Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

Kenneth Meyer, Glen Hall, Dan Offin (auth.)9780387097237, 0387097236

This text grew out of graduate level courses in mathematics, engineering and physics given at several universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. Topics covered include a detailed discussion of linear Hamiltonian systems, an introduction to variational calculus and the Maslov index, the basics of the symplectic group, an introduction to reduction, applications of Poincaré’s continuation to periodic solutions, the use of normal forms, applications of fixed point theorems and KAM theory. There is a special chapter devoted to finding symmetric periodic solutions by calculus of variations methods.

The main examples treated in this text are the N-body problem and various specialized problems like the restricted three-body problem. The theory of the N-body problem is used to illustrate the general theory. Some of the topics covered are the classical integrals and reduction, central configurations, the existence of periodic solutions by continuation and variational methods, stability and instability of the Lagrange triangular point.

Ken Meyer is an emeritus professor at the University of Cincinnati, Glen Hall is an associate professor at Boston University, and Dan Offin is a professor at Queen’s University.

Table of contents :
Front Matter….Pages I-XIII
Hamiltonian Systems….Pages 1-25
Equations of Celestial Mechanics….Pages 27-44
Linear Hamiltonian Systems….Pages 45-68
Topics in Linear Theory….Pages 69-115
Exterior Algebra and Differential Forms….Pages 117-132
Symplectic Transformations….Pages 133-145
Special Coordinates….Pages 147-173
Geometric Theory….Pages 175-216
Continuation of Solutions….Pages 217-230
Normal Forms….Pages 231-270
Bifurcations of Periodic Orbits….Pages 271-299
Variational Techniques….Pages 301-327
Stability and KAM Theory….Pages 329-354
Twist Maps and Invariant Circle….Pages 355-387
Back Matter….Pages 389-399