# Accession Number:

## AD0706899

# Title:

## REGULARIZATION OF VECTOR FIELDS BY SURGERY,

# Descriptive Note:

# Corporate Author:

## BROWN UNIV PROVIDENCE R I DIV OF APPLIED MATHEMATICS

# Personal Author(s):

# Report Date:

## 1970-05-01

# Pagination or Media Count:

## 16.0

# Abstract:

Regularization is a common procedure in the study of differential equations. The usual method is to multiply the vector field by a suitably chosen positive scalar function which vanishes on the set of singularities of the vector field. The new vector field or differential equation thus obtained may have no singularities and thus may be easier to study. The qualitative behavior of solutions off the set of singularities is the same for both vector fields. Another method of regularization involves surgery and roughly, the idea is to excise a neighborhood of the singularity from the manifold on which the vector field is defined and then to identify appropriate points on the boundary of the region. The regularization of the planar 2-body problem on surfaces of constant negative energy was first discussed by Levi-Civita. The 2-body problem is regularized using surgery. This method gives a new way of looking at the classical result and makes apparent the geometrical reasons for its success. It is possible to regularize the three body problem on surfaces of nonzero angular momentum using the present methods. This result will be treated in a subsequent paper. Author

# Descriptors:

# Subject Categories:

- Celestial Mechanics
- Theoretical Mathematics