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Unformatted text preview: ODE - Pleiades problem II-6-1 6 Pleiades problem 6.1 General information The problem consists of a nonsti system of 14 special second order di erential equations rewritten to rst order form, thus providing a nonsti system of ordinary di erential equations of dimension 28. The formulation and data have been taken from [ HNW93 ]. E. Messina contributed this problem to the test set. Comments to [email protected] . The software part of the problem is in the le plei.f available at [ MI03 ]. 6.2 Mathematical description of the problem The problem is of the form z 00 = f ( z ) ; z (0) = z ; z (0) = z ; (II.6.1) with z 2 IR 14 ; t 3 : De ning z := ( x T ; y T ) T , x; y 2 IR 7 , the function f : IR 14 ! IR 14 is given by f ( z ) = f ( x; y ) = ( f (1) ( x; y ) T ; f (2) ( x; y ) T ) T , where f (1 ; 2) : IR 14 ! IR 7 read f (1) i = X j 6 = i m j ( x j x i ) =r 3 2 ij ; f (2) i = X j 6 = i m j ( y j y i ) =r 3 2 ij ; i = 1 ; : : : ; 7 : (II.6.2) Here, m i = i and r ij = ( x i x j ) 2 + ( y i y j ) 2 : We write this problem to rst order form by de ning w = z , yielding a system of 28 non-linear di erential equations of the form z w = w f ( z ) (II.6.3) with ( z T ; w T ) T 2 IR 28 ; t 3 : The initial values are z w = B B @ x y x y 1 C C A ; where 8 > > < > > : x = (3 ; 3 ; 1 ; 3 ; 2 ; 2 ; 2) T ; y = (3 ; 3 ; 2 ; ; ; 4 ; 4) T ; x = (0 ; ; ; ; ; 1 : 75 ; 1 : 5) T ; y = (0 ; ; ; 1 : 25 ; 1 ; ; 0) T : 6.3 Origin of the problem The Pleiades problem is a celestial mechanics problem of seven stars in the plane of coordinates x i , y i and masses m i = i ( i = 1 ; : : : ; 7). We obtain the formulation of the problem by means of some mechanical considerations. Let us consider the body i . According to the second law of Newton this star is subjected to the action F i = m i p 00 i ; (II.6.4) where p i := ( x i ; y i ) T . On the other hand, the law of gravity states that the force working on body i implied by body j , denoted by...
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