| fall | time | place | lectures | tuesday 14:00 - 17:00 | MI 611 |
ECTS : 8 credit points
Classical Mechanics deals with dynamical systems without damping or friction. Here think of ideal pendula, springs or tops, the Solar system, etc. We deal with the Newtonian, the Lagrangian (or variational) and the Hamiltonian theory of such systems.
Important issues are the Noether Theorem, that links conservation laws to symmetry and the Liouville Theorem stating the conservation of volume by the phase flow. As a direct consequence of the latter, Hamiltonian systems can't have attractors.
Main part of the theory deals with symplectic manifolds, that also has applications in other parts of physics, like in optics. The language introduced in this course is needed for all reading of modern Dynamical Systems and Mathematical Physics.
Ordinary Differential Equations,
Manifolds (a bit),
Introductory Dynamical Systems.
Presentation and home work excercises.
Tuesday 7 september. Introduction and overview, systems with 1 degree of freedom, harmonic oscillator, systems with 2 degrees of freedom, energy surface, accessible region in configuration space. Exercises (.pdf, .ps).
Tuesday 14 september. Central force field, effective potential, 2-body-problem, variational principle, Euler-Lagrange equations. Exercises (.pdf, .ps).
Tuesday 21 september. Mathematical pendulum - Lagrangean viewpoint, Legendre transformation, Hamiltonian equations, mathematical pendulum - Hamiltonian viewpoint, conservation of phase space volume, theorem of Liouville, Poincaré Recurrence Theorem. Exercises (.pdf, .ps).
Tuesday 28 september. Small oscillations, coupled harmonic oscillators, Lagrangean mechanics on manifolds, tangent bundle, Noether's theorem. Exercises (.pdf, .ps).
Tuesday 5 october. Natural mechanical systems, principle of d'Alambert, Lagrange multiplier, parametric resonance, exterior algebra. Exercises (.pdf, .ps).
Tuesday 12 october. Exterior product, differential form, path integral, exterior derivative, Stokes theorem, free rigid body, symplectic manifold, symplectic form, canonical 1-form, Hamiltonian vector field. Exercises (.pdf, .ps).
Tuesday 19 october. Hamiltonian flow preserves the Hamiltonian, Poisson bracket, Noether's theorem revisited, Lagrange top, Lie bracket, Lie derivative. Exercises (.pdf, .ps).
Tuesday 26 october. Cartan's magic formula, Hamiltonian flow preserves the symplectic structure (is canonical), theorem of Liouville, Poincaré mappings preserve volume, fluid dynamics, linear symplectic structures, symplectic geometry. Exercises (.pdf, .ps).
Tuesday 2 november. Linear symplectic mappings, spectrum, Darboux's theorem, Hamiltonian systems on S^2, Euler's equations. Exercises (.pdf, .ps).
Tuesday 9 november. Integral invariants, geometric description based on vortex lines, principle of least action. Exercises (.pdf, .ps).
Tuesday 16 november. Generating functions, the Hamilton-Jacobi method, Poisson structures, action angle variables, integrable systems, Liouville's theorem. Exercises (.pdf, .ps).
Tuesday 23 november. Proof of Liouville's theorem, the Korteweg-de Vries equation, averaging. Exercises (.pdf, .ps).
Tuesday 30 november. Adiabatic invariants, Birkhoff normal form theory. Exercises (.pdf, .ps).
Tuesday 7 december. Fundamental problem of classical dynamics, Diophantine conditions, perturbation theory, the double pendulum, the geometry of reaction dynamics, transition state theory in two degrees of freedom. Exercises (.pdf, .ps).
Tuesday 14 december. Transition state theory in more degrees of freedom, Huygen's priciple and geometric optics, systems with symmetries.