International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September ISSN
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1 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- bstct: Jcbin intgl nd Stbility f th quilibium psitin f th cnt f mss f n xtnsibl cbl cnnctd stllits systm in th lliptic bit. Vijy Kum ssistnt Pfss, Dpt. f Mthmtics, Vidydn Institut f Tchnlgy & Mngmnt, ux- 89(IHR, INDI (Emil: vijy.ism@gmil.cm Nikky Kumi Rsch Schl, Dpt. f Mthmtics, Dylbgh Eductinl Institut g-8 (Kumi.nikky9@gmil.cm Duing th mtin f cbl cnnctd stllits systm, t lst n quilibium pint xists whn ptubtiv fcs lik i-sistnc, sl pssu, shdw f th th du t sl pssu, mgntic fc nd bltnss f th th ct simultnusly. W hv btind tw quilibium pints in cs f ptubtiv fcs lik shdw f th th du t sl pssu nd bltnss f th th cting tgth n th mtin f tw xtnsibl cbl-cnnctd stllits. Lipunv s thm hs bn usd t xmin th stbility f th quilibium pints. Kywds: Equilibium Pints, Lgng s Thm, Lipunv Thm, Stllits, Stbility, Sl Pssu, Obltns f Eth.. Intductin Th psnt pp is dvtd t xmin th stbility f th quilibium pints f th cnt f mss f systm f tw stllits cnnctd by light, flxibl nd xtnsibl sting und th influnc f shdw f th blt th du t sl pssu in lliptic bit. ltsky V.V is th pin wk in this fild. It is th gnlistin f th wks dn by ltsky V.V, Singh R., Sinh S.K, nd Singh C.P,. Equtin f mtin Th qutin f mtin f n f th tw stllits mving lng kplin lliptic bit in Nchvill s cdints cn b btind by xpliting Lgng s qutin f mtin f fist kind in th fm f tw dimnl cs. x x y x Ψ " ' cs cs l ( ν x y l y x Ψ " ' cs ( ν y (
2 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- wh, csν x y v Tu nmly f th cnt f mss f th systm P Sl pssu pmt µ m m k Obltnss fc pmt. P Ψ Shdw functin pmt. p µ ( m m m m Ntul lngth f th sting mdulus f lsticity H dshs dnt th diffntitin with spct t tu nmly v Th cnditin f cnstint is givn by x y ( T find th Jcbin intgl f th pblm, th vgd vlus f th scul tms du t pidic tms psnts in th qutins f mtin ( cn b dducd s givn fllw : dv dv dv ( dv cs cs( v dv dv ( ( Ψ cs cs( v dv ( cs ( cs Ψ cs cs( v dv
3 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- nd cs ( v dv Ψ cs ( v dv ( cs ( Ψ cs ( v dv ( Wh is tkn t b cnstnt Ug ( in (, w gt x" y' x ( x ( ( cs cs ( ( ( ( x nd y" x' y ( ( cs ( ( ( ( y Wh, x y Th cnditin f cnstint givn by ( tks th fm x y ( ( Fm qutins f mtin ( it fllws tht th tu nmly v ds nt pp xplicitly in th qutins f mtin, s th must xist Jcbin intgl f th pblm. Multiplying fist nd scnd qutins f ( by x nd y spctivly nd dding nd intgting, w gt th Jcbin intgl in th fm. x y x ( ( ( x y ( x Wh, h is th cnstnt f intgtin. ( xcs y ( ( ( x y h ( cs ( y cs ( (6
4 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- Th cuv f z vlcity is btind by putting x x y ( ( ( x y ( ( xcs cs ( ( ( x y ( x y in (6 s h ( y cs ( ( Hnc w cnclud tht th stllit by mss m will mv insid th bundis f diffnt cuvs f z vlcity psntd by ( f (6 f diffnt vlus f Jcbin cnstnt h..equilibium psitin f th systm W hv btind th systm f qutins ( f th pticl f mss m f th systm in tting fm f fnc. It hs bn ssumd tht th systm is mving with ffctiv cnstnts nd hnc th sting cnncting th tw stllits f msss m nd m will min lwys tight Th quilibium psitins f th systm givn by th cnstnt vlus f th cdints in th tting fm f fnc Nw, lt x x nd y y giv th quilibium psitin wh x nd y cnstnts. x x nd y y Thus, qutins givn by ( tk th fllwing fm: x ( x ( ( cs cs ( ( ( ( x y ( ( ( ( ( ( cs y (8 Wh x y ctully, it is impssibl t find th slutin f th lgbic qutins (8 in its psnt fm. Thf, f u futh invstigtin, it hs bn ssumd tht This mns tht th sun ys is in th lin f th pig f th llipticl bit f th cnt f mss f th systm. Putting, th qutins f mtin givn by (8 tk th fm:
5 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- ( ( ( ( ( ( ( x x nd y ( ( ( ( y (9 Fm (9, w gt th quilibium pints s [, ], ( ( ( ( ( [ ] (, b c ( ( ( ( ( ( ( ( ( Nw, it cn b sily sn tht th quilibium pint givn by ( nly givs mningful vlu f, th Hk s mdulus f lsticity if. Stbility f th systm < < 9 W shll study th stbility f th quilibium psitin givn by ( f th systm in th sns f Lipunv. F this, lt us ssum tht th smll vitin in th cdint t th givn quilibium pint [, ]. lt nd b smll vitin in x nd y-cdints spctivly f th givn psitin f quilibium. x nd y x nd y ' " ' x y " (
6 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- 6 Putting th vlus f x, y, x, y, x nd y fm ( in (, w gt n putting nd, cs s ( ( ( ( ( ( ( ( ' " n ( ( ( ( ' " nd ( Wh, ( T btin Jcbin intgl f th qutins f mtin (, w multiply fist qutin f ( by ( n nd scnd qutin f ( by nd dd thm thth, w gt ft intgting ( ( ( ( ( ( ( { } ( ( ( { } h l ( Wh h is th cnstnt f intgtin T tst th stbility in th sns f Lipunv, w tk Jcbin intgl ( s Lipunv s functin (,,, V nd is btind by xpnding th tms f ( s (,,, V ( ( ( ( ( ( ( ( ( ( ( ( 8 6
7 Intntinl Junl f Scintific & Engining Rsch, Vlum, Issu 9, Sptmb- ( ( ( h ( ( ( Wh ( stnd f th thid nd high d tms in th smll quntitis nd. Nw, by Lipunv thm n stbility it fllws tht th nly citin f th givn quilibium psitin (, t b stbl is tht v dfind by ( must b psitiv dfinit nd f this th fllwing cnditins must b stisfid: 6 ( ( ( (i 8 ( ( ( ( (ii ( > ( ( ( ( (iii > ( ( (6 Sinc [, ] givn by ( is th quilibium pint nd s > nd s putting th vlus f, it cn b sily sn tht ll th cnditins f (6 idnticlly stisfid. Cnclu: Thus, w cnclud tht th quilibium psitin [, ] f th systm is stbl in th sns f Lipunv. Rfnc: ( ltsky V.V. : but th Rltiv Mtin f Tw Cnnctd dis in bit. Ksmichskiy Issldvni, vl., N. 6, p.p. 8-8, 969 (Russin ( Singh R..: Th mtin f tw cnnctd bdis in n llipticl bit. ulltin f th Mscw stt univsity mthmtics-mchnics. N., P.P.8-86, 9 (Russin ( Sinh S.K.: Effcts f sl pssu n th mtin nd Stbility f th systm f tw int-cnnctd stllits in n llipticl bit. stphysics nd spc scinc. N., P.P. 9-, 988 ( Singh C.P.: Mtin nd stbility f int-cnnctd Stllits systm in th gvittinl fild f blt th. Ph.D thsis, submittd t.r.. ih Univsity, Muzffpu, 98.
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