UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Periodic orbits of the second genus for the crossed orbit problem of the helium atom Milley, Hermon Reginald 1941-11-10

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
831-UBC_1941_A8 M5 P3.pdf [ 1.46MB ]
Metadata
JSON: 831-1.0080556.json
JSON-LD: 831-1.0080556-ld.json
RDF/XML (Pretty): 831-1.0080556-rdf.xml
RDF/JSON: 831-1.0080556-rdf.json
Turtle: 831-1.0080556-turtle.txt
N-Triples: 831-1.0080556-rdf-ntriples.txt
Original Record: 831-1.0080556-source.json
Full Text
831-1.0080556-fulltext.txt
Citation
831-1.0080556.ris

Full Text

L£ [fur A % . n PERIODIC ORBITS OJ? THE SECOKD GEMJS FOR THE GROSSED ORBIT PROBLEM 01 THE HELIUM ATOM Herman. R e g i n a l d M i l l e y A T h e 3 i s Submitted i n P a r t i a l F u l f i l m e n t of the Requirements for t h e Degree of M&STER OF ARTS i n t h e Department of MATHEjIAT IC S THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1941 CONTESTS I . PRELIMIMARY: THE FIRST GENUS ORBITS. § 1. I n t r o d u c t i o n Page (1) § 2. The D i f f e r e n t i a l E q u a t i o n s " (2) § 3. The E q u a t i o n s of Displacement " (4) I I . THE SECOND GENUS ORBITS. - § 4. D e f i n i t i o n of Second Genus O r b i t s Page (6) § 5. The D i f f e r e n t i a l E q u a t i o n s " (7) § 6. The E q u a t i o n s of V a r i a t i o n and t h e i r S o l u t i o n s " (8) § 7. N o t a t i o n !l (15) g 8.. The P e r i o d of the Second Genus O r b i t s 11 (17) I 9. The S o l u t i o n s « (17) Ref e r e n c e s » (28) - I - PBRIODIC ORBITS: OF THE SBOOHD GENUS • " i FOR THE GROSSED ORBIT PROBLEM' OF THE HELIUM ATOM I . PRELIMINARY. §1. I n t r o d u c t ion« It i s proposed h e r e t o construct the second genus orbits f o r a special case of the problem, d i s c u s s e d by Dr. Buchanan i n h i s p a p e r 1 ) "Crossed O r b i t s i n the Restricted Problem, of Three Bodies w i t h Repulsive and Attractive. F o r c e s . " The case d e a l t with i s that designated i n the l a t t e r as Part I I , Case I . The problem considered d e a l s with the m o t i o n of two i n f i n i t e s i m a l b o d i e s which a r e attracted by a f i n i t e body but r e p e l l e d by each, other, the nature of the f o r c e s involved b e i n g Newtonian ( i . e . , , obeying: the inverse square l a w ) . For simplicity, the two i n f i n i t e s i m a l bodies w i l l be called " e l e c t r o n s " and the f i n i t e body the " n u c l e u s " . A particular solution of the problem i s that i n which t h e electrons revolve i n c i r c l e s with the nucleus as centre and r e m a i n diametrically opposite. Two tjrpes of orbit are obtained when the electrons are displaced from their c i r c u l a r motion. In part I the electrons remain diametrically opposite and equidistant from the nucleus. In part II the distances of the electrons from the nucleus are equal, t h e i r longitudes d i f f e r by 180°, but their latitudes are the same. The particular case which i s common to parts I and I I — that i n w h i c h the l a t i t u d e s a re zero — I s c o n s i d e r e d i n p a r i I I and i t i s d e s i g n a t e d as case I.. I t i s i n the v i c i n i t y of these, o r b i t s t h a t we s h a l l make our c o n s t r u c t i o n of t h e second genus o r b i t s , . I n s e c t i o n s 2 and 3 a b r i e f o u t l i n e of the r e s u l t s ob t a i n e d i n "Grossed O r b i t a . . . . " w i l l be made, showing the method used i n c o n s t r u c t i n g the. f i r s t genus o r b i t s upon w h i c h t h e work of t h i s t h e s i s i s to- he based- §2* The D i f f e r e n t i a l Equations.. T a k i n g a r e c t a n g u l a r system of axes w i t h o r i g i n at the nucle u s and d e s i g n a t i n g the. c o o r d i n a t e s of the el e c t r o n s * by xl» Yl» z-i 2:2* J2* z2> w e have f o r the f o r c e f u n c t i o n . "U, of the system: •a- where k^ = r a t i o of r e p u l s i o n to. a t t r a c t i o n , , r± * ( x L 2 + y L 2 4 2 ^ 2 i t - r 2 - ( x 2 2 + ? 2 2 * z 2 2 ) * r i 2  s | ] x l - x 2 ) 2 i C j i - y 2 ) 2 4 C ^ l - 2 2 ) 2 ] * and. the u n i t s o f s-paae and tIme have been so chosen t h a t t h e g r a v i t a t i o n a l c o n s t a n t equals u n i t y * The e q u a t i o n s of t h e mot i o n a re thus From these e q u a t i o n s and a c o n s i d e r a t i o n of the. con s t r a i n t s on t h e mot i o n of the e l e c t r o n s as o u t l i n e d i n the i n t r o d u c t i o n , we o b t a i n the two d i v i s i o n s of the problem, v i z ; y 2 " ~ y l ' E l ~ r 2 = ^ r 1 2 ' z 2 = ~ z l * P a r t l i t S£ = — i ^ , y 2 - - y x» z 2 ~ z i - As we a r e not concerned w i t h the development of p a r t I we s h a l l c o n s i d e r i t no f u r t h e r . By v i r t u e of r e l a t i o n s ( 3 ) , we need c o n s i d e r the motion of one e l e c t r o n o n l y . On s u b s t i t u t i n g these r e l a t i o n s i n e q u a t i o n s ( 2 ) E and t r a n s f o r m i n g the r e s u l t i n g e q u a t i o n s and the v i s - v i v a i n t e g r a l t o c y l i n d r i c a l c o o r d i n a t e s by the sub s t i t u t i o n s : : we o b t a i n ; ,* A ' A- 0 " -t- 3~A-'& ' — O 3." = — /• the i n t e g r a l of (5 b) b e i n g h^B~ & • U a l n g t h i . last r e l a t i o n to a l i e n a t e ff' f l o . ( 5 ) . we g e t : /i," •= g 2~ — A __ These equ a t i o n s have the p a r t i c u l a r s o l u t i o n where ^ =• / - ^ a </*-*-< I ~4~ §3. The E q u a t i o n s of Displacement and t h e i r S o l u t i o n s . By means of the s u b s t i t u t i o n s we o b t a i n from e q u a t i o n s (6) the "equations of di s p l a c e m e n t " , t h a t i s , . the equ a t i o n s g i v i n g the di s p l a c e m e n t s from the plane o r b i t g i v e n by s o l u t i o n s ( ? ) . They a r e : 'f t-o*he - f % ^ r ' j - 1 1 Y f V f f V • • • J i + p < W - ^ c i * i j [ e t » r r j - s , Y ^ ' T " - i r 5 ; * • • - J The s o l u t i o n s of these equations w i l l g i v e the p e r i o d i c o r b i t s of the f i r s t genus of which t h e r e a r e t h r e e t y p e s , each type b e i n g c h a r a c t e r i z e d by i t s p e r i o d . The p e r i o d s are determined by a c o n s i d e r a t i o n of t h e equ a t i o n s of v a r i a t i o n o f ( 9 , : . h f - ° ; -° which have the t h r e e s e t s of g e n e r a t i n g s o l u t i o n s , v i z : Case I : f> - f\Onit -+ f^A^1^ P e r i o d : ^ jr Case I I : ^ x ] ) o d ^ i -f- £ ^ P e r i o d : ^ J^L i r r a t i o n a l * p e r i o d :  T = M ^ ) = ^ ( - f Case I I I : -5- Gorresponding t o case I , a s o l u t i o n of equations (9) i s found t o e x i s t o n l y when ^ ~ £> . T h i s s o l u t i o n forms the " g e n e r a t i n g s o l u t i o n " f o r t h e second genus o r b i t s t o be con s t r u c t e d . The f i r s t genus o r b i t s f o r case I are ob t a i n e d by s e t t i n g - + - f, £ 1- where the P- are v a r i a b l e s and the f* are c o n s t a n t s . V s On s u b s t i t u t i n g i n equations (9) and e q u a t i n g c o e f f i c i e n t s of £^ on each s i d e of the r e s u l t i n g e q u a t i o n , a s e r i e s of d i f f e r e n t i a l e q u a t i o n s of the form w i l l a r i s e , which, when i n t e g r a t e d s e q u e n t i a l l y , , determine the v a r i o u s P• and <T„- . V * The i n i t i a l c o n d i t i o n s f'(o) = / ; f>(o) ~- o serve t o e v a l u a t e the c o n s t a n t s a r i s i n g from the i n t e g r a t i o n s , The r e s u l t s a r e : ^ Ccru Z 4- £ ( % C "~ i Caj-^T -o (10) -6- I I - THE SECOffl) GEMJS ORBITS 34. D e f i n i t i o n of Second Genus O r b i t s . 2 ^ Suppose we have a set of d i f f e r e n t i a l e q u a t i o n s i n w h i c h the are a n a l y t i c i n the. arguments, do not c o n t a i n t e x p l i c i t l y , and are p e r i o d i c w i t h p e r i o d T. The p e r i o d i s , i n g e n e r a l , a f u n c t i o n of the parameter £ , I f these equations admit the p e r i o d i c s o l u t i o n s \ - 9- ( € ; t) h a v i n g the. p e r i o d T, t h e n such s o l u t i o n s are s a i d t o be of the f i r s t genus. Mow l e t where £.'0 i s c o n s i d e r e d as a f i x e d c o n s t a n t and A as a v a r i a b l e parameter. When these s u b s t i t u t i o n s are made i n the above d i f f e r e n t i a l e q u a t i o n s we o b t a i n a s e t i n y^ i n which t h e r e are no terms independent of y^ or i \ . I f t h i s set admits the p e r i o d i c s o l u t i o n s ^ = ; -t) h a v i n g the p e r i o d H b e i n g an i n t e g e r , t h e n the s o l u t i o n s %. = 0- ( 8 O J - t ) + ( € a ^ A j ± ) are s a i d t o be. of the second genus. S i n c e the v a n i s h w i t h "A , the second genus o r b i t s approach t h o s e of the „7- f i r s t genus as X approaches z e r o . §5. The D i f f e r e n t i a l E q u a t i o n s . I n e quations (9) make the s u b s t i t u t i o n s £ ^ C„ (' + X j where £ occurs e x p l i c i t l y , where the zero s u b s c r i p t i s a t t a c h e d to f, j f , £, £• merely t o i n d i c a t e t h a t t h e y are t h e o r i g i n a l v a l u e s of these quan t i t i e s as g i v e n by e q u a t i o n ( 1 0 ) . (Mote: t h e r e are no s o l u t i o n s f o r second genus o r b i t s i n the plane.) We thus o b t a i n the e q u a t i o n s : +-fi 13*J<+v-'x f . C o n A W o V ^ 3 j j -8- the terras Independent of p and q ha v i n g dropped out by re a s o n of equations (9)„ §6. The E q u a t i o n s of V a r i a t i o n and T h e i r S o l u t i o n s . The. equations of v a r i a t i o n of (11) a r e obtain e d by equa t i n g t o zero the l e f t s i d e : / r/tt(r3c^rj^i-(^3'=m^ • • - J ( 1 2 ) S i n c e these two equa t i o n s are independent, t h e i r s o l u t i o n s w i l l be c o n s i d e r e d s e p a r a t e l y . C o n s i d e r f i r s t , e q u a t i o n (12,a). By the t h e o r y of l i n e a r d i f f e r e n t i a l e q uations w i t h p e r i o d i c c o e f f i c i e n t s , % e may get p a r t i c u l a r s o l u t i o n s ' t o t h i s e q u a t i o n by d i f f e r e n t i a t i n g p a r t i a l l y i t s g e n e r a t i n g s o l u t i o n (10,a) w i t h r e s p e c t t o the a r b i t r a r y c o n s t a n t s which are c o n t a i n e d i n the l a t t e r b ut which do not appear i n the o r i g i n a l d i f f e r e n t i a l e q u a t i o n . I n t h i s case, two such a r b i t r a r i e s occur: t 0 and £ . The two p a r t i c u l a r s o l u t i o n s are u o How, s i n c e . the n 2 [ £ e ) = 1(^J . ^  j ^f\^0)]^(^^) * ^ o A l s o , j -(12^ +2^rci-f-^^)£ -9- We t h e r e f o r e o b t a i n Sir] - Cc<r»*f**-+J • [~ ^ r " *<<^ T h i s b e i n g a p a r t i c u l a r s o l u t i o n , we may d i s r e g a r d the con s t a n t f a c t o r , s i n c e i^) w i l l be m u l t i p l i e d by an undetermined c o n s t a n t l a t e r , and take 1r i as our p a r t i c u l a r s o l u t i o n c o r r e s p o n d i n g to ^ . The o t h e r p a r t i c u l a r s o l u t i o n of (12,a) w i l l c o n s i s t of a p e r i o d i c f u n c t i o n p l u s ~C times another p e r i o d i c f u n c t i o n . F o r , s i n c e £ e n t e r s i n t o /° b o t h e x p l i c i t l y and i m p l i c i t l y ( i n Z by ), we have 2 where the b r a c k e t s e n c l o s i n g ^ £ denote e x p l i c i t d i f  f e r e n t i a t i o n o n l y . On p e r f o r m i n g the i n d i c a t e d d i f f e r e n t i a t i o n s , t h i s s o l u t i o n becomes: y t r ) 1- A t here ^Lt) Is g i v e n by and X appears i n the d i f f e r e n t i a t i o n 2£ £ 3 y (13.3 ) -10-. I n order t h a t (13.1) and (13.2) may c o n s t i t u t e a funda mental s e t -of s o l u t i o n s ~ • of e q u a t i o n (12 , a) , t h a t i s , i n order t h a t the g e n e r a l s o l u t i o n of (12,a) s h a l l he Kj_ and b e i n g a r b i t r a r y c o n s t a n t s , the fundamental d e t e r  minant of the set must not be z e r o . T h i s determinant i s : ^CrJ ^Cr) + /\<p(V f- A C -4>Lx) 5) How i t has been shown t h a t the v a l u e of the determinant of such a fundamental set i s c o n s t a n t . We can t h e r e f o r e compute the v a l u e of D most c o n v e n i e n t l y by s e t t i n g f equal t o z e r o . Thus ± +- - f •»--- - O 1 • , I t t h e r e f o r e f o l l o w s t h a t the most g e n e r a l s o l u t i o n of e q u a t i o n (12,a) i s and Lp b e i n g , of co u r s e , p e r i o d i c ( ~x.Tr) The above method cannot be used i n the s o l u t i o n of e q u a t i o n (I2,b) because of the absence of a g e n e r a t i n g s o l u t i o n , -11- or r e t h e r , because i t s g e n e r a t i n g s o l u t i o n i s i d e n t i c a l l y 6 1 z e ro. T h i s • e q u a t i o n , a form of H i l l ' s equation,, ' w i l l be s o l v e d by making the standard t r a n s f o r m a t i o n (14.1) i n which / , ^ * . *• /• p V - « v 4 > " • a. The v a r i o u s and /Z£ w i l l be determined so as to f u l f i l l t he f o l l o w i n g two c o n d i t i o n s : ( i ) The p e r i o d i c i t y c o n d i t i o n (14.2) ( i i ) The i n i t i a l - v a l u e c o n d i t i o n : (o)-=t j n^r°) = ^  ^ i — The s u b s t i t u t i o n of (14.1) i n e q u a t i o n (12,b) y i e l d s : or s- As b e f o r e , we equate t o zero the c o e f f i c i e n t s of £ . Each such e q u a t i o n w i l l have f o r i t s complementary f u n c t i o n the s o l u t i o n of f "J ^ I T .J C o e f f i c i e n t of £ : - . __2_ ^' _ „ ^ ^* • p ^ -o -12- whence a. A p p l y i n g c o n d i t i o n s (14,2), we get 0 / bmce .— , i n g e n e r a l , i s not r a t i o n a l , we must have G 2 = 0. Theref ore C o e f f i c i e n t of £ nr0 - i or •t • • lip*. f ifii / ' ^ t r 0 y How i n order t o i n t r o d u c e no terms p r o p o r t i o n a l to ' c I n t o our s o l u t i o n , i t i s ev i d e n t t h a t the con s t a n t term on the r i g h t s i d e of t h i s e q u a t i o n must v a n i s h . That i s , . A r ° I n t e g r a t i o n g i v e s i « , . . _ _ ! _ g « 1 and a p p l y i n g i n i t i a l and p e r i o d i c c o n d i t i o n s we o b t a i n ±IT - whence, C-? = 0,. and 0 o = • <-> ' 2 4 — Th e r e f o r e -13- Coef f i e l e n t of fi*.. + > ^ ( l ^ ^ c * J c ^ ^ J - j^/ir - a S u b s t i t u t i o n f o r , ^  , /zr y i e l d s To ensure t h a t v 2 s h a l l have no n o n - p e r i o d i c terras, we cause the cons t a n t terms on the r i g h t s i d e of t h i s l a s t e q u a t i o n t o v a n i s h . That i s , — /S - — 1 - - 1 » ^ whence ^ 3 J^- ( > Then ; 3(^+3)(^-«J c r S(tJ^.-3)(^-H] + UL t M - J*- e~x^ /(fO+fc) /(>/* As b e f o r e , i t can be shown t h a t 0 ^ = 0 , and t h a t has the v a l u e 33/* 5 - 264/U2 j 264/u. + 288 3 8 / ^ ( l — / < ) ( 4 — / ^ ) 2 I n s i m i l a r f a s h i o n we can f i n d as many more ji- S and fifj -14- as w i l l determine f3 and if t o any d e s i r e d degree of approx i m a t i o n . Having now obtained the: p a r t i c u l a r s o l u t i o n q of (14.1), t h e other p a r t i c u l a r s o l u t i o n i s obtained by r e p l a c i n g i by - L i n it and & , t h a t i s , the c o n j u g a t e of q i s a l s o a 7 ) p a r t i c u l a r s o l u t i o n Thus, the g e n e r a l s o l u t i o n of e q u a t i o n (12,b) i s _ (fj . s,-':AZ/7,,(-' Tr 'J where and v ^ 2 ) d i f f e r s f r om v ^ " ^ o n l y i n the s i g n of <^  .. I f the s i g n s of b o t h 'C and ^ are changed i n (14.4), q does not change. T h e r e f o r e , because of the p a r i t y of cos 'C and s i n Z, i t must be e v i d e n t t h a t the c o e f f i c i e n t s of the c o s i n e terms i n are always r e a l and those of the s i n e terms always p u r e l y i m a g i n a r y . The fundamental determinant of t h i s s e t of s o l u t i o n s w i l l be computed f o r l a t e r use. I t i s -. -15- 4Tl /A b e i n g c o n s t a n t I n v a l u e , i t s e v a l u a t i o n w i l l be most ea s i l y - e f f e c t e d by s e t t i n g % = 0. How v ^ 1 ) (0) = v ^ 2 ) ( 0 ) = 1,- by e q u a t i o n (14.2) , i i , and v ( D ( o ) = - v ^ ( o ) = ^ • t ( C + t , V ^ •• • j by e q u a t i o n (14.4). Hence a power s e r i e s i n £ (14 T h i s completes the s o l u t i o n s of the equations of v a r i a t i o n ( 1 2 ) . These s o l u t i o n s are the complementary f u n c t i o n s of equations (11) whose p a r t i c u l a r i n t e g r a l s are next to be determined as power s e r i e s i n °\ . §7. d o t a t i o n As the a l g e b r a i c e x p r e s s i o n s become too" unwieldy and so obscure the methods of a t t a i n i n g c e r t a i n r e s u l t s i n the subsequent c o n s t r u c t i o n s , we s h a l l employ the " f o u n d a t i o n - l e t t e r " n o t a t i o n of Dr. Buchanan The n o t a t i o n c o n s i s t s of symbols of the form. g( > K ) S ^ ' )( J and r e p r e s e n t s power s e r i e s i n )\ h a v i n g f o r t h e i r c o e f f i c i e n t s sums of c o s i n e s or s i n e s r e s p e c t i v e l y . Of the two parentheses -16- I n the s u p e r s c r i p t s , the f i r s t has two e n t r i e s : an i n t e g e r 0, 1» 2, ..., f o l l o w e d by the l e t t e r e or a. The i n t e g e r d e s i g n a t e s b o t h the lowest power of £ i n the s e r i e s and the p a r i t y i n £ , w h i l e t h e l e t t e r e or o denotes t h a t the arguments of the c o s i n e s or s i n e s a r e even or odd m u l t i p l e s of "5 r e s p e c t i v e l y . The second p a r e n t h e s i s c o n t a i n s an i n t e g e r which denotes the amount by which the h i g h e s t m u l t i p l e of i n the arguments of the t r i g o n o m e t r i c terms o c c u r r i n g i n the c o e f f i c i e n t of any power of £• exceeds t h a t power. I n t h e f o l l o w i n g work we s h a l l use the m o d i f i e d n o t a t i o n o b t a i n e d by d e l e t i n g the l e t t e r s e and o i n the f i r s t p a r e n t h e s i s s i n c e i n bur case the arguments of the s i n e s and c o s i n e s are n e i t h e r e x c l u s i v e l y even nor e x c l u s i v e l y odd m u l t i p l e s of T> * We may have o c c a s i o n a l s o t o a d j o i n a s u b s c r i p t to a g i v e n f o u n d a t i o n l e t t e r , i n which case we understand t h a t we are c o n s i d e r i n g a p a r t i c u l a r s e r i e s of t h a t type. An Example. • ' . " d 0 ) ( o ) = e ( c ^ c ^ ^ C ^ c ^ . c ^ ^ r " /V/ & K 1<'1C *" " " ' " «3K-—j %~~~\ ^ For f u t u r e r e f e r e n c e , we may note t h a t i n t h i s n o t a t i o n , -17- §8. The P e r i o d of The O r b i t s of the Second Genus. For t h e f i r s t genus o r b i t s , the p e r i o d Is air i n x> i c 3- w h i c h i s Tjr- — K}+b)2~- T I n t . The p e r i o d of the second genus o r b i t s i s t o be,by d e f i n i t i o n , N ' T 5 ( I + a power s e r i e s i n A J IT b e i n g i n t e g r a l . How t h e p e r i o d of the s o l u t i o n q i s T h e r e f o r e i n order t h a t our f i n a l s o l u t i o n s be p e r i o d i c { i n IS , we must hare the p e r i o d ^ ^ * - l f r i n Z ~ • T, a power s e r i e s i n ^ J as r e q u i r e d by d e f i n i t i o n . A l l v a l u e s of £ f o r which t h i s does not h o l d a r e e x c l u d e d . §9. The S o l u t i o n s of E q u a t i o n ( 1 1 ) . A f t e r s u b s t i t u t i n g f o r the v a l u e of f> ( e q u a t i o n ( 1 0 ) ) . i n e q u a t i o n ( l l ) r we. a r r i v e at the f o l l o w i n g : V -18- where 'oo f - £> (i + \. ^ ^ ) ^ ^ j + - - - In order t o i n t e g r a t e e quations (15) as a power s e r i e s i n A , we put > On e q u a t i n g c o e f f i c i e n t s of l i k e powers of }\ i n the r e s u l t i n g e q u a t i o n we o b t a i n a s e r i e s of d i f f e r e n t i a l e q uations i n pj and q.. h a v i n g f o r t h e i r complementary f u n c t i o n s the s o l u t i o n s (13.4) and (14.4) r e s p e c t i v e l y . I t wi11 be shown t h a t the ~Y- and the c o n s t a n t s of i n t e g r a t i o n w i l l be so 4 determined t h a t pj and q. w i l l have the p e r i o d T 2- We proceed then as i n d i c a t e d . -19- Knowing the complementary f u n c t i o n f o r the f i r s t of these e q u a t i o n s , we may s o l v e i t c o m p l e t e l y by the method of v a r i a t i o n of parameters. Thus, where the a r b i t r a r i e s have been a s s i g n e d the s u p e r s c r i p t s 1 f i ) m order t o a s s o c i a t e them w i t h p^ and q^. I n g e n e r a l k| J ' and k ^ w i l l be a s s o c i a t e d w i t h the s o l u t i o n s of p. and q 1 * Here D i s the fundamental determinant of § 6 and has the v a l u e D = c o n s t a n t = 1 4 a power s e r i e s i n £ . Employing the f o u n d a t i o n - l e t t e r n o t a t i o n , these equations may be w r i t t e n : -20 3> /,„ tJ"-> whence, upon I n t e g r a t i o n , 3- k!" - 4J^^y,(^^r'j] t h e d's r e p r e s e n t i n g power s e r i e s i n £ w i t h c o n s t a n t c o e f  f i c i e n t s . S u b s t i t u t i o n of these e x p r e s s i o n s i n the complementary f u n c t i o n y i e l d s ZD +• A ft^sTj *-ffi<?? On r e d u c i n g , and n o t i n g t h a t the terms i n At c a n c e l o f f , The complete s o l u t i o n a t t h i s stage w i l l now be -21- I n order t h a t be p e r i o d i c , terms c o n t a i n i n g T as a f a c t o r must be e l i m i n a t e d . That i s , T h e r e f o r e the p e r i o d i c s o l u t i o n s f o r p^ and q-j_ are f i y The a r b i t r a r y c o n s t a n t s KV 1 may be e v a l u a t e d by the use of the I n i t i a l c o n d i t i o n s A p p l y i n g these c o n d i t i o n s to the above equations I t i s found t h a t K*1* s' 0; = - The p e r i o d i c s o l u t i o n s thus become ' 1 - 3 " I n e q u a t i o n (18) we have a l i n e a r r e l a t i o n c o n n e c t i n g K g 1 ^ and ^ . Another such r e l a t i o n s h i p w i l l be found i n the process of I n t e g r a t i n g p^ and q 2 which w i l l u n i q u e l y determine these two c o n s t a n t s and hence f u r t h e r s i m p l i f y ( 2 0 ) . (20) -22- C o e f f i c i e n t of A We o b t a i n the s e t : i n which ~? f". otSj [& +f, E £ * ^ Upon varying the parameters, there result D and A b e i n g the determinants g i v e n i n §6. C o n s i d e r the l a s t two of these e q u a t i o n s . We s h a l l I n v e s t i g a t e those p a r t s of o j 2 ) which c o n t a i n K g 1 ^  and y, , w i t h the view to o b t a i n i n g a supplementary r e l a t i o n t o ( 1 8 ) . (2) The terms of Q, i n w h i c h we are i n t e r e s t e d are those -23- c o n t a i n i n g £ and , and are shown i n the f o l l o w i n g t a b l e C o e f f i c i e n t of£.('+*) C o e f f i c i e n t of fiO + S) f ' ^ ^ Our two equations may now be w r i t t e n Aki* Q,[2^ and Q^2^ b e i n g those p a r t s of not i n c l u d e d i n the above t a b l e . Removal of c o n s t a n t terms g i v e s or O (23) -24- s i n c e ^ rjlb 0, o t h e r w i s e we would have the case f o r a n a l y t i c a l c o n t i n u a t i o n of the f i r s t genus o r b i t s . E v i d e n t l y we must exclude a l l v a l u e s of £ f o r which the determinant O 7- a < of e q u a t i o n s (18) and (23) i s equal to z e r o . We have, therefore,. (') (24) K 2 1 ^ ~ d 3 1 ^ a n d % ~ d 4 1 ^ w h i c h g i v e s f o r equations (20), (2) When these v a l u e s have been s u b s t i t u t e d i n Pv ' and (2) Q , t h e r e r e s u l t : The c o e f f i c i e n t of here Is the same as t h a t of 1^ i n t h e c o e f f i c i e n t of A . The s i g n i f i c a n c e of t h i s f a c t w i l l be apparent i n the l a t e r development. We s h a l l now complete the i n t e g r a t i o n s of the l a s t p a i r of e q u a t i o n s (22) . They are of the form: -25- e On i n t e g r a t i n g , -kr^^rvr'tv-^v.'sr''' (25) S u b s t i t u t i n g these r e s u l t s i n the complementary f u n c t i o n we a r r i v e a t the p a r t i c u l a r i n t e g r a l Thus the complete s o l u t i o n f o r q 2 i s : ^evys-"l • ^ )fe'rM~r^• <26) C o n s i d e r now the f i r s t p a i r of equations ( 2 2 ) . They w i l l y i e l d , on s o l u t i o n , a r e l a t i o n i n v o l v i n g V _ We have, I n t e g r a t i o n g i v e s -26- The p a r t i c u l a r I n t e g r a l i s , therefore,. •f- or and we get f o r the g e n e r a l s o l u t i o n : : The c o n d i t i o n f o r p e r i o d i c i t y w i l l g i v e the r e l a t i o n a -t- K + ^ « = ° (28) (27) A second s i m i l a r r e l a t i o n which, w i t h ( 2 8 ) , w i l l unique- (2) l y determine E 0 " and / , w i l l a r i s e i n the next i n t e g r a t i o n 2 i n the same manner as d i d e q u a t i o n '(25). Moreover, the coef f i c i e n t s of and ~Y w i l l he the same as those of ^ and ~Yf i n ( 2 3 ) . T h e r e f o r e the f u n c t i o n a l determinant of these two eq u a t i o n s w i l l he the determinant ( 2 4 ) . Thus we have K 2 2 ) = a n d ^ = d 4 2 ) ' Completing the i n t e g r a t i o n s i n (27),. we f i n a l l y a r r i v e a t the g e n e r a l s o l u t i o n s f o r p 2 and q 2 ; ~h c -27- On Imposing I n i t i a l c o n d i t i o n s i n equations ( 2 9 ) , the i n t e g r a t i o n c o n s t a n t s may be e v a l u a t e d . Proceeding s i m i l a r l y , we can c o n t i n u e the process of d e t e r m i n i n g the succeeding p. and q. , the two unknowns .... J . J K 0^ and ~Y- b e i n g determined by txva r e l a t i o n s s i m i l a r to those of equations (18) and (23).. Moreover, the f u n c t i o n a l determinant w i l l be ( 2 4 ) . H -28- REFEREHCES 1) - D. Buchanan: "Crossed O r b i t s i n the R e s t r i c t e d problem, of Three Bodies w i t h R e p u l s i v e and A t t r a c t i v e F o r c e s . " — R e n d i c o n t i d e l C i r c o l o Matematico d i Palermo, Tomo LV, (1931). 2K D- Buchanan: " P e r i o d i c O r b i t s of the Second Genus near the S t r a i g h t - L i n e E q u i l i b r i u m P o i n t s i n the Problem of Three B o d i e s . " — P r o c e e d i n g s of the Royal S o c i e t y , A, V o l . 114, (1927) 3) . H. P o i n c a r e : "Les Methodea U o u v e l l e s de l a Mecanique C e l e s t e . " —Tome I , Chapter IV. D. Buchanan: "Asymptotic S a t e l l i t e s near the S t r a i g h t - L i n e E q u i l i b r i u m P o i n t s i n the. Problem of Three B o d i e s " — A m e r i c a n J o u r n a l of Mathematics, V o l . 4.1, (1919). 4) . F.R.Moult on: " D i f f e r e n t i a l E q u a t i o n s " — Chapter IV. " P e r i o d i c O r b i t s " — Chapter I . 5) . F.R.Moulton: " P e r i o d i c O r b i t s " - Chapter I , § 18. 6) . F.R.Moulton: " P e r i o d i c O r b i t s " — Chapter I I I , P a r t I I . 7) F.R.Moulton: " P e r i o d i c O r b i t s ; l — Chapter I I I , § 51. 8) . See 2) above. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080556/manifest

Comment

Related Items