Lal, Roshan

Mishra, S C

IACS Institutional Repository

Indian J- Phys. T 7 B  (5 ) ,  5 6 7 -5 7 0  (2 0 0 3 ) o M to eUP\Integrability of two coupled oscillators using complex coordinatesRoshan Lai' and S C Mishra*O c p a r l m e n l  o f  P h y s i c s ,  K u n i k s h c l r a  U n i v e r s i t y ,  K u i u k s h e l r a - 1 3 6  1 1 9 ,  H a r y a n a ,  I n d i a* D e p a r t m e n t  o f  C o i n p u l e i  sS c je n e c .  G o v e r n m e n t  C o l l e g e ,  K a l k a - 1 3 3  3 0 2 ,  H a r y a n a ,  I n d i aE - m a i l  r o s h a n h i r a n w a l @ y a h o o  c o m  H ccetvt'd  25  N ovem het 2002. accep ted  25 June 2005\ b s l r a c t  A  g e n e r a l  m e t h o d  i s  u s e d  t o r  f i n d i n g  t h e  s e c o n d  c o n s t a n t  o f  m o t i o n  o l  f o u r th  o r d e i  in  c o m p l e x  c o o r d i n a t e s  f o r  t w o  d i m e n s i o n a l  c l a s s i c a l  AsiL'ins A g e n e r a l  p o t e n t i a l  e q u a t i o n  is  o b t a i n e d  w h i c h  p r o v i d e s  a l a r g e  c l a s s  o t  s e c o n d  c o n s t a n t  o f  m o t i o n  f o r  *wo d i m e n s i o n a l  s y s t e m s  H o w e v e r ,  i t  lids been o b s e r v e d  t h a t  t h e  a r b i t r a r y  c o n s t a n t s  a r e  c h o s e n  i n  s u c h  a  w a y  th a t  s y s t e m  b e c o m e s  e x a c t l y  i n l e g i a b l cK e y w o r d s  C l a s s i c a l  i n v a r i a n t s ,  i n t e g r a b i c  s y s t e m s ,  s e c o n d  c o n s t a n t  o f  m o t i o nl»A('S N o .  0 2  3 0  I KWhittaker 11J first investigated the problem of the construction o( an invariant other than the total energy which will be called (he second constant of motion for a system-V, = - -dVdx,d vd x . 'where V = V(a, .a'2 ).H is Studies were however restricted to the invariant of first or second order in momenta. In recent years, although there have been several attempts 1 1 -9 ] made to construct the second 1)1 der invariants, not much efforts have been made to obtaining ihc higher order invariants for such systems.The utility of the second or higher order constant of motion, they can be constructed for a system, it has been noticed that may reduce some nonlinear dynamical problems to a tiuaclrature. In fact, there already exists scarcity of classically integrabic systems in higher degrees of freedom.by introducing the complex variable z = Ai + U2 and ^ V| - u s , some simplifications [5 ] were achieved in the ilenvation and analysis which turn out to be more transparent.t ’) ) rTcspond in g  A u t h o rWe have examined the time-independent systems for the potential V = lz^ mz~z^  ^ nz^ in two dimensions.A simple analysis reveals an interesting case of two coupled oscillators which somehow could not be explicitly investigated earlier using complex coordinates. Here, we derive a general potential equation and construct the second constant of motion for the said potential where /. m. n arc arbitrary constants. Using rationalisation method, we obtain I = n = 1, m = 6, which is a case of two coupled quartic oscillators.We consider the dynamical system whose Lagrangian is(1)with the concomitant equations of motiondv  - „(?Vz = - 2 -t— , z = - 2 -—r .  (2 )dz dzAssuming the existence of the second constant of motion upto fourth order in momenta in the form of(3)where/./ k. 1= 1.2. ,5, = z. = z and arc functionsof z ,z .©2003IACS568 Roshan Lai and S C MishraThe invariant/implies d l / dt ~ 0 and using (3) we get To solve these equations, we notice from (8) and (13),have flu,, = fl,,,,(z ) = cr,(z) and 02222 =Now differentiating eq. (9) with respect to z to eliminate(4)After accounting for the proper symmetrisation of the coefficients and since (4) must hold identically in we can obtain the following relations .i^jkljn ^ ^^ Unnj ^^ lnuj,k n^njKJ ~ (^)^ i l , k  ^  jk , i  ~~«(U ■»■«/>‘S/ =0 . (7)Hqs (5)-(7) yield the following set of partial differential equationsda“ “ - 0 ,dad lda^ ^^^dz■f 4- 1112 = 0 ,^ ^^ 1^122 9 ^^1112 _ Qdz  dz^ A/||22 ^  ^ ^^ 1222 Qd z d zdai-}-i2 da'^ '^ ^^ ^4 : f -— =0, d z  dz= 0^r/2222dzdfi, I d V  d V+ 2 tf,|,2 .dz  dz  d z^da. .  ^ d V  ^ d V3 --—---- Zc/t')!') — h a-^ /pT» ““  dz  dzd zda^,  ^da^  t- + 2d l dz-:2a, d Vdz+ 2 a d V1122^ 2 2  . T * 'l 2 _ o—— 4- 2 _ — 2 r/p>>2 .. _ + 2 rV| pn ~ _ »dz  dz  d z  d z-I ^ ^  -I ^—^ = 2 a . , ---- + 2 flp^ II I.= 2 o,2 — + 2«22 3 :r- d z  d z  d z(8)(9)( 10)(1 1 )(12 )(13)(14)using eq. (8) we have d^a in 2 /^ z ^  = 0, which admits the(20)(16)(17)(18)(19)solution as‘>\{i2=c^2(z)z + <J^(z).Similarly, eqs. (12) and (13) will lead to the solution as^2222 ~ Xz U)z + X?^iz). (21)On differen tiating  eq. (10) with respect to z and correspondingly subtracting the results and making use of cq (20) and (2 1 ), we obtainc j ^dz" dz-Now, we fix ct3 = and X}> ” above equationreduces to the form1 ^/“cTt 1 x^r —r r ^  “ -------“ Constant (D j ,z air z dz'^(T2 = (1 / (i)D^Z^ 4 DJ^ Z 4;C2 =(1/6)D 3z’ +Z?,z +/>7.Substituting these values of (T2,;|^2,O’3 and in eqs. (20) and (21), wc haveI p ------ D-^ ZZ D ^zz 4  /7«jC 4  / 7 | ,0a  j 222 “  — 2". Z "t" /^ f, ZZ 4  Z 4  O 26 ^(22)(23)Similarly, we find CTj and X] starting from eq. (9) and (12) asCT| — «!  11 j “  ”  — D 3Z —2 D^ z  — ^ D^ z  ^oX\ ” ^2222 “ 4 D7Z + /79.oUsing eqs. (22) and (23) in eqs. (10) and (11), we obtain= -(2  / 3 ) z ^  = -(1 / 3)D,zz^ -  (2 / y)D^z d z  d zor fl, = - ( 1  / 6)D3Z^z  ^ -  (1 / 3)D4Z^  - ( 1  / 3)D<iZ^  -  +Ao Finally, the solution of eqs. (8)-(13) yielda,,,, = - ( 1 / 6)D3z‘' - 2 D4Z ^ - 4 D5Z + D^, <^4)<*1112 = ( 1 / 6)D3Zz ’ +D 4ZZ + D j Z+D, , (“*’*«1I22 = - ( 1 / 6)D3Z^Z^-(1/3)D 4Z^-(1/3)0^2- + D,o.Integrability o f  two coupled oscillators using complex coordinates« I 2 2 2  =  f  6 )D ^ z^ z  +  Df^zz  +  D iZ  +  D j ,^^ 2222 “  -(1 /6)D 3Z ^-2D 6Z *-4D v2 + D, .569(27) substituting these coefficients, the potential equation reduces to a simpler form(28)In order to eliminate a , ,, eqs. (14)-( 17), wejilterentiate equation (16) with respect to 2 and making use of (l4 )fo r^« i,! d z  we obtain’^ '^12 _ ^' d T ~ J zd V d V^ 1112“ ;^ ■•■‘*1122" ^  d z  d z■ I f  i  D , r .  *  - 1 ( 1  t  D , :  +  a )i \ 2I d  d V  d V(29)Similarly, on differentiating (17) with respect to and making use oi (15) for dci22 / 5 z , we obtaind d V  d VI d f  d V  3 d z [a->22i —— + a o z 12227 ^d zd*V  5 (1  W V-H 6 J d z d z  3 dz3 U(30)N o w  to eliminate we again differentiate twice eq. (29) \^iili respect to z ^^ nd equation (30) with respect to :: using/ d z “dz~ -  I d z ^ d z ^ . we finally obtainf  2 /942" + 4 D5Z (32)I ^^1122 1 ^^ ^^ 1112 ^^ "^ 1222 3  ^ ^^^2222i J:c)W ~3 dz^ d z d z '  ^ 3  dz'^f .r-.\ d z ‘dVdzFor a given form of potential V(Zy ^), the unknown constants D^ 's can be determined by rationalising the potential equation, subsequently, the determination of other coefficients a^ j can be obtained which would lead to the final form of the second constant of motion.t2 ^2 (/ rrj'i-n o d - v____4. du 2222 1222 d^Vd z d z  d z “ j d z ' [ <?: d z  , dz^We consider the potentialV{z, z) = -H tnz^V  + . (33)dz2. - a .V (  ^«1I22 '^“l 222 d ^ V 2■J‘^ 1222d z d z  J d z d z -d * v f^« lll2 dtt\ 122 d ^ v'lll2 ..J.dzdz^ d z dz  J dzdz~d-*ydzdz^^'^1112  ^ 1 d ' 222 d' 1^122, 1 <?d z d l ^ ^ 3  <?z’ d z d z ' 3^^«1112 5^a||22 d'’“« .i ,B d - vd z d z dz^ d i ^  J d eAjfljj2 da, d^V 1 d ^ vdz .'~dz' f d z ^ 3dzOn substituting the derivatives of this potential in eq. (32) and rationalising the resultant equation, we find that / = n = 1 and HI = 6,Dj^=D^ = D^ = D^ = 0. For this potential, the value of fljI and a ,, is obtained from eqs. ( 14) and ( 15) respectively«ii = |0 |Z '^  + |f ^ z B + 4 D ,z B ^  + j l \ z h + ^ D ^ z \  (34)«22 =■^^2^^ +4D2Z'?" + ^ ^ 9Z^Z+y02Z^ (35)and the value of obtained from eqs. (16) and (17) and putting D, = D ,^ £>g = as^12 = + 6D^yZ‘z^ -IDgZ^z^= 0. (31) ^his is a potential equation corresponding to the fourth- invariant which involves the potential derivatives and •^ nown coefficients through unknown constant D/s. On+ | d 2z ’z - ^ D , z"+D ,„Z ^We obtain the value of a^ using eqs. (18) and (19) as(36)flo = D^oZ* - ^ D g z ^  + ^ D 2 Z ^ z  + ^ D q Z ' z ^  +12Dfoz^z^5 7 0 Roshan Lai and S C Mishra- 5  1 , T o n  . 4  =  4  , 26 ^  _ 4 _ 4  , 112 „  _ j _ 3+ — + 38Z),oz“r  + j  + —  D jz 'z+  - -  A / z ^  +  12D,oz‘’z- + -  D . J z ' '  + D,oZ« - - -  D,z*3 3(37)T h e  u t i l i t y  o f  t h e  s e c o n d  c o n s t a n t s  o f  m o t i o n  i f  c o n s t r u c t e d  f o r  a  s y s t e m ,  h a s  b e e n  n o t i c e d  i n  r e c e n t  y e a r s  f r o m  s e v e r a l  p o i n t s  o f  v i e w ,  p a r t i c u l a r l y  s o l v i n g  s e v e r a l  n o n l i n e a r  p r o b l e m s  o f  p l a s m a  p h y s i c s  a n d  h y d r o d y n a m i c s ,  a n d  a l s o  i n  t h e  s t u d y  of  a  c l a s s i c a l  a n a l o g  o f  Y a n g - M i l l s  f i e l d  e q u a t i o n s  w i t h  r e f e r e n c e.  . , „ ,  . , , ,  t o  t h e  g e n e r a t i o n  o f  p o t e n t i a l s  f o r  b o t h  t i m e - i n d e p e n d e n t  anHl i m e - d e p e n d e m  b y  c h o o s i n g  t h e  s u i t a b l e  g a u g e s .  T h i s  invarian, a l s o  h e l p s  u s  t o  f i n d  a  g e n e r a l  c l a s s  o f  s o l i t o n  s o l u t i o n s  where s o m e  i n t e g r a l  c o n s t r a i n t s  o n  t h e  f i e l d s  e x i s t  w h i c h  a r e ,  in turn t h e  c o n s e q u e n c e s  o f  a p p r o p r i a t e  p h y s i c a l  c o n s e r v a t i o n  laws s u c h  a s  c h a r g e ,  i s o - s p i n  e tc  [ 1 1 ] ,  T h e  h i g h e r - o r d e r  invariants d e t e r m i n e  t h e  i n t e r n a l  s y m m e t r y  o f  a  p h y s i c a l  s y s t e m  particulaily i n  m o l e c u l a r  d y n a m i c s  1 1 2 | .the corresponding invariant becomes— D,(z V ’ + z"z‘ ) +12 + z^ z^+ I H a ( , - V  + Z V )  4 38D,„z-‘z^ + y  D ,z \-“-  D.z* + ^  DqZ’z + 2 A 5 ’z“ + ^D qzz’ + -  D r JAcknowledgmentsAulhors wish to thank Prof. R S Kaushal for his interest in tins work. We are also extremely grateful to the referee for his valuable suggestions.-2D,z^z^ ----X 1^4: X---+— DtiZ + - A r  z24(38)Here, we have studied the classical integrable systems using complex coordinates. We establish a general potential equation of fourth order. Rationalising the potential eq. (32), one can construct the second constant of motion. Interestingly considering a potential of the type V(z, z) = Iz^ + mz^z^ + we find the relation between the arbitrary constants I, m and n as I = n = 1, m = 6 which satisfy for the construction of the second constant of motion (38).The usefulness of the potential equation (32) would provide a large class of potentials whose second constant of motion can be constructed. One can also construct the second constant of motion using generalised coordinates [ 10 |.Kcl‘tTeni:«s[IJ E T Whittaker A Treatise on the A n a ly t ic a l D w a rn u s  o f k in u h ' a n d  R ig id  B o d ie s  (London : Cambridge University press] (I960)12J J Hietannta r/iv.v Rep 147 87 (1987) ; B Grammuiieos. B I)nn//i and A Ramani J Math. PIm 24 2289 (1983)13 J L S Hall P hysica D8 90 (1983) . S C Mishra Rep Math. Phys 32 217 (1992) , R S Kuushal and S C  Mishra P ra in ana -J  P liw  2b 109 (1986)[4] H Lewis and P G Leach J Math Pliys 23 237 (19K2i B Grammatico.s and H Dorrizzi J Math Phy.s. 25 2194 (I9W|15 J S C Mishra, R S Kaushal and K C Tnpathy J  Math. Phy.s 25 22P(1984) ; R S Kaushal and S C Mishra Phy.s Uit A102 7 (19K4)[6] R S Kaushal, S C  Mishra and K C  Tnpathy J Math Phv.s 26 420(1985)(7J M Lakshmanan and R Sahadevan Phy.s. Rep 224 1 (1993)[8] R S Kaushal Classical and Quantum Mechanics of Notucniinl Potentials (New Delhi . Narosa) (1998)[9] K J Whiteman Rep. Prog. Phys 40 1033 (1977)110] S C Mishra, Roshan Lai and Veena Mishra Proi 22nd Intcnmil Colloquium on Group Theoretical Methods in Physics (Hobari Australia) p360 (Boston, USA : International press) (1999)111] R Friedberg, T D Lee and A Sirlin Phys Rev D13 2739 (1976) [12] R Rajaraman and E J Weinberg Phys Rev Dll 2950 (1975)

Integrability of Two Coupled Oscillators using Complex Coordinates

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Integrability of Two Coupled Oscillators using Complex Coordinates

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