Regularization of certain systems of differential equations
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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.