A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering Sciences and Technlgy Tpi, Swabi, NWFP, Paistan Email: shabbir@giiedup Abstract: The Classical Newtnian prblem f describing the free mtins f N gravitating bdies which frm an islated system in free space has been cnsidered It is well nwn frm the Pincare s Dictum that the prblem is nt eactly slvable Sets f N bdy systems cmpsed f masses having spherical symmetry, apprpriate angular velcities (< rad/s) and bunded psitin vectrs are eamined A prcedure has been develped which yields epressins apprimately defining the trajectries eecuted by the masses Intrductin: The prblem has been slved in a tw dimensinal plane Sets f N bdy systems in which the gravitating masses pssess spherically symmetric mass distributins, small angular velcities (< rad/s) and bunded psitin vectrs have been cnsidered Other appraches can be fund in [-5] The fllwing figure shws ur cnfiguratin fr the system f N bdies
e y e e m m e N m K mk e N e e here vectr alng Figure is the psitin vectr f mass m, is the rtatin angle f, e is the radial unit and e is a unit vectr perpendicular t e N We assume that the th bdy having mass m and psitin vectr remains apprimately in tw bdy mtin with a bdy f mass psitin vectrs f M m and = m ( n ), placed at a pint given by M = N n= n N m ( ) n n n n = N m ( ) n n n = The M are assumed t remain apprimately cllinear fr all time in accrdance with this assumptin It then fllws that the respective unit vectrs in thse directins als remain apprimately cllinear fr all time, that is e e M t Since we are free t scale each psitin vectr alng its directin, the bundedness f the psitin vectrs ( () t < N ) can be mdified as () t N The clsed frm apprimatins that we will find fr the case f N bdies will be valid fr the case () t < rad/s & () t N Nte that these cnstraints include almst all practically encuntered situatins, in the sense that angular velcities encuntered in celestial mtin are nrmally much less than rad/s and psitin vectrs are bunded These cnditins therefre negligibly limit the applicatin f ur results
Main Results: bdies Figure presents the cnfiguratin f ur tw bdy apprimatin f the system f N y e e M M e M Figure Here e M is a unit vectr in the directin f M and e M is a unit vectr perpendicular t e M The vectrs and M are taen t be apprimately cllinear fr all time, resulting frm the tw bdy mtin analgue in figure This in turn implies that () t = () t + π () t = M () t () t = M () t t Using Newtn s Laws t mdel the M e M system in figure we get m M Gm M () = e M M Gm M = e M () M where G is the universal gravitatin cnstant The representatin f vectrs and M as (t) e and (t) e M respectively, will find use in the scalar representatin f () and () M which can be shwn t be 3
= M GM (3) + = 0 (4) Gm = (5) M M M M M + = 0 (6) M M M The system f equatins (3), (4), (5) and (6) is nt eactly slvable We recall that the prblem is being slved fr the case () t < rad/s and () t ± ( N) We culd find an apprimatin t equatin (3) by nting that 0 t 0 t (since () t < rad/s and ( t) ± ) This implies that and hence the fllwing is fund t be an apprimatin t (3) It can be shwn that if GM (7) M N, then M N Als, if () t < rad/s, then M () t < rad/s Using these arguments, just lie we fund (7) t be an apprimatin t (3), we can find (8) t be an apprimatin t (5) We nw define the vectr = M M Gm = (8) M and therefre = M M Since is a vectr alng the directin f, it fllws that a unit vectr alng the directin f is the same as a unit vectr alng the directin f M We define e t be a unit vectr alng the directin f Hence it fllws that e = e Als, and can be represented as prducts f scalar time functins and respective unit vectrs M M (9) 4
= te () = () t e (0) M = () t e M M and e = e =e () M Using the first equatin in relatin () we can rewrite () as = e = M = () Rewriting (7) and (8) while maing use f () we get Adding the abve tw result we get GM = Gm M = Gm ( + M) + M = Maing use f (0) and () it can be shwn that (3) (4) (5) () t = () t + () t (6) M Substitutin f equatin (5) in equatin (6) then yields Gm ( + M ) = We nw integrate (3) and (4) twice with respect t time t get (8) and (9) respectively t t () = + ( ) () t0 t0 t t t GM t dt dt t t M () = ( ) () M + M 0 t t t t t Gm t dt dt (7) (8) (9) where = (0), = (0) M 0 M = (0), M = M (0) and t 0 is the starting time In, bth (8) and (9) the epressin t t t t () tdtdt is unnwn We can integrate equatin (7) twice with respect t time t find an alternative frm fr the abve epressin as 5
( t t ) + ( t) ( ) t t 0 () t dt dt = t t Gm + M (0) where 0 = (0) and = (0) Nw substituting equatin (0) int (8) we get M () t = + ( t t) + [ () t ( tt) ] () ( m + M) In rder t find the analytic epressin fr the trajectry f m by use f equatin (), we must find t( ) Therefre we rewrite () as M M t ( ) = 0 + ( m + M) ( m + M) M M + t( ) + ( ) () ( m + M) ( m + M) In rder t find ( ) eplicitly, we need t find t( ) and ( ) and substitute these epressins int equatin () Using equatins (), (), (9), (0) and (), we can shw that Gm ( + M) e = The abve equatin can be represented in its scalar frm as Gm ( + M) = (3) + = 0 (4) where is the rtatin angle f vectr, the same as the rtatin angle f vectr abve tw equatins can be slved t give ( ) = 4 Gm ( + M) c cs + c sin + M The where c and c are cnstants f integratin which can be determined by incrpratin f the (5) initial cnditins Nw since = π, we can write ( ) ( π) and shw that M 6
( ) = as 4 Gm ( + M) c cs csin ( ) = where = (0) This can be simplified and written cs( φ ) + (6) where Gm ( + M) c =± c + c = and = Having fund ( ), what, φ tan 4 c remains t be dne is t slve fr t( ), which can be shwn t satisfy t d t (7) ( ) = ( ) + where = (0) A substitutin f (6) in (7) yields = + cs( φ) + where = (0) Evaluatin f the integral gives t( ) d t (8) + tan [ ] (0 5φ 0 5 ) t( ) ( )( ) + = tan (05φ 05 ) + + ( )tan(05 φ 05 ) (( + )( )) tan(0 5φ 0 5 ) + tan (( + )( )) ( )tan(0 5 05 ) tan (0 5φ 0 5 ) tan φ + (( )( )) tan (0 5φ 0 5 ) tan ( )tan(05 φ 0 5 ) (( )( )) tan ( ) tan(0 5φ 05 ) (( )( )) + tan [ ] (0 5 0 5 ) ( )( ) φ + tan (05φ 05 ) + + 7
( )tan(05 φ 05 ) (( + )( )) tan(0 5φ 0 5 ) + tan (( + )( )) ( )tan(05 05 ) tan (05φ 05 ) tan φ + (( )( )) tan (0 5φ 0 5 ) tan ( )tan(05 φ 05 ) (( )( )) tan ( ) tan(05 φ 0 5 ) (( )( )) + (9) Substitutin f (6) and (9) int equatin () wuld then yield the fllwing epressin fr ( ) M M M ( ) = + t + m + M m + M m + M { M } + cs( φ) + m + M + tan [ ] (0 5 0 5 ) ( )( ) φ + tan (0 5φ 0 5 ) + + ( )tan(05 φ 05 ) (( + )( )) tan(0 5φ 0 5 ) + tan (( + )( )) ( )tan(05 φ 05 ) + tan (0 5φ 0 5 ) tan (( + )( )) ( )tan(05 φ 05 ) tan (0 5φ 0 5 ) tan (( + )( )) ( ) tan(05φ 05 ) tan (( + )( )) + tan [ ] (0 5 0 5 ) ( )( ) φ + tan (05φ 05 ) + + * t 8
( )tan(05 φ 05 ) (( + )( )) tan(0 5φ 0 5 ) + tan (( + )( )) ( )tan(05 φ 05 ) + tan (0 5φ 0 5 ) tan (( + )( )) + t 0 ( )tan(05 φ 0 5 ) tan (0 5φ 0 5 ) tan (( + )( )) ( ) tan(0 5φ 05 ) tan (( + )( )) Nte that the result presented abve is fr the generalized case, and therefre it fllws that the abve result hlds valid fr the th bdy f mass m N (30) Summary: A set f N gravitating masses was cnsidered and a different prcedure was develped t find epressins defining the eecuted trajectries It was shwn that apprimate slutins can be fund in clsed frm, in spite f the fact that eact slutins t such a prblem d nt eist We assumed the N bdy system t be cmpsed f masses having spherically symmetric distributins, small angular velcities (< rad/s) and bunded psitin vectrs N number f tw bdy mtin analgues were then used t apprimately replicate N bdy interactin The slutins apprimately describe the trajectries assciated with N bdy mtin in free space References: H Airault, H P Mcean and J Mser, Ratinal and Elliptic slutins f the Krteweg-de Vries equatin and related many bdy prblem, Cmmun Pure Appl Math 30 (977) 95-48 F Calger, Eactly slvable ne dimensinal many bdy prblems, Lett Nuv Ciment 3 (975) 4-46 3 F Calger and A M Perelmv, Prperties f certain matrices related t the equilibrium cnfiguratin f the ne dimensinal many bdy prblems with the pair ptentials V () = - ln sin and V () = /sin (), Cmm Math Phys 59 (978) 09-6 4 H W Braden and V M Buchstaber, Integrable systems with pair wise interactins and functinal equatins, Rev Math & Math Phys 0 (997) 3-0 5 M Bruschi and F Calger, Slvable and/r integrable and/r linearizable N bdy systems in rdinary (three dimensinal) space I, J Nnlinear Math Phys 7 (000) 303-386 9