Dynamics and Hydrodynamics - ASTR 506

Lectures for the course Dynamics and Hydrodynamics PhD course at NMSU.


  • I: Solar System Dynamics

    • Class 1: Two-body problem recap.

      Orbital elements, angular momentum, energy, semimajor axis, barycentric orbits.
      Notes on 2-body problem.
      Notes on barycentric orbits.

    • Class 2: Three-body problem.

      Tisserand Parameter, Triangular Lagrangian Points

    • Class 3: Lagrange points, Jacobi constant.

      Lagrangian Points and the Jacobi Constant, Colinear Lagrangian Points

    • Class 4: Stability/Instability of Lagrangian points.

      Stability Of Lagrangian Points - Linear analysis, Instability of collinear points and stability of triangular points, Tadpole and Horseshoe Orbits

    • Class 5: Hamiltonian formulation.

      Hamiltonian Formulation, Angle-Action Variables: Intro, Angle-Action variables as Polar Coordinates

    • Class 6: Hamiltonian formulation.

      Poincare Invariant Theorem, Green's Theorem, Hamilton-Jacobi Equation and the Generating Function, Harmonic Oscillator via Angle-Action variables

    • Class 7: Delaunay variables.

      Delaunay variables: Intro, Kepler problem in angle-action variables. I. Actions, Kepler problem in angle-action variables. II. Change of actions, Kepler problem in angle-action variables. III. Angles: Longitude of Ascending Node, Kepler problem in angle-action variables. IV. Angles: Argument of Pericenter, Kepler problem in Angle-Action variables. V. Angles: Mean Anomaly.

    • Class 8: Perturbation Theory.

      Splitting the Hamiltonian, Force splitting, The Disturbing Function. Hierarchical Triples: The Hamiltonian in orbital elements.

    • Class 9: Kozai-Lidov oscillations.

      Averaging over the outer binary: conservation of vertical angular momentum; Averaging the Hamiltonian over the inner binary: conservation of energy Kozai-Lidov resonance: Oscillations in eccentricity and inclination, Critical Inclination for KL cycles and libration of argument of pericenter, Critical Angle and Examples of Kozai-Lidov oscillations.

  • II: Galactic Dynamics

    Notes on GitHub (python notebook).

    • Class 10: Potential theory.

      Gravitational Potential, Poisson Equation, Newton First Theorem, Newton's Second Theorem

    • Class 11: Halo and disk potential.

      Gravitational Potential Energy, NFW Dark Matter Halo, Non-Spherical Potentials - Thin Disk, Thick Disk Potential

    • Class 12: Potential of ellipsoids.

      Homoeoid, Elliptic Coordinates, Potential of Homoeoids, Potential of Homogeneous Ellipsoid, Hernquist potential and Exponential Disk

    • Class 13: Orbits.

      Orbits in a dark halo potential; epicyclic approximation.

    • Class 14: Virial Theorem and Dynamical Friction.

    • Class 15: Dynamical Friction.

      Two-Body Scattering, Relaxation Time, Dynamical Friction.

  • III: Hydrodynamics

    Notes on GitHub (python notebook).

    • Class 16: Boltzmann equation.

      Boltzmann equation, conservation theorem, continuity equation.

    • Class 17: The Navier-Stokes equation.

    • Class 18: Numerical hydrodynamics.

      Discretization, stencil diagrams, von Neumann stability analysis, donor cell, second order schemes, slope limiters.

      Notes on GitHub (python notebook).

    • Class 19: Primer on tensors.

      Notes on GitHub (python notebook).

    • Class 20: Acoustics.

      Linearization, wave solution, dispersion relation.

    • Example of Euler method in Fortran90 and in C++