Mechanics, Dynamics and Chaos
Class: Tuesday and Thursday 1:30-3:00, 4E19
Instructor: Ravi K. Sheth (
Office Hours: Tuesday and Thursday 4-6pm, 4N3 DRL
Textbook: Galactic Dynamics (Binney and Tremaine, Princeton Series in Astrophysics)
Grades: 6 Homework sets (each 12.5% of total grade) + 1 in class 45 min presentation (25% of total grade)

Some notes

Astrophysics background (modified from Theoretical Astrophysics, Padmanabhan, Cambridge Univ. Press, 2000). I've included these notes particularly for those unfamiliar with astrophysics, not because I expect you to know this stuff by the end of the course, but because they provide a handy reference to some of the physical phenomena I may refer to during the course.

Lagrangian and Hamiltonian Mechanics (from J. Binney)

An analytic model for spherical galaxies and bulges (Hernquist 1990)

Spherical evolution models: describes models for the formation of spherical objects by gravitational instability in an expanding Universe. The density profile of the final object depends on the initial density profile: tophat and cone-hat models are considered. Some discussion of 'secondary infall' models is also included.

Ellipsoidal collapse models (White and Silk 1978)

The frozen potential approximation (Bagla and Padmanabhan 1994): includes a nice discussion of extensions to the Zeldovich approximation, and describes a nice model of nonlinear clustering.

The adhesion approximation (Gurbatov, Saichev and Shandarin 1989): a beautiful relation between Burgers' equation and Zeldovich's approach to nonlinear large-scale clustering.

Heating of disks by infalling satellites (Benson et al. 2005)

"If I had been rich, I probably would not have devoted myself to mathematics" (Lagrange). Click here for more about Lagrange himself, and here for more about Hamilton ("Well, Papa can you multiply triplets?").


Problem Set 1 (due 1 February 2005).
Problem Set 2 (due 18 February 2005).
Problem Set 3 (due 04 March 2005).
Problem Set 4 (due 01 April 2005): 2.10, 3.2 and 3.19, 3.4, 3.5, and 3.10 from the textbook.

Presentation dates (each talk 45 mins)

19 April: N-body simulations (Jonathan); Chaos (Michelle)
21 April: MOND (Yan, Matt); Black Holes (Mike)
26 April: Black Holes (Chris)

Possible presentation topics/links

In general, a good place to search for astrophysics literature is the NASA-ADS project. Once there, click on Search References, which will allow you to search by author name, paper title, keyword(s), etc.

Exactly solved N-body systems:
This reference (Lynden-Bell and Lynden-Bell) does the classical case. The Lynden-Bell's also studied the relation between these classical systems and a special set of quantum mechanical systems.

Modified Newtonian Dynamics:
This is the original paper by Milgrom (1983). More recent interest was generated by Bekenstein (2004).

Binary Black Holes:
It is thought that there is a black hole at the center of most galaxies (another idea due to Lynden-Bell!). There has been some recent discussion of what happens if there are two massive black-holes at the center. We discussed the restricted three-body problem in class, which provides a good introduction to some of the main ideas. The question is: Do the black holes coalesce? If so, what is the main physical mechanism? Ejection of nearby stars? Gravitational radiation? Something else? And how long does this take? Start here, and follow the reference trail.

Numerical Simulations I:
Discuss different techniques (mesh versus tree-based algorithms) that are used to solve the N-body problem numerically; a good option if you are more interested in N-body systems in general than in astrophysics. Methods include mesh-based, tree-based, and symplectic integrators.

Numerical Simulations II:
The idea is to run some simulations of galaxy collisions using a code called GADGET. This project is more computationally intensive than the others, but the upside is that having an expert in its use here in the department would make you invaluable!

The Once and Future Universe:
Current data indicate that the scale factor of our universe is accelerating. One consequence of this is that objects will eventually become isolated from one another. What is required for objects to remain bound against this cosmic accelerations? What will the structure (e.g. density run) of such objects be? How does this structure compare to that of the model equilibrium systems we discussed in class? Click here to play.