Regularization of certain systems of differential equations Benson, Dean
dc.contributor.department Mathematics 2018-08-23T08:27:00.000 2020-06-30T07:24:33Z 2020-06-30T07:24:33Z Fri Jan 01 00:00:00 UTC 1954 1954
dc.description.abstract <p>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 &gE; 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 &gE; 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 &lE; 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.</p>
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dc.identifier.articleid 13889
dc.identifier.contextkey 6866446
dc.identifier.s3bucket isulib-bepress-aws-west
dc.identifier.submissionpath rtd/12890
dc.language.iso en
dc.source.bitstream archive/|||Fri Jan 14 19:32:20 UTC 2022
dc.subject.disciplines Mathematics
dc.subject.keywords Differential equations
dc.title Regularization of certain systems of differential equations
dc.type article
dc.type.genre dissertation
dspace.entity.type Publication
relation.isOrgUnitOfPublication 82295b2b-0f85-4929-9659-075c93e82c48 dissertation Doctor of Philosophy
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