Convex Functions Harmonic Maps, and the Stability of Hamiltonian Systems.
Abstract
Trajectories of conservative dynamical systems are particular examples of harmonic maps. If Y is the configuration space of a dynamical system, then a trajectory of the system is a harmonic map from the real line into Y. More generally, let X and Y be Riemannian manifolds with X compact. It is shown that the image of any harmonic map f from X to Y cannot be contained in domains which are too small; specifically, that the image of any such f cannot be contained on any domain which supports a convex function. From a modification of the proof it is shown that, except in the neighborhoods of certain exceptional points, a trajectory of a dynamical system cannot lie entirely in any such domain. This fact leads to criteria for the growth and instability of dynamical systems. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Sep 24, 1970
- Accession Number
- AD0713885
Entities
People
- William B. Gordon
Organizations
- United States Naval Research Laboratory