Lectures & Tutorials

  • Fall 2020/2021
  • When and where = Fridays, 12:20-14:45 via Zoom. A link will be available on MS Teams.
  • Language: Czech or English
  • Requirements for exam: To pass the exam, students should prove understanding of the fundamental parts of the theory and ability to solve typical problems.
  • To get to the exam, students have to solve a project assignment, pass the midterm (písemka v půlce semestru) and solve the homeworks.
  • Assignments and study materials will be available in MS Teams.
  • Some content is based on our book.

Lectures

Approximate syllabus (can be changed if students wish):

  • Principle of least action, Hamilton canonical equations. Cornerstones of differential geometry, Lie groups and algebras, duals of Lie algebras. Euler-Poincare equations, rigid body rotation. Infinite-dimensional Lie groups and fluid mechanics.
  • Lagrangian and Eulerian fluid mechanics, solids, viscoelastic fluids.
  • Principle of maximum entropy. Liouville and Boltzmann entropy. Sackur-Tetrode relation for ideal gases.
  • Reversibility and irreversibility. Dissipation potential. Energetic and entropic representations. Second law of thermodynamics.
  • General Equation for Non-equilibrium Reversible-Irreversible Coupling (GENERIC).
  • Liouville equation and kinetic theory. Electromagnetic fields and their interaction with matter.
  • Mixtures, hyperbolic evolution and Maxwell-Stefan equation, Fick's and Ohm's law. Hyperbolic heat conduction and Fourier's law.
  • Fluctuations, Wiener processes, fluctuation dissipation theorem, Fokker-Planck equation, Ito calculus.

Lectures

  1. 2.10.: Introduction to GENERIC. Principle of least action. Hamiltonian mechanics. Hamilton-Jacobi equation. Assignment: Geometric optics
  2. 9.10.: Tensor fields, pull-backs, Lie derivative.
  3. 16.10.: Lie derivative and Jacobi identity. Functional derivatives. Noether (and her inverse) theorem, symmetries and conserved quantities. Assignment: Kelvin-Helmholtz theorem in fluid mechanics
  4. 23.10.: Lie groups. Hamiltonian rigid body mechanics by reduction, Euler-Poincare equations.
  5. 30.10.: Hamilton canonical continuum mechanics on the material manifold (elasticity).
  6. 6.11.: Reductions of Poisson brackets. Eulerian Hamiltonian continuum mechanics, complex fluid mechanics, superfluids.
  7. 13.11.: Reductions of Poisson brackets. Eulerian Hamiltonian continuum mechanics, complex fluid mechanics, superfluids.
  8. 20.11.: Entropy. Shannon information entropy, Liouville entropy. Geometric principle of maximum entropy (MaxEnt). Entropy of ideal gas.
  9. 27.11.: Gradient dynamics, dissipation potentials.
  10. 4.12.: Hydrodynamic dissipation. Maxwell model of complex fluids, Navier-Stokes-Fourier. Relation with natural configurations. Reversibility and irreversibility. Dynamic MaxEnt reduction technique. Onsager-Casimir reciprocal relations.
  11. 11.12.: Test (Pisemka)
  12. 18.12.: Random processes, Langevin equation and fluctuations.
  13. 8.1.: Fluctuation dissipation theorem in GENERIC.