New Mexico State University

Department of Astronomy

PO Box 30001, MSC 4500 Las Cruces, NM 88003

wlyra - nospam - nmsu.edu

+1 575-646-1400

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

Notes.**Class 3: Lagrange points, Jacobi constant.**

Lagrangian Points and the Jacobi Constant, Colinear Lagrangian Points

Notes.**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

Notes.**Class 5: Hamiltonian formulation.**

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

Notes.**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.

Notes.**Class 8: Perturbation Theory.**

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

Notes.**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.

Notes.

### 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++