Trimester 03 · Course detail

Continuous Dynamical Systems & Wave Fields

Ordinary differential equations, Fourier analysis and distributions, classical thermodynamics and statistical kinetics, wave dynamics and physical optics

Credits: 15
Sequence: 3 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Ordinary differential equations, Fourier analysis and distributions, classical thermodynamics and statistical kinetics, wave dynamics and physical optics

Evidence of mastery

Dynamical-systems & wave-field study + research brief

Subject 01

Ordinary Differential Equations & Dynamical Systems

Official syllabus code: MIMUW 1000-213bRR

The qualitative theory of ODEs: existence and uniqueness of flows, linear systems, and the stability machinery that classifies non-linear behaviour.

Existence, Uniqueness & Flow Topology

Picard–Lindelöf, Peano & Grönwall

Construct local solutions by successive approximation, contrast with Peano's non-unique existence result, and bound solution divergence through the Grönwall inequality.

Linear Systems & Matrix Exponentials

Solve x′ = Ax via the matrix exponential e^{tA}, decompose the flow through Jordan structure, and introduce periodic linear systems with Floquet theory.

Autonomous Systems, Phase Portraits & Invariant Manifolds

Analyze autonomous systems in the phase plane, classify critical points, and construct stable and unstable invariant manifolds of equilibria.

Non-Linear Stability: Lyapunov, LaSalle & Poincaré–Bendixson

Prove stability with Lyapunov functions, apply the LaSalle invariance principle to limit sets, and use the Poincaré–Bendixson theorem to constrain planar dynamics.

Subject 02

Fourier Analysis, Integral Transforms & Distributions

Official syllabus code: MIMUW AM III / Metody

From the Hilbert-space geometry of L² to Schwartz functions, Sobolev spaces, and tempered distributions — the functional-analytic backbone of wave fields.

Hilbert Space Geometry & Series

L² Hilbert Space Geometry & Sturm–Liouville Eigenfunction Bases

Treat L² as a Hilbert space and derive complete orthonormal eigenfunction bases from regular Sturm–Liouville problems.

Fourier Series, Fejér Kernels & Dirichlet Kernel Singularities

Prove convergence of Fourier series through summability with Fejér kernels and expose the singularity structure of the Dirichlet kernel.

Transforms, Sobolev Spaces & Distributions

The Continuous Fourier Transform & the Schwartz Space S(ℝⁿ)

Define the transform on rapidly decreasing functions, prove its continuity, inversion, and mapping properties on the Schwartz space.

Sobolev Spaces Hˢ(ℝⁿ) & Tempered Distributions S′(ℝⁿ)

Develop the Sobolev scale Hˢ and extend the Fourier transform to tempered distributions, the natural setting for generalized wave solutions.

Subject 03

Classical Thermodynamics & Statistical Kinetics

Official syllabus code: FUW 1100-2INZ23

Thermodynamics as a differential-geometric theory of state space, then microscopically grounded through kinetic theory and the Boltzmann equation.

Thermodynamic Structure

Exact Differentials, Pfaffian Forms & Carathéodory's Formulation

Formulate heat and work with exact and inexact differentials and derive the second law from Carathéodory's axiomatic inaccessibility principle.

Thermodynamic Potentials, Legendre Transforms & Maxwell Relations

Build the internal-energy-to-Helmholtz-to-Gibbs ladder via Legendre transforms and generate the Maxwell relations from state-function symmetry.

Phases & Kinetics

Phase Transitions, Clausius–Clapeyron & Landau Free Energy

Derive coexistence curves with the Clausius–Clapeyron relation and model continuous phase transitions through Landau free-energy expansion and symmetry breaking.

Kinetic Theory, the Boltzmann Equation & the H-Theorem

Move from distribution functions to the Boltzmann transport equation and prove Boltzmann's H-theorem as the microscopic origin of irreversibility.

Subject 04

Wave Dynamics, Physical Optics & Symplectic Lab

Official syllabus code: FUW 1100-2INZ13 / COMP

Wave propagation from d'Alembert's solution through dispersive and vector optics, closed by an in-silico laboratory on symplectic integration.

Wave Equations & Dispersive Media

Hyperbolic d'Alembert & Inhomogeneous Wave Equations

Derive d'Alembert's general solution of the 1-D wave equation and solve inhomogeneous problems with retarded Green's functions in higher dimensions.

Dispersive Media: Phase Velocity, Group Velocity & Kramers–Moyal

Distinguish phase from group velocity in dispersive wave packets and model their transport with Kramers–Moyal-type expansions.

Vector Optics & Geometric Limits

Fresnel Equations, Polarization Tensors & the Geometric Eikonal

Derive Fresnel reflection and transmission amplitudes, encode polarization states in tensor form, and recover ray optics as the eikonal limit of the wave equation.

In-Silico Lab

Symplectic Integrators: Verlet, Ruth 3rd/4th & RK4 Stability

Implement and benchmark Verlet and Ruth third- and fourth-order symplectic schemes against Runge–Kutta 4, comparing long-term energy behaviour and linear stability regions.