7, sections MC1,...,MC5).
3. Fluid Mechanics
Euler’s Equations. Conservation of mass; Balance of Momentum; Conservation of energy; Transport Theorem; Incompressible flows; Isentropic fluids; Bernoulli’s Theorem; Example: Euler flow in a channel; Rotation and Vorticity; Kelvin’s Circulation Theorem; Evolution equation for the vorticity; Stream lines and Stream function; Example: Stream function with radial symmetry. [CM] chapter 1.
Navier-Stokes Equations. Stress tensor; Cauchy’s Theorem; Navier-Stokes Equations in dimensionless variables; Reynolds number; Helmholtz–Hodge Decomposition Theorem; Stokes Equations; Dissipation of kinetic energy; Example: flow in a channel (Poiseuille flow). [CM] chapter 1.
Variational principle for Euler Equations ([MP], page 5); Examples: Couette flow, gravity waves [N].
Potential Flow. Complex velocity and complex potential; Blausius Theorem; Kutta–Joukowski Theorem; Examples, flow around a disk, potential vortex; D’Alembert’s Paradox in Three Dimensions (see also [MP] page 52); Point vortex Hamiltonian system (see also [MP] chapter 4); Rigorous derivation of the point vortex system ([MP], page 157, and [N]); Boundary layer; Prandtl equations. [CM] chapter 2.
Stability of stationary solutions. Liapunov function; Arnold stability theorem. [MP] chapter 3 (pages 102-110).
References
[C] E. Caglioti, Lecture Notes at https://sites.google.com/site/ecaglioti/didattica/MR
[T] Teta, A., Notes on Hamiltonian Mechanics (16/4/2024)
[E] R. Esposito, Appunti dalle lezioni di Meccanica Razionale, ed. Aracne
[T1] Teta, A Mathematical Primer on Quantum Mechanics, Springer 2018
[H-S] M. Hirsch and S Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press 1974
[FM] A. Fasano and S. Marmi, Analytical Mechanics, Oxford Univ. Press 2006
[H] K. Huang: Statistical Mechanics, Wiley, 1991
[CM] A. Chorin, J.E. Marsden: A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 2000.
[BCM] P. Buttà, G. Cavallaro, C. Marchioro: Mathematical Models of Viscous Friction, Lecture Notes in Mathematics (Springer), 2015.
[Ce] A. Celletti, Esercizi e complementi di meccanica razionale, ed. Aracne
[CM] A. Chorin, J.E. Marsden: A Mathematical Introduction to Fluid Mechanics, Springer-Verlag, 2000.
[MP] C. Marchioro, M. Pulvirenti: Mathematical Theory of Incompressible Nonviscous Fluids, Springer-Verlag, 1994.
[N] notes available on elearning.
The exam consists of an interview on the most relevant topics presented in the course.
DATES OF THE EXAMS
EXAM RESERVATION DATE START EXAM RESERVATION DATE END EXAM DATE
27/12/2023 10/01/2024 15/01/2024
03/06/2024 20/06/2024 24/06/2024
27/06/2024 11/07/2024 16/07/2024
26/08/2024 06/09/2024 12/09/2024
01/11/2024 17/11/2024 21/11/2024
27/12/2024 10/01/2025 15/01/2025
For further information on the course see: https://corsidilaurea.uniroma1.it/it/corso/2023/32384/programmazione