Regularization of certain systems of differential equations

1954
Benson, Dean
Organizational Units
Organizational Unit
Mathematics
Welcome to the exciting world of mathematics at Iowa State University. From cracking codes to modeling the spread of diseases, our program offers something for everyone. With a wide range of courses and research opportunities, you will have the chance to delve deep into the world of mathematics and discover your own unique talents and interests. Whether you dream of working for a top tech company, teaching at a prestigious university, or pursuing cutting-edge research, join us and discover the limitless potential of mathematics at Iowa State University!
Abstract

Certain types of systems of differential equations of the form d2x dt2-2l x,ydy dt=Wx , d2ydt2 +2lx,y dxdt=W y, 5.1 have been regularized in this paper. The regularizations by both Levi-Civita and Birkhoff for the equations of motion in the restricted problem of three bodies were discussed;Hill's equations arising in a problem of celestial mechanics were of the form of equations (5.1). This system of equations has one singular point and was regularized by a transformation similar to that used by Levi-Civita. The equations of motion for the restricted problem of n-bodies, for n ≧ 4, were regularized for three of their singular points by using transformations of the type employed by Birkhoff two times in succession. For n = 4 the system was completely regularized, since there were only three singular points;In section IV the system of equations d2x dt2-2l x,ydy dt=-nxr n+2, d2ydt2 +2lx,y dxdt=- nyrn+2 , 5.2 with the integral dxdt 2+d ydt 2=2rn- C 5.2' was considered. It was determined that transformations of the type z = wk would regularize equations (5.2) for values of n in the interval -2 < n < 2 when the proper choice of k was made. It was shown that one should choose k = 2 when -2 < n < 0, and k ≧ 42-n for 0 < n < 2. While this selection for values of k regularizes equations (5.2) for all values of n in the interval -2 < n < 2, there is a better choice for k if n is in the interval 1 ≦ n < 2. When n lies in this last interval, choose k by equation (4.4) which gives k = 22-n . Examples using these transformations in removing singularities in systems of differential equations of the type (5.2) have been given.