Celestial mechanics has always been a source of interesting dynamics problems. Many ideas in dynamical systems theory (and the rest of mathematics, for that matter) were first developed in an attempt to understand the Newtonian n-body problem. I will try to present a mix of practical topics of physical interest and others of purely mathematical interest. Topics will include as many of the following as time permits:
- Basic Newtonian, Lagrangian and Hamiltonian mechanics
- Two-body problem and Kepler problem: how to solve, hidden symmetry, regularization of collisions, conjugacy to geodesic flow
- Perturbations of Kepler: satellite problem, precession of earth
- Restricted Three-Body Problem, Lagrange points, stability
- Central configurations and relative equilibria: basic properties, existence, finiteness, Morse theory, linear stability
- Blowing up singularities: total collision, parabolic motions, simultaneous collisions
- Variational methods: calculus of variations, existence of the figure-eight orbit for the three-body problem
- Chaos in the three-body problem: symbolic dynamics for near collision orbits in the isosceles three-body problem
None required but here are some recommended books and a link to some lecture notes of mine, including notes for this course called “Topics in Celestial Mechanics”
- Mathematical Methods of Classical Mechanics (2nd edition), by V.I. Arnold.
- Intro. to Hamiltonian Dynamics and the N-Body Problem, by Meyer, Hall and Offin
- Math. Aspects of Classical and Celestial Mechanics, by Arnold, Koslov and Neistadt
- Lectures on Celestial Mechanics, by Siegel and Moser
- Analytical Foundations of Celestial Mechanics, by Wintner
- Notes on Dynamical Systems, by Moser and Zehnder
Here is a link to the PDF file of the latest version of the notes for this course: Topics in Celestial Mechanics. Other lecture notes can be found here: Various Lecture Notes
Grades will be based submission of a few homework assignments
- Homework I, Due Monday, February 24. Choose any four problems from Section 1-4 of the Lecture notes and write them up.
- Homework II, Due Friday, April 10. Choose any three problems from Section 6 of the Lecture notes and write them up.
- Homework III, Due Friday, May 8. Choose any three problems from Section 6.4 or later in the Lecture notes and write them up.
From time to time, Mathematica notebooks relevant to the course will be added here
Hidden Symmetry. This notebook explores the hidden symmetry of the planar Kepler problem.
Central Force Problems. This one computes orbits for some central force problems, including the Kepler problems and other power-law problems.
Restricted Three-Body Problem. This one has several demonstrations about the planar, circular, restricted three-body problem (PCR3BP)
Collinear Three-Body Problem. This one has several demonstrations about the collinear three-body problem.