# Accession Number:

## AD0713885

# Title:

## Convex Functions Harmonic Maps, and the Stability of Hamiltonian Systems.

# Descriptive Note:

## Interim rept.,

# Corporate Author:

## NAVAL RESEARCH LAB WASHINGTON D C MATHEMATICS RESEARCH CENTER

# Personal Author(s):

# Report Date:

## 1970-09-24

# Pagination or Media Count:

## 11.0

# 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

# Descriptors:

# Subject Categories:

- Celestial Mechanics
- Theoretical Mathematics