Asymptotic Solution to the Scalar Version of the Two Body Problem
When the Two Bodies Collide - A Case Study
Ahmed Bakheet1,∗, Ali Abdulhussein1, Laheeb Muhsen Noman1
1Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq
Emails: ahmedbakheet@uomustansiriyah.edu.iq; aaabdulhussein@uomustansiriyah.edu.iq;
laheeb-muhsen@uomustansiriyah.edu.iq
Abstract
The main goal of this paper is to obtain a special form of asymptotic solutions to the scalar version of the two body problem
whenever the two bodies collide on the real line at the collision time. It has been shown that the desired asymptotic solution
maintains certain properties when t approaches the collision time. However, it is not easy to Handel such a mission without
the employment of successive approximations technique. The successive approximations technique has been modified and
adjusted to serve as the main tool in the process of obtaining such solution. Moreover, it has been shown that the series
of successive approximations converges absolutely and uniformly to a continuous function that approaches to 0 when t
attains the collision time in a certain interval. The problem of one dimensional collision between the two bodies has been
solved asymptotically at the collision time.
Keywords: Two Body Problem; Collision; Asymptotic; Successive Approximation