Learning About the Three Body Problem & Hamiltonian Systems

[zanazzi78]

I`m reading up on the three body problem, since today we covered the two body problem in Dynamics classe.

The problem is I don`t know what a Hamiltonian is, the sense refers to a hamiltonian system with 2 degrees of freedom!

Could someone please explain what a hamiltonian/ Hamiltonian system is?

edit : I think this needs to be moved to Celestial Mechanics, sorry!
- - no problem! Phobos

 

[pervect]

If you are already familiar with the Lagrangian, the Hamiltonian formulation is a slight "tweak" of Lagrange's formulation.

In the Lagrangian formulation, one writes a single function called the Lagrangian, L, in terms of positions and velocities which determine the equations of motion of the entire system. The equations of motion are expressed as partial differential equations of the Lagrangian which are always the same (except for the exact form of the function L) and are known as "Lagrange's equation".

The Hamiltonian formulation modifies this so that one writes the function in terms of position and momenta rather than positions and velocities. The Hamiltonian approach generates a system of first order differential equations, while the Lagrangian approach generates a second order system.

If you are not already familiar with the Lagrangian formulation, this answer sadly might not make a lot of sense. The Lagrangian formulation is well worth learning, but it's probably outside the scope of a single post on a discussion board to explain it.

One web reference that might be interesting because it talks about the Lagrangian formulation of mechanics, the Hamiltonian formulation, AND the three body problem is:
http://alamos.math.arizona.edu/~rychlik/557-dir/mechanics/mechanics.html

The goal of this tutorial is to present the Lagrangian and Hamiltonian formalism of mechanics. After reading this tutorial the reader will be able to write down equations of motion for various conservative mechanical systems. In particular, the reader will learn how to write the equations of motion in a rotating coordinate system (section 5) and will learn the canonical (Hamiltonian) form of the equations of motion in the restricted circular three-body problem (section 6).

This URL would probably work best in conjuction with a textbook, though. A standard graduate level textbook is Goldstein's "Classical mechanics", there are probably simpler undergraduate treatments (but I don't know of any specific titles to recommend).