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An illustrative example of the ellipsoidal pendulum Mellish, Arthur Preston 1928

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U . B . C . LIBRARY CAT. MO. L£ìg>V •CC. WO. lllw AN ILLUSTRATIVE EXAMPLE of the ELLIPSOIDAL PENDULUM by Arthur Preston Mellish X X X X A Thesis submitted for the Degree of MASTER OF ARTS in the Department of MATHEMATICS The University of British Columbia X X X x April, 1928. CONTENTS. Part I •Introduction.. Page 1. The Problem Considered. 1 Part II Resume of Dr. Buchanan's paper on the Ellipsoidal Penduluia, 1. The Differential Equation. 2 2. Spherical Pendulum. 3 3. The -z.- equation. 4 4. The Equation of Variation. 4 5. The Integration of Equation (ll). 5 6. The Horizontal Motion. 6 Part III The Algebraic Expressions for Certain Series. 1. Introduction. 9 2. A Number of Auxiliary functions of "X- • 9 3. The Series C,0 , CAoj C o a , C ,, . 12 Part IV The Construction of the Orbit. 1. 2. The Values assigned the Constants. The Numerical Coefficients of Certain 3eries. 14 14 Pa. ; a The X, - equation. 15 The X - and ^ - equations. 18 The Orbit. 19 UBC Scanned by UBC Library AN ILLUSTRATIVE EXAMPLE of the ELLIPSOIDAL PEiiDULUM Part I Introduction. 1. The Problem Considered. In a paper to be presented to the Royal Society of Canada in May,1928, Dr. D. Buchanan has obtained equations which define the periodic orbits of a particle constrained to move, under gravity, on the surface of the ellipsoid of revolution X S ^ *"•(!+0 - O , ( 1 ) £ being a parameter and JL a constant. It is assumed that the surface is smooth and conforms somewhat closely to a sphere. For € = O the problem reduces to that of the spherical pendulum which is considered by Moulton in his memoir on "Periodic Orbits." For £ ^ 0 Dr. Buchanan has designated the problem as the "ellipsoidal pendulum." The object of this thesis is to construct the orbit determined by assigning numerical values to the various constants on which the solution depends. It is first necessary to give a brief outline of Dr. Buchanan's paper. This is done in Part II. In Part III several series which enter into the solution of Jrhe z - equation are computed. In Part IV numerical values are assigned to all constants and the result-ing orbit eventually constructed. F Part II Resume of Dr. Buchanan's Paper on the "Ellipsoidal Pendulum'; 1. The Differential Equations. As suggested "by the form of equation (l) the origin of co-ordinates is taken at the centre of the ellipsoid with the z-axis vertical and positive upwards. The particle is assumed to be of unit mass and to move on the ^iven surface without friction. The differential equations are then found to be < £ V * i > o + + + ¿ ¡ > ' 1 [ X o + x . e + x ^ v + ) (2) + where X . ^ I j ^ . ^ - C g ) ^ = 1 ( s x ^ c , * - * ^ (3) (4) - 3 -The constants ^ and C v are the acceleration due to gravity and the energy constant respectively. 2.The Spherical Pendulum. For <c - o equations (2) reduce to those of the spherical pendulum. We shall require here only the % -equation, viz: <«> and its solution as obtained by Moulton. Equation (5) admits the integral ( 6 ) where Cg, is the constant of integration. If the right side of (6) has the roots , ^ j^jjhaving «(^»i^: ""'j } then there are made in (5) the substitutions , On solving the resulting equation for uu the following solutions are obtained for 3. and "b : a^ yCO - + (i-Cfo + TYu.1^ • • • • 1 ( b ) <> = — U + i i Ij. V i.r If the cases of the simple pendulum and of revolution with infinite speed in the _ plane are excluded, then <x oi- . . . C, v satisfy the relations > ) 3 I 1 ) Û , I , > (9) - 4 -3. The -^-equation. The equation of (2) is transformed "by means of (7c) and by putting % = This equation then "becomes 'i £ ^  + era a T-yU 3l (Lni+t')-- J L^ » t C < o + C-u>C„ + u9-A(Toa< C - Y i 0 + C - j ' V ^ -+• terms of higher degree in<£^u>-, (11) where thé primes signify a ifferentiation with respect to t. The various C'a denote series similar to ^ ) being power series inyj- with cosines of even multiples of t in the coefficients. In the sequel, as here, such series tire uenotea by the founda-tion letter C with subscripts, superscripts or strokes. 4.The Equation of Variation ana its 3olution. If we equate the left member of (ll) to zero we obtain the equation of variation, + ly-or*** yut-^i^ -2>CI*XT H- t ) * • - • o ( 12 ) of which the generating solution is X = ^  . Applying the method of solution developed by Poincare' for this type of equation, we find that its Tolution is u>--f\ cf+ft(X (13) where ft and Ô are constants of integration,dn4 Cp - a.T j_ ul /xUo + c -+• ul ( - d3 /»¿o a^  32 /o-U^  ^  C 1 + / • • • - ' + i + + / ( . i n » I 9 (V* 2,C +- ^  if. ^Th-îj -f-6.The Integration of Equation (11). liquation (ll) will now "be integrated a« a power series in . Accordingly the «"ubstitution i uy -- (15) } «« 0 K ' is made in (ll). On equating coefficient?! of like powers of 6 in the resulting equation, cited as (ll'), a sequence of differential equations is determined which give successively the values of u>- . The initial condition U-'Co^- 0 is imposed. 0 It then follows from (15) that 0 , Cj--.,.. 00.) ( i 6 ) Coefficients of e. The terras in £ in (ll') yield the equation - c^a. • - W , (17) whera "W, - C ( 0 in (15). The complementary function of (17) is U>, * a, f -ujk, i-X-H<r<]p) . (18) The particular integral is obtained by varying the parameters CL, and Xr, . It is found that a , * ^ [- fa^i r - k z f q w, 9 w, ¿ r ^ r j -fr, U 9 ) the determinant A being constant. It is shown that />•= "X (o ) . (20) The p r o d u c O C Wi gives rise to constant terms of the type When the integrations in (19) are performed and the resulting values of O-. and substituted in (18) the complete solution for fy becomes _ A , f + L f + C - 6 -where $ , and "S i are the constants of integration ana C? , i* a cosine series described in ^ 3. As a/, is to be period-ic we must have K ib , + ^ , o ; or (Ò, - ^ (22) î »; where c*- is a power series in yU similar to . From the initial conditions (16) it follows that H , - 0 . Hence the desired solution at thin step is M 1 V ' Coefficients of € , j> The coefficients of £ in (11') give + • -- C ^ . (24) The integration of this equation is similar to that of (17)and a a. a ^ The succeeding steps are similar to the preceding and the db solution at the 'H. step is This completes the solution of (11). On substituting the resulting value of in (10) the solution of (10) for the vertical motion becomes Z - ^ i ^ c ' ^ c ' - i (26) m 6.The Horizontal Motion. As the oc- and ^-equations in (2) are the same it is necessary only to consider the X-equation. When the F - 7 -transformation» (7c), (26) and are applied to (2a) it becomes ^ ( K U a l C t f * A T-e « * 0 M * , 2 7 ) + where ¿3 -"o 0 - Ù U ( o /o -+- Ck^ /<J • • - ' -h 'VU c o - a 6 o t a i . y + û ^ ^ -The constant term and the terms i n i n (27) are identical with the corresponding terras in the - equation of the spherical pendulum. The solutions of (27) will be similar in form to those of the spherical pendulum ana with the exception of two slight variations rioted below there will be no difference in the constant term and the terms i n y ^ O in the resulting equation. Making use of the solutions ob-tained by Moulton, we have ^ - ft» £ * » ^- /a-c»O/3~0 * •J * [*» A-v-y* ^ Ot\ CtfT^S ~ J 0 i - 8 -/a - J S — [ f ^ ^ * , -• m- * • • - • 1 ' K - * 3 L 4fZ^7ToT7 ^ J> oc, ' / - C ^ + - • • •) <+(°<>+"i) y B N A C i + c - ) Zio)1-, The constants A* and »3*, differ from the corresponding constants in the spherical pendulum as the expression for Fl^ 3. X contains (. 1 + a n d t h 0 9 e f o r k^* 1 fli a n d ^a. contain which in this problem has a slightly different value to that of the spherical pendulum. This completes the solution. Part III The Algebraic Expressions for Certain Series. 1, Introduction. The object of this part is to compute all coefficients in equation (ll) which contribute terms in e and to (11'). These coefficients are C(0 j C a Q } Cô ^  ) C(l ;that is those the sura of whose subscripts is not greater than two. It will be remembered that these coefficients resulted from the transformation of (2c) by (7c) and (10). Hence we have _ 2 je H * 3 , ( 2 Q ) " a c - i , - ^ ) ^ <5-^  u>- ^  £ , J ^ Coi, - O These series and all others contributing to the "Z -equation will be carried out to and including terms in^tx 2.A Number of Auxiliary Functions of • With the exception of C 6 ^ the coefficients (29) are - 10 -the suras of a number of different functions of~X- - accord-ingly the computation will be simplified considerably by first computing the auxiliary functions ty *, ( ^ C ^ f r ) j It is found that X, CO) • c, * Z.L + )1 - J_ r / \ /• A (30 ) )j* coil wfui) y -1 o where 11 & ¿ i - <2.|J 4 ¿ I ) S i = a. Û * I J -= a,. « ¿ ^ A S i f l ^ + a ^ û ^ j a » / - - 0-n + ) S a ; a 3 3 ; -Ti -' a3JT) Ö-i'l = ¿ i I ; t, - aa.i ; 77 -t ; - 12 -3. The 3eries C , 0 , C t 0 J C 6 4 J C l ( . We now find from (29) and (30) that 0 , 0 * * 4 + ^ 13 -where S . + «V. , S / M -- - j Part IV The Construction of the Orbit. l.The Values assigned the constants. V When Or - o t jz. is the length of the stpherical pendulum. It is natural, therefore, to put A - / . The expression^,— ^ o c c u r s frequently in the computation so it might be well to assign values to U, and which will give this expression a simple value. Suitable values are found to be 1.4 for 0/, and -.6 for o( ^ . These give o{ d^ r £ . Then we have from (9b), (9c) and (7b) that C, * <-a. , y U » T h e resulting value for vJ is one of the most suitable we coula choose. / It is not too small ana yet is small enough to make the various series in ^ converge with sufficient rapidity. We must choose <= so that (26) will converge. j?or this to be true — ^ must be less than unity, that is Hence we may take G - ' o o I . If length is measured in feet and time in seconds will be equal to 32.2. Tabulating these values, we have - 2 " /, (33) yU. - -a. > fc - • ool y Cj = 3a.. a • J , 2. The Numerical Coefficients of Certain Series. From (11') it is found in (24) that e C i 0 ^ C -y*.*C C' JC„ . (34) As we are not taking the series further than terms in ^ ^ we will not need the series and only the first term of the - 15 -series C(( . By noting this fact now quite a bit of needless computation will be ^aved. We therefore will only need the auxiliary constants in (31) and (33) which contribute to Cl0 and to the first term of Cu , The expansion for Coa_ contains none of those constants. The required auxiliary constants are , a , ^  ) i ^  , > ) G ,, , • It is found that * = a. , i-'ii 40,0.^ - w 2 o o ; G-O * i-3sr o} a, H - -2.-000 ; a(y. - -^'oo ¿ w * - - / f T t / o " - " ' 2 ' ; CLkt -li'flj " 1' , --K^ f . Prom these we find that C, o * - • if >'3» + ( - ^  < cjr ^  i + 3 • 1 / if C ^ a r) + C—6"'' b'-i- "7- £ V 3 a. - ¿•aoa a. " " J .The "X -equation. We now proceed to find the value of the ^-equation. It will be necessary to go through most of the steps which are done in terms of general symbols in ^ 5 of Part II. First it is found from (20) and (14) that - 16 -A s - a.- a a j and J c ** • 3 (36) Next we have from (18), (19) ana (21) that K i P c t r ^ where the three integrals do not contain constants of integra-tion, as the*e constants are expressed in £]t and 13 , . «.s these integrals are later multiplied b y y ^ w e are only in-terested in the term independent o f i n the third integral. In the first we must also consider the terms contributing to Co, . AS K is proportional to ^ the second will drop out altogether. Considering the product where X is defined in (14) and Cl0 in (35), and using (36) it is found that ci ( ^  - -X (a - i o bfo • " • ) = a- • Next, from (22) and (14) We now wish the first terms of the series - <pJ%c(o <xZ and - i x j c i X ( the sum of which gives the first term of C\ in (21). Using (14), (35) and (36) we find that r - L (^JC-.y-i-j/croar-H. Jetr*. X "-Ca'oS C I - c ^ u - c ) 4- - - • - . .'Oi'd $ (' - -- - '/ o I b -C t (39 ) Using this result with (14) and (38) we now have - 17 -t>, - ft,-* + C , 7 3 6 -- - o / a s e ^ u j } j (40) and C • (41) On completing this part of the solution it is found from the above result and from (33) that 5 C = ' O i i ' C ' = - ' O o ^ i t . -Jyuo i'oo-j - 'ooi'fttra- a t" — ' o o i^ -c J -/• a 1 (,'0<jy--'ot>y. a, C — • o o 2 v - • O 0 0 "i Although the computation is carried to three terms it cannot be relied on to give accuracy beyond two significant figures. It will be found from (26), (35) and (8) that these two figures are the first two places of decimals. Hence only the first term of (42) will enter into the final solution. This result is to be expected from the very small value <r r <•') assigned to & . From the small value obtained for -r»-^ it might safely be assumed that all terms contributed to the solution by C L X ^ i n (26) are too small to be counted. How-r c») ever, to make the solution complete#we will compute . From (34) we have in (24) that Using (40), (41) and (32) it is found that C a.a.a (.,-eC(P h-X ) C '¿aa ¿ - T b T i c ^ _ - 1 8 --k-xa. l i t i a ^ i ' I / «a. i> c O b T j ìa.7- C " & i ' ^ - ' 8 6 i r » 4 -+ • 11 a. i C j to t t • I 6 b Cera { * ' - i i ' j r c o ì X 3 o that era, ¥ -h ^  a. a a. - • yyy 6 a t" 4- 1 11 ft t> c-tr^ - b U + • i t<o 1 S- "Ej ^ ( i J The remainder of the computation of ^ is similar to that of C1'^ . The following results are obtained: C C - -D c - O a t C - • a S"-«- ' O i l t r v -a-I H- '0 / <2. l J 0 i ¡ftf. + • tf C/a a. t_ ' o o ¥ ¿'c-^  »4- c • o i 6 ^  6 j - 1 - ; M d ^ C " ^ . . . . « : " ' = ' o o o V t ^ M , (_ - , o(7t t y v • <j a 0 ^ ^ a c 4 ' f t O O ^ U J C ' O o o ^ '(540-i a-<r 3 This completes the solution of the 7> - equation. Taxing all numbers to two places of decimals we have from ( 3 5 ) , (42), ( 4 3 ) , and (8) that ^ - ' 0 1 Z-C^ a. ~ _ • I f - • SL I X f •*• • " , ( 4 4 ) .The X- and ^-equations. The solutions for the *x- and equations will not be - 19 -carried beyond terras in y u o • Thus they may be obtained directly by substituting for the iiiiown constants in ( 2 8 ) . It is found successively that nc, - j(<j-cpir) ,>(45) % '-»latt^ 'wc-fe'^ iarii i -6 r +-- & a4f -j, 6 + - - j uj - 'I O b ' / ^ ' ^ C + ' T ^ ^ f ^ l ' t t + ' O l A v w ' i ' 6 ^ 4 -To find the period of C* and ^ we raust find the period of OC, C & ^ t in (28). If'?, is the period of and the period of C then P. -- Tf 7 \ » , £JT • Hence T'-YP, - = 5'Tf. (46) It may be noted that is not. exactly commensurable with , but is only so because we take i to a finite number of places of decimals. 5. The Orbit. Equations (45) and (44) give Or.^.zfor any time T . The appended table gives , X for values of i -if differing by from O to a complete period. A T ' column is added giving the value of which gives a check as to the accuracy of the computation as far as it is taken. From (l) the value of this expression - 20 -should he JL / . The diagram is self-explanatory. In making the ** diagram OC was plotted against 7L , that is for 2,--i>| ( j ^ was so taken that \) ?c N- 1 = ' 1 , ' 1 % } respectively. This makes no essential difference in the orb it merely serves to lessen the slight error introduced by neglecting the coefficients of in the X - and ^ -/ equations. 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