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Classical Mechanics [MA3673]

Wintersemester 2016/17

Prof. Dr. Robert König

  • Donnerstags,14:15--15:45
  • Raum MI 02.10.011
  • erstes Seminar: 20. Oktober 2016
  • 2. Teil: Blockseminar: Samstag, 3. Dezember 2016, Beginn 9.00 Uhr. Faculty Club (4. Stock), IAS Pfeil
  • Hauptseminar Classical Mechanics: Topics

    Topics

    1. Newtonian mechanics: spacetime, reference frames, Galilei-transformations. The two-body problem. Kepler's laws and motion in a centrally symmetric potential. Motion of a point particle in 1 dimension, phase portraits. (Cuong Pham, Mona Muhr, 20.10.2016)

    2. The general N-body problem. Integrals/constants of motion: momentum, angular momentum, (kinetic) energy. The 10 classical constants of motion for a closed system. Rotating/moving reference frames, Coriolis and centrifugal force. (Jan Tabeling, Lilly Palackal, 27.10.2016)

    3. Oscillations: linearization, first-order linear inhomogeneous differential equations, propagators/flows, spectral decompositions and solution by exponential map. Stability. Damped and undamped oscillators, coupled oscillators. Parametric resonance, trace criterion for 2-dimensional systems. (Ludwig Kraft, Emrullah Akbas, 3.11.2016)

    4. Phase space of N particles, vector fields, dynamical systems, integral curves (trajectories), Hamiltonian vector fields and Hamiltonians, integrals of motion, local flows, autonomous and non-autonomous systems, existence of integrals. (Efdal Kalkan, Falko Spaeh, 10.11.2016)

    5. Lagrangian mechanics: Lagrangian, uniqueness/equivalence/gauge invariance, transformation under coordinate changes, holonomic/nonholonomic constraints, variational calculus, virtual displacements, Lagrangian equations of the first kind, D'Alembert's principle. (Christine Spieker, Tobias Haugeneder, 17.11.2016)

    6. Lagrangian equations of the second kind, examples. Principle of least action. Continuous symmetries, Noether's theorem, derivation of the 10 classical integrals of motion. (Robert Tandler, Alexandra Starostina, 24.11.2016)

    7. Hamiltonian mechanics: canonical equations of motion, Legendre-transformations, Hamiltonians, equivalence of Lagrangian and Hamiltonian formulations, transformation of Hamiltonians under diffeomorphisms. (Ludwig Kraft, Robert Tandler, 1.12.2016)

    8. Motion of rigid bodies. Euler-Lagrange equations, moments of inertia and angular momentum. The spinning top: motion of an axially symmetric rigid body in a uniform force field. (Jan Tabeling, Christine Spieker, 3.12.2016)

    9. Symplectic group, phase space as a symplectic vector space, linear Hamiltonian systems, canonical (symplectic) transformations and their generators, characterization of Hamiltonian vector fields and canonical flows. Hamilton's equations in terms of the Poisson bracket. (Falko Spaeh, Efdal Kalkan, 3.12.2016)

    10. Poisson brackets, continuous symmetries and integrals of motion. Liouville's theorem on phase space volume. Poincaré's theorem. Schwarzschild's capture theorem. (Tobias Haugeneder, Emrullah Akbas, 3.12.2016)

    11. Numerical examples (Alexandra Starostina)

    12. Deterministic chaos (Lilly Palackal, 3.12.2016)

    13. Special relativity: limitations of non-relativistic mechanics, postulates of special relativity, structure of the (in)homogeneous Lorentz group, covariance of equations of motion, boosts and addition of rapidities, proper time and relativistic equation of motion (Cuong Pham, Mona Muhr, 3.12.2016)

    References

    We will mostly follow Straumann's book [1] and Graf's lecture notes [5]. Additional references: [4] (a classic), [3] (mathematical) and [2] (physics-oriented).

    1.
    N. Straumann. Springer, 2015. Available here

    2.
    F. Scheck, Mechanik.Springer, 1996. Available here

    3.
    G. Gallavotti, The Elements of Mechanics, ser. Theoretical and Mathematical Physics.Springer, 1983. Available here

    4.
    V.I. Arnold, Mathematical Methods of Classical Mechanics, ser. Graduate Texts in Mathematics.Springer, 1978. Available here

    5.
    G. M. Graf, ``Allgemeine Mechanik,'' Vorlesungsskript ETH Zürich. Available here