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An illustrative periodic orbit of the second genus Gage, Walter Henry 1926

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A« fllusmsriYc / s is \j . OR a/r Of +hc ecotfD Genus, IA] kJ(JCo^- Uit-v^/vr, l ~ f ^ - & -C*S^ U.B.C. LIBRARY CAT. WO. LfaAi.tm/l*- G*. k-ACC. NO. STi^JTo AS ILHJSSBAfm PERIODIC OBBIS? OF THE SECOBD GESOS. 1>y WALTER HEUET 04®: A fheeie submitted for the Degree ©f MASTER OF ARTS • in the BAf&ai&'PXcs THE BKiVSESlKT OF BRITISH COLUMBIA AJR1L 1926. f * 4 :y, ^ v^^^iivU^ii^"*^ AB XLMBTFATXYB PffBIOlSC ORBIT OP THE SKJOTD GEMJS. 1* The Problem £ . The Iqaations of Motion. S. The P i n t t e a m Perlodle Orbit*• 4. The Perlodle Orbits of the Seooaa Genoa 5. the Equat lore of Variation. 4* the Complete Bqoatlomt The Coefflclente of A The Coefficients of A" 7. Vonetiooa Bjcenple for Bqoillbrinav PoU* <a) fable S eonstante Table XI.. first genu* orbite Table 1X1 second geans orbite Illustration Aff ILLUSTRATIVE PERIODIC ORBIT OP THE SECOJTD GEEUS. 1 . The Problem The object of this paper is to construct algebraically and graphically a periodic orbit of the second genus. In 1772 Lagrange showed that if two finite spheres revolve in circles about their commom centre of mass, then there are three points on the line joining their centres at which an infinitesimal body would remain if given an initial projection so as to be instantaneously fixed with respect to the moving bodies* When the infinitesimal body is given an initial displaoe-rent from the point of equilibrium, it is possible, by proper choice of initial conditions, to construct closed orbits relative to the moving system. Orbits known as periodic orbits of the first genus have •2 been discussed by Poincare , Darwin, Flummer, and others, including Koulton, whose "Oscillating Satellite3" (Chapter T, "Periodic Orbits") is quoted la thle paper. Second genus periodic orbits are defined by Poincare. A discussion of such orbits, a proof of their existence, and the theory for their construction, are given by Dr. B. Buchanan in a paper entitled "Periodic Orbits of the Second Genus Sear the Straight-Line Equilibrium Points.** Our construction is made following the method suggested by this latter dissertation. - z * 2. The Equations of lotion* Let the motion of the infinitesimal "body be referred to a set of rotating rectangular co-ordinates £,H.,S of which the origin is at the centre of mas of the finite bodies, the % axis is the line joining the finite bodies, and the £-*l- plane Is the plans of their motion. The units of length* mass* and tine will be taken so that the distance between the finite bodies, the sum of their masses, and the constant of proport 1 onality respectively shall each be unity* Let the masses of the finite bodies be denoted by jx andl^U., ( O<JU. %\ ), 0n denoting the co-ordinates of the infinitesimal body by €,»M a»d differentiation with respect to 4 t by primes, the differential equations of metleu are c^v = fs K " +:.•*-€' r 3 « > |.H. \ (I) 3 M • n > • s -The equllibrlwi potato are given %gr the solutions of the a§ a \ M of the too seta of points satisfying these squatIons we art oonoornod i s thfto discuss ion only with those points which l i e on the straight lino joining the f inite bodies. Zn this set there are three points, which are denoted by (a)* (bj# («)• where (a) l i e s between +- *> end tho f inite snssysv, (b) between^ and l-^o, and C between l-j» and-etw The eo-ordinates of those points are represented by £ # , 0, 0 where tho particular value of £ 0 depends upon the point In question* $» The f irs t Qens Periodic Orbits* To obtain the f irst genus periodic orbits wo displace the Inf lniteeinal body so that £ = €.+ex, l ( = 0 « t | , f = 0 + * £ , (-O where £ is an arbitrary parameter, and transform the time by putting t -tQ - 0-r^)T, whore & la determined as a function of £ so that the solutions in x*y and i shall bo periodic with period 2 TT inT. On denoting differentiation with respect to T by a dot over the variable* equations become fcr~» j . •I ,(3J •J. • 4 • There X , T , and 2 , are homogeneous polynomials of degree K in s^ y and z. 5h© values needed in the practical constructions are y3 = icf^^3^), • ' • • z» = - ^ j . ^ = 3 E A > ; • -7 C4; t l^ ff. + / L . Cc; The upper* ssiddle or lower signs are taken in 3B» according as the equilibrium point is t sJ , (&}, or (o) respectively^ ^he periods of the periodic solutions of ( 3 ) are determined from the solutions of the l inear tensa. these eolutlOBs^are L<yc , -car . fz , -i°T - 5 -where X», — - X ^ , are constants of integration, and cr** and /»** are the negative and positive roots of the quadratic In 41"; -One real peri©d(for which K f ^ - 0 ) is 2 \t /JA* a»^ orbits with this period are called orbits of Class A. Our construction concerns this class only, The periodic solutions of Class A are 6 ^ - oe + (c, ^^>. i /A r j £ - f o f V - - , (£> where « 4-A O+iA) ' ' H-Ab-rA+tiA*-) The init ial conditions chosen are £3-(°J = °, £*6>) = e - • • 4« Forloolo Orbits of tho S«ooa4 Ooaa*. Followlas; tho aothaa la tho *?orlo41o Orbits of ths Sooaa* 1 I — I , " 6 — rotara to oqaatloaa (1) and aako ths M M trmaoformtloaa o* boforo with tho oxooptlaa of pottlay t - t 0 - (.'«-« i+y) r , CO vaors S sad V' oro as rot uadaflaod* to aozt safest! tato t0 t \ f*r £ . aharaf, aa4 /^ara arbltrsry- paraaatora* Kqaatleao similar to (3) rooalt, sxoopt that ( i+S ) ia roplaead by { H - ^ l ^ l n 4 £ ay ( £ 0 4 ^ I . A dloplaooaaBt of tta laf laltsalaal body **•• tha psriodle orbit af Ou fIrat aoaao U thorn gtraa, aw that x~Xt+}>, £ = * , + £ , } a i ( f A ( T ; •hora Sc X f / d (^ ; aa« 60 } , doflao orblto ahooo solatloaa (alaa* wit* <£ ) ara daflaod ay IS), whoro£6+^roplscoo ft. law* W = A~fl • *• obwiooaly laws oqastloaa (S), ami oo—oqusutlr tho oqaatloaa doflalas; p,<j tad r sro i(*,*f>)^,+tr-3U1t/.;(J,Kn)V j - 7 --G+ONe, +• (•t.t-<&•«>,+-*)+^  v*-)\j+ - - J *(>,*•*) V - - - - J 4-•* 3 ~" where 6 ®i . = 3 8r„/><3. + — , K, = 38&JM. + - , 6. The Sanations or Variation* if tie linear terns of (8) only are considered we obtain the "equations of rarlation"* whose solutions6 are shown to he f^kx+ '^(+kz* *M.x+k9*. U^^'UJL^ A 9 -IT,-* X.+ O O J - 10 -4* The Coapleto Equations. la order t e solve the couple te equations we put6 We now solve the differential equations obtameo b<f<a,uatiNCj- the same powers of A Is equations (2) a f t e r t h e above transformation Is made* Sinee A oen he taken arbitrarily strut 11 the construct ions a r c a«de for the f irs t power of A onl^an-d, f o r t h c second LTI £0 - C e r t a i n terns* however, in the equations for /I are considered in the determination She, PoofflOltfttB tf A,, the equations involving the coefficients of A In equations (6) are ef"« fe,^-2^04+(H34-- } C<0; * 11 -SM « 2 ( i n A ) a , + $U= tC,** -^A + 2(i+2A)tr,f 36 2h s - — ZA S « 3A-C- **£+ U 3 * i + 3 _§J^ l8£i"" 3 8_^^ 3iba,a 7A* 3Cc, Sax- 3 &<*,<:,+ j 5 3 | = - A ^ fA 5 3 ^ • .c r^  «,, 3C S ^s 3 6a, 3 - ^ iTA c 3jk) 35* JlfA 3C 9C fA * - 2 2 -fh© complementary functions of these differential equations are similar to (9) wltht of course* different arbitrary multipliers, fhe particular Integrals are obtained by the veil known method of the "variation of parameters". fhe steps is the solution for the particular integrals are explained fully in the "Periodic Orbits of the Second &en«s% sal therefore the results only are given in this paper* As explained in tile aforementioned paper the arbitrary multipliers (denoted by h>j fcg i can be dealt with In pairst and from each pair there results a part of the particular Integral • 14 -Oj 3 ( _ g;i-I " - V 3 t-o; -13-^ V 0-, (H-v; <r,( ic- «# *Va f H-v; ^ t ^ ) / ^  " ' \ <rt (it-^) foot-^U)J J < ^ J . ( i) (i) *8 « * * * fi^-Hl /^''tJt + a9iD-v»i3.tL- IJXT lev-r,+3c J +> 14 -^ . = _3_ £** + ^•5 .v*^*) /?(/?vO xLc*, 3 ^Aljr0 + fc^1* and * g ( l ) g ive*-fet' <w^r-(^J^-Cert ZJAT, +• £ {r,(*,tw*et-)+ » a f » } T us + - ] ' 1 4-£}vrt i H-SP J ( ' 3 ; H i " ^ j ^12. * o • > .2-£L = A ("MO"-*/3) - 1 6 -F5- = F3s^, +- * i s £' _ £ * $31 .2, ^ J F f = t r < S t ^ ^ -Z- ^ 3A~,+ £ ^ S3 *f t ° /S r «- A ^ ^ ' 3 ,^ J A A, > -2- 3 j _ «2_» ^ , £ , ^3 = E l 3 ^ , +-E-i+Sll-EiLL3z._tEnS3^ElSs3l +£,,3*, - IT • X Z. su JL ' -Z. it** E,*~ -*G(<r*^x)Xi£— + ~X^- ( - I S *f fa/°-hj\f-JL*r*V~-*/9) Thus the particular integrals Of the equations in Pftqf* sad rt are found by summing the various p,*s, q / s , or r ,*s , in ( l l )« (12), (13) The complementary functions are similar to the solutions (9), where we denote the arbitrary oonstants by Xj , K2 * *6 It will he noticed that the particular integrals of the complete P | , 4 | . a*4 f» eqsatioas are made up of periodic terras, sons of which are multiplied hyVj* and ales non^periodic terns ( i . e . terras multiplied J y * j * both with and withoat ~^, * Let us writs the particular integrals of the p, , qT and r,. equations as <)-, ?i •*- £,+- (x,)-, + - ^ ) r vs-i ft S3 4- s Y +- (*, h ^ ) r w*-., respectively' ^l • c 2 # 3 1 • s 2 • 8 3 snf i s 4 denote simply cosine or sine series* as the letter indicates, while the subscript means the particular series, determined from ( i l l , (12) and il&l* Combining the particular integral and the complementary function we can now writs the complete solutions for p , , qj,and r j , as l a order to obtain periodic solutions i t is necessary t o (1) ( l ! choose *3 a * 4 = 1 ^ and eoaditiOM tOT symmetrical Orfcite (p, (e*s q2 M r » j i©J } dessand that % = o t and % " K,*1 *h* solutions are zwt yet period!* ttalese She eolations at th i s step "become where Ci • c 2 • Sj , S2 t S3 « and S4 are the periodic terns of equations ( i l l * (12) ant ( i s ) . At this step we have the condition (14) ehieh leaves % ' arMtrary and gives one rela tion connecting J% and Y[ • In order to find - fO -• eeoond rail tlon we proceed to oonelder the teres is A .Of A . *he aaaplaaeaUif fanotleas of ta l i eet of equation* art i l s l l t r to thOM far p} , q} Tj . with different arbitrary constants * i ^ - - * « ^ 8 ' We consider the particular lntegralo, which are again found by the "variation of paraaetero". According; to thle mat hod aa suppose (2) (2) (2) kj « *2 k j t o k the arbitrary nnltlpllere* *he f. <2) (2) differential eqnatlons^or the anltlpllere 1) ' and ha. are of the where Pj and »• denote the total ity of teraa la the equations. Siaee the differentlei equations for 1% t eg and r2 IHTOITS PI , Qj,and n and their poeere i t le easily eeaa that Dlcontains terna Of the form /£-€. ' « where Z ia a constant •Consequently the integration far H yielda Bon-periodio teraa. These teres* whloh involve >; and *6*1! noa* he equated to aero la order to have 0' _ . (1) periodic eolations, and wa have a second relation la *6 and y, } sanely UT *"V « which ia arbitrary ia taken different froa aero, ir» (it) c t = n _aJ_ i-uea,jE,-bO«,*.f, *-3J±<n n • J , + « , ' » ' Taa aoaatlaa la k^*1 «traa taa aaaa ooodltloo vi ta taa »lft« I M B aqaatSaaa (14) aad (16) ara aalva* aa b r a «6 * Aa appraodaatlaa hl£Mr taaa taa f l ra t paaav la £a far K^1* aad V^  , larolraa taa oaaatrootlam «f •olatleaa far taa taird and a? f * la a l l taa aquatlaaa. Tala vamld aaaa aa - 22 -exceedingly large amount of computation which would be unnecessary on account of the value of 60 and ) \ • She complete solutions of the second genus orbits Involve two periods: £tf />J~A * and ZXT /<TX * She value ©f tfy is gives by 6 0 > *-+•£' lt4o-3(U-<rt) ex3 It Is possible, by the choice of £ Q .. to make <Xj very nearly equal to 0~* . Moulton states that there are infinitely many values of nfor which i//\ aa^ <P are commensurable. We can thus construct solutions Sn which the above periods are commensurable, or very nearly so* T* Btaaarical Example for Equilibrium Point (a J * She following is a numerical example of second genus orbits* She construction is carried out to the second power of ^ tor the first genus orbit* and to the first power in £0 and A for the second Sable I gives the values of the constants computed in this or ether papers** Sables H and W give the values for T of the first genus and second genua orbits respectively. Ci* ICj 4 *aa arbitrary constant, is chosen equal to unity, while the values Of f0 and A are chosen as 0.1 and 0.01 respectively. Shis -waltm of f makes the ratio ^/fA very nearly equal to * '/jJ 2 - 23 -fhe illustration pictures the first genus orbit in dotted lines, and the second genus in fall lines* The second genus orMfc makes several revolutions before re-entering, owing to the complexity Of Its " eriod. Our computation is carried ©ttt for oss revolution, stroke lines* however* Indicate the path of the infinitesimal bed? as its period approaches the close* - 24 -AI ILLUSTRATIVE PERIODIC ORBIT OF THB SBCOHD 0BJ08 TABXB I. Constant A o-"1-r K. Tn, e c « • . * , c, a, %» K <r, C, Talne 2.648 2.811 8*869' —« 2.667 - 0*747 6.548 18.288 - 0.316 0.161 - 0.112 - 0.0 37 0.164 - 0,013 1.767 1.690 Constant fi C , - % F+ H-, Hi l+r Hi^ , Hr ffs M-iu £*, d , d * A Talne -0.849 X e o 0.924 X J/£0 81.604 3.886 75-661 • 44.902 89.681 29.242 67.746 - «**.6*6 36.604 - 260.728 - 181.070 129.467 1/11 - 26 -AI ILL0S7BATITE FEE IODIC OBBIT OP THE SKOOB) OEIDS TABU I I . £ -=*.l o * .0 •l •2 •8 •4 .6 •6 .7 •8 .9 U 0 1*2 1.4 *o X< -.00166 -.00178 -.00196 -•00229 -•00272 -•00820 -•00367 -•00409 -.00442 -.00462 -•00467 -•00438 - 00362 « 0 * . -00000 -•00086 - 00067 -.00092 -.00107 -.00112 -.00106 -,00088 -.00062 - 00030 00006 00071 00109 M. •00000 •0O99C •01962 •02881 •0373 •0448 •0811 •0662 •0699 •0620 •0625 •0669 .1493 t 1.6 1.6 2.0 2.4 2.8 8.2 8*4 3*6 3.8 4.0 4.2 4.4 Vi -.00268 -.00186 -.00166 -.00267 -.00449 -.00422 -.00387 -.00244 -•00179 - 00168 -.00216 -•00302 Vt v . •00108 .00057 -.00011 -.00110 -700062 •00080 .00111 -.00099 .00047 -.00022 -.00081 - 00112 •0847 •0166 -.0082 - 0898 -.0607 -.0677 -.0478 -.0820 -.0186 •0068 •0266 •0421 1 - 26 -m ILLUSTRATIVE PSBIOBIC OBBIT OP THE SBCOHD GEHOS. TABLE III, % . 0 .1 .2 •9 . 4 . 6 . 6 . 7 •8 . 0 1*0 1*2 **4 1.6 l 1.8 2 . 0 2 .4 £ o (*.-*-!>) - . 0 0 0 3 1 - . 0 0 0 4 0 -#00066 -00106 -00158 -00218 i*00282 -00345 -00408 -00458 -00492 -00582 -00525 -00480 -00446 ••©0417 * 00412 v-*. Co(^+ts) - •00000 - . 0 0 0 8 5 - 00189 -00275 -00352 -00418 •00470 -00508 -00529 -00588 -00520 -00449 -00322 -00158 00024 .00203 •00466 6o~.l» AF«01« V^P-BmUft-q] t So(x^-t) .00000 .0109 •0214 .0314 .0407 . .0488 •0558 •0613 .0653 •0676 •0682 •0644 •0839 .0380 .0181 - . 0 0 3 7 - . 0 4 3 9 2.8 3 .2 3*4 3 . 6 3 .8 4*0 4 . 2 4*4 109 .6 109.8 110.C 110.2 110*4 - . 0 0 3 9 3 - .00258 - 00165 -#00090 - . 0 0 0 6 0 - , 00092 - •00184 - 00316 - 00304 - 00178 - .00077 - . 00032 - .00088 ,— i 3 1 / 2 8 . 8®hrT+ <j) •00514 00318 00152 - .00034 - .00219 - 00376 - 00487 - .00537 .00486 •00378 •00221 •00036 «»0O155 — ., „, , ,, 8&izrtr\ —.0666 - . 0632 - . 0 5 1 9 • •0353 - . 0 1 5 2 •0065 .0275 •0457 - •0576 - tASS - . 0246 - . 0 0 3 3 •0182 \ y y v Ua) ~'01 (a) X • Znd Genus • End of period First Genus AI ILUJSTRATITE PERIODIC ORBIT OF THE SECOBD 0SBD8. 1* Lagrange, "Coll •eta* ?/©rlns," Tol TI pp 229-824. 2 . Polncare, "Las Bethodee Boarellee da la Keoanlque Celeste," Vol I (1892) p 109* Sir George H. Darwin, "Aeta Kathetnatloa" T o l . XXI (1897) p.99 Planaer, "Monthly Botloes, Royal Astronomical Society, • T o l . LXIII (1902!) p.436. and T o l . LXIV (190S)p 98. 9 . Moulton, "Periodic Orbits", Cbaptar T. 4* Moalton, "Introduction to Celest ial Mechanics". 5 . Buchanan "Periodic OrMte of too Second Gams Bear the Straight-Line Bqnllibritun Points." (In Preee) 8 . Buchanan, "Asjnptotle Sate l l l tee Bear the Straight-Cine Equilibrium Points." American Journal Of Mathematics, To l . XLI, Bo. 2 April,1919. 

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