Recursive Generation of Isochronous Hamiltonian Systems

June 16, 2018 10:35 WSPC/INSTRUCTION F ILE iso˙jnmp Ar ticle Journal of Nonlinear Mathematical Ph ysics, V ol. 1 , No. 1 (2 009) 1–6 c  M. LAKSHMANAN RECURSIVE GENERA TION OF ISOCHRONOUS HAMIL TONIAN SYSTEMS V. K. CHANDRASEKAR † , A. DUR GA DEVI ∗ , M. LAKSHMANAN ‡ Centr e for Nonli ne ar Dynamics, Scho ol of Physics, Bhar athidasan University , Tiruchir app al li - 620 024, I ndia. † sekar@cnld.b du. ac.in ∗ dur ga@cnld.b du.ac.in ‡ lakshman@cnld.b du.ac.in W e prop ose a simple proced ure to identify the collectiv e coordinate Q which is u sed to generate t he isochronous Hamiltonian. The new iso chronous Hamiltonian generates more and more isochronous oscillators , recursively . In recen t years considerab le in terest has b een sh own to identify and classify iso c hronous systems. In this direction Caloge ro an d his cow ork ers ha ve in tro du ced a n u mb er of sys- tematic pro cedures to generate isochronous oscilla tor systems [1–7]. In a differen t d irec- tion the existence of amplitude ind ep end ent frequency of nonlinear oscillators ha ve b een iden tified using nonlo cal tr ansformations [8, 9]. Recen tly Calogero and Leyvraz [6, 7] p ro- p osed a new p ow erful tec h nique to generate isoc hronous Hamiltonian systems. In th is tec h- nique they ha ve shown that the real autonomous Hamiltonian H ( p, q ) can b e transformed to an Ω-mo dified Hamiltonian, that is, H (1) = 1 2 ( H ( p, q ) 2 + Ω 2 Q ( p, q ) 2 ), wh ic h has the iso c hr onous prop ert y . Here H b eha ves as th e new momen tum and Q is th e ca nonically con- jugate/col lectiv e co ordinate conju gate to the Hamiltonian, su c h that the Poisson brac ket { H,Q } =1. Ω is an arbitrary constan t. Due to the nature of the Ω-mo dified Hamiltonian system, no w the new momentum H and co ordinate Q ev olv e p erio dically with p erio d T = 2 π / Ω, and so the m omen tum p and co ord inate q also evo lve p erio dically with the same p erio d. In [10], the authors ha ve shown the int er esting connection b et we en symplectic rectification and iso chronous Hamiltonia n systems. In this b rief communicatio n, we pr op ose a simple pro cedure to iden tify the collecti v e co ordinate Q whic h is u sed to generate the iso chronous Hamiltonian. W e also p oin t out the further in teresting p ossibilit y to generate r ecursiv ely more iso chronous oscilla tors fr om the newly constructed iso chronous Hamiltonian. W e also illustrate this p ossibilit y with an example. T o iden tify Q w e start with the follo wing theorem: Theorem 1. If H = H ( p, q ) is the Hamiltonia n of a given system suc h that it c an b e inverte d to find a single value d q or p in terms of the other variable and H explicitly, then the system admits at le ast one inte gr al of the form I = − t + Q ( p, q ) , and Q ( p, q ) i s a 1 June 16, 2018 10:35 WSPC/INSTRUCTION FILE iso˙jnmp 2 V. K. Chandr asekar, A. Dur ga Devi and M. L akshmanan c ol le ctive c o or dinate c onjugate to the Hamiltonian in an appr opriate phase sp ac e (avoiding multivalue dness and singularities). Pro of. The Hamilton equ ations of the Hamiltonian H ( p, q ) are ˙ q = ∂ H ∂ p = f 1 ( p, q ) , ˙ p = − ∂ H ∂ q = f 2 ( p, q ) . (0.1) No w inv erting the Hamiltonian H ( p, q ) in terms of p or q and substituting the resultant expression in to the right hand side of the ˙ q or ˙ p equation we get ˙ q = f 1 ( p, q ) = f 3 ( q , H ) , or ˙ p = f 2 ( p, q ) = f 4 ( p, H ) . (0.2) In tegrating the ab o ve equation we get I + t = Z dq f 3 ( q , H ) = Q 1 ( p, q ) , or I + t = Z dp f 4 ( p, H ) = Q 2 ( p, q ) (0.3) where I is the in tegration constan t and Q 1 and Q 2 are tw o of the p ossible collectiv e co or- dinates (where the ph ase space c hosen su c h that the co ord inates are single v alued and non-singular). The latter fact can b e easily pr o ve d by noting that the tota l differentia tion of an y one of the Q ( p, q ) yields on using (0.1), dQ dt = ˙ q ∂ Q ∂ q + ˙ p ∂ Q ∂ p = { H , Q } = 1 . (0.4) Th us Q 1 or Q 2 is canonically conjugate to H a nd can serve as the requ ired collectiv e co ordinate. No w considering the ab o ve Hamiltonian H as a new momentum and Q as a new collectiv e co ordinate and sub stituting these into the Ω-mo dified iso c hronous Hamiltonian H (1) giv en b y C alogero and Leyvraz [6, 7], H (1) = 1 2 ( H ( p, q ) 2 + Ω 2 Q ( p, q ) 2 ) , (0.5) one can sho w that th e dynamical system in p and q is ind eed iso c h ronous. T his has b een pro ved in [6, 7]. Example:1 Let us consider H = pq , then Q = log q . Then the Ω-mo dified Hamiltonian H (1) = 1 2 ( p 2 q 2 + Ω 2 (log q ) 2 ) is iso chronous. Next we n ote that the collectiv e co ordinate Q (1) = 1 Ω tan − 1  Ω Q H  , confined to the principal branc h of the right h an d side, is conju gate to the Hamiltonian H (1) whic h is obtained using Theorem 1, an d Eq.(0.3). Then w e n ote the follo win g theorem. Theorem 2. L et H (1) − a (1) and Q (1) = 1 Ω tan − 1  Ω Q H  , wher e the latter is c onfine d to the princip al br anch of the ar ctan function, b e the new momentum and the c ol le ctive c o or dinates, r esp e ctively, then the Ω (1) - mo difie d Hamiltonian H (2) = 1 2 [( H (1) − a (1) ) 2 + Ω 2 (1) Q 2 (1) ] also has iso chr onous dynamics, wher e a (1) and Ω (1) ar e su itable arbitr ary p ositive system p ar ameters. June 16, 2018 10:35 WSPC/INSTRUCTION FILE iso˙jnmp R e cursive Gener ation of Iso chr onous Hamiltonian Systems 3 Pro of. F rom the nature of H (2) , the solutions for H (1) − a (1) and Q (1) are written as H (1) − a (1) = A (1) cos[Ω (1) t + δ (1) ] , Q (1) = A (1) Ω (1) sin[Ω (1) t + δ (1) ] , (0.6) where A (1) and δ (1) are arbitrary constants. Sub stituting equation (0.6) into the exp ressions for H (1) (equation (0.5)) and Q (1) and in verting we get the solutions for H and Q in the forms H = q 2 a (1) + 2 A (1) cos(Ω (1) t + δ (1) ) cos[ Ω A (1) Ω (1) sin(Ω (1) t + δ (1) )] Q = 1 Ω q 2 a (1) + 2 A (1) cos(Ω (1) t + δ (1) ) sin [ Ω A (1) Ω (1) sin(Ω (1) t + δ (1) )] . (0.7) These solutions ev olv e p eriod ically with p erio d T = 2 π / Ω (1) , f or a (1) > | A (1) | so that the quan tit y inside the s quare ro ot r emain p ositiv e for all times. Th e expressions for p and q can b e obtained u p on in verting H and Q . No w, as we already kno w that H and Q ev olv e p erio d ically , it is ob vious that p and q must al so evo lv e p erio dically with the same p erio d , namely T (1) = 2 π / Ω (1) , but in general d ifferen t from T (0) = 2 π / Ω. F rom the ab o ve Theorem 2, one can ident ify recurs ively the Ω ( i ) mo dified Hamiltonian from H ( i +1) = 1 2 [( H ( i ) − a ( i ) ) 2 + Ω 2 ( i ) Q 2 ( i ) ], i = 0 , 1 , 2 ......n , where H (0) = H , Q (0) = Q , Ω (0) = Ω, and a (0) = 0. Here H ( i ) − a ( i ) and Q ( i ) are the new momen tu m and its corre- sp ond ing canonically conjugate/coll ectiv e coordin ate, resp ective ly . All the ab ov e systems yield p erio dic solutions with p erio d T ( i ) = 2 π / Ω ( i ) and they can b e deduced u s ing th e relations, H ( i ) = a ( i ) + q 2 H ( i +1) cos[Ω ( i ) Q ( i +1) ] , Q ( i ) = 1 Ω ( i ) ( q 2 H ( i +1) sin[Ω ( i ) Q ( i +1) )] , i = 0 , 1 , ...n and also H ( n ) = a ( n ) + A ( n ) cos[Ω ( n ) t + δ ( n ) ] , Q ( n ) = A ( n ) Ω ( n ) sin[Ω ( n ) t + δ ( n ) ] . (0.8) Here A ( n ) and δ ( n ) are arbitrary constan ts, Ω ( i ) and a ( i ) , i = 0 , 1 , 2 ......n , are system (arbitrary) parameters. Note that for the solution to remain real, on e has to imp ose the cond ition a i − 1 > p 2( a i + X i ) = X i − 1 and a ( n ) > | A n | = X n , i = 1 , 2 , ...n . Usin g the ab o ve p erio dic s olutions, one can easily see that the canonical v ariables p and q also evo lve p erio d ically with p erio d T n = 2 π / Ω ( n ) . W e now illustrate the ab ov e recurs iv e pro cedure with an example. Example:2 Let us consider the Hamiltonian H = p n g ( q ), where g ( q ) is an arbitrary f u nction of q , for whic h the Hamilto n ’s equations can b e w r itten as ˙ q = ∂ H ∂ p = np n − 1 g ( q ) , ˙ p = − ∂ H ∂ q = − p n g ′ ( q ) . (0.9) June 16, 2018 10:35 WSPC/INSTRUCTION FILE iso˙jnmp 4 V. K. Chandr asekar, A. Dur ga Devi and M. L akshmanan Here g ′ ( q ) = dg dq . Note that the in tegration of the equation dp/dq = − pg ′ ( q ) / ( ng ( q )) (vide equation (0.9)) gives th e in tegration co nstan t I = p n g ( q ) w hic h is nothin g but the Hamilto- nian H . F rom the Hamilto nian we get p = ( H /g ( q )) 1 n and substituting th is expression in to the ˙ q equation, w e obtain ˙ q = n ( H /g ( q )) n − 1 n g ( q ) = n ( H ) n − 1 n g ( q ) 1 n . (0.10) In tegrating (0.10) w e get I + t = g ( q ) 1 − n n np n − 1 Z g ( q ) − 1 n dq . (0.11) No w th e collec tiv e co ord inate Q ( p, q ) is of the form Q ( p, q ) = g ( q ) 1 − n n np n − 1 Z g ( q ) − 1 n dq , (0.12) whic h is conju gate to the Hamiltonia n, that is { H , Q } = 1. This is in conformit y with Theorem 1. F or simplicit y let us co nsider the case n = 1 and g ( q ) = q . I n this case the n ew m omen- tum and the collect iv e co ord inate are written as H = pq and Q = log( q ). Su b stituting these in to th e Ω - mo difi ed Hamilto nian H (1) giv en in (0.5) we ge t H (1) = 1 2 (( pq ) 2 + Ω 2 log( q ) 2 ) . (0. 13) Using th e pro cedu re giv en in [6, 7] or follo wing our pr o cedure given ab o v e the solution for p and q now b ecome p ( t ) = A cos(Ω t + δ ) e − A Ω sin(Ω t + δ ) , q ( t ) = e A Ω sin(Ω t + δ ) (0.14) whic h are p erio d ic with p erio d T = 2 π / Ω, so the system for p and q is iso c h ronous. No w consider the Ω - mod ified Hamiltonian H (1) − a (1) as the new momen tum and Q (1) = 1 Ω tan − 1  Ω Q H  as the collec tive coordin ate in the Ω (1) - mo difi ed Hamiltonian H (2) , that is, H (2) = 1 2  1 2 (( pq ) 2 + Ω 2 log( q ) 2 ) − a (1)  2 + Ω 2 (1)  1 Ω tan − 1  Ω log( q ) pq  2  . (0.15) No w we can obtain the solutions f or p and q as p ( t ) = q 2 a (1) + 2 A (1) cos[Ω (1) t + δ (1) ] cos [ Ω A (1) Ω (1) sin(Ω (1) t + δ (1) )] /q ( t ) , q ( t ) = e 1 Ω √ 2 a (1) +2 A (1) cos[Ω (1) t + δ (1) ] sin[ Ω A (1) Ω (1) sin(Ω (1) t + δ (1) )] . (0.16) Cho osing the arbitrary parameters su c h that a 1 > | A 1 | , the system for p and q is iso c h ronous since the solution (0.16) is p erio dic with p erio d T = 2 π / Ω (1) . This is in conform it y with Theorem 2. June 16, 2018 10:35 WSPC/INSTRUCTION FILE iso˙jnmp R e cursive Gener ation of Iso chr onous Hamiltonian Systems 5 Then we m a y extended the ab ov e analysis to the Ω (2) mo dified Hamiltonian. In th is case H (2) − a (2) can b e tak en as the momen tum and Q (2) = 1 Ω (1) tan − 1  Ω (1) Q (1) H (1) − a (1)  as the conjugate co ord inate and therefore H (3) = 1 2  Ω 2 (2) Ω 2 (1) tan − 1 [ 2Ω (1) tan − 1 [ Ω log ( q ) pq ] Ω(( pq ) 2 +Ω 2 log( q ) 2 − 2 a (1) ) ] 2 + 1 4  Ω 2 (1) Ω 2 tan − 1 [ Ω log( q ) pq ] 2 − 2 a (2) + 1 4 (( pq ) 2 + Ω 2 log( q ) 2 − 2 a (1) ) 2  2  . (0.17) The solutions for p and q can no w b e written as p ( t ) =  √ 2 q a (1) + f (1) cos( f (2) ) cos( f (3) )  /q ( t ) , q ( t ) = e 1 Ω √ 2 √ a (1) + f (1) cos( f (2) ) sin ( f (3) ) , (0.18) where f (1) =  2( a (2) + A (2) cos[Ω (2) t + δ (2) ])  1 / 2 , f (2) = A (2) Ω (1) sin[Ω (2) t + δ (2) ] / (Ω (2) ) and f (3) = (Ω / Ω (1) ) f (1) sin[ f (2) ]. W e assume h ere again that a 2 > | A 2 | and a 1 > p 2( a 2 + A 2 ) so that p and q are real. Here also the canonical v ariables p and q are p erio dic with p erio d T = 2 π / Ω (2) confirming the iso chronous c haracter of the dynamics. F ollo wing a similar analysis, one can generate more and more iso c hr on ou s Hamilto nians. T o conclud e, we ha ve prop osed a simple p ro cedur e to id entify the collectiv e co ordinate Q w hic h is co njugate to the giv en Hamiltonian H in order to generate isoc hronous systems. Using th e kn o wn Hamiltonian H an d collectiv e co ord inate Q , we hav e pro v ed the p ossibilit y of generating more and more iso c hronous oscillator s y s tems recursively . The work is sup p orted by a Department of S cience and T ec hn ology (DST), Go vernmen t of Ind ia, Ramanna F ello wship p rogram and a DST–IRHP A researc h pro ject, Go v ernment of India. References [1] Ca logero F 2008 Iso chronous Systems (Oxford: Oxford Univ ers ity P ress) [2] Ca logero F 1997 A Class of integrable Hamiltonian systems whose solutions a r e (perha ps) all completely p erio dic, J. Math. Phys: 38 5711-9 [3] Ca logero F and Leyv r az F 2006 Is o ch ronous and partially-iso ch ronous Hamiltonian systems are not rare, J. Math. Phys: 47 042 901:1- 23 [4] Ca logero F and Ley v raz F 200 6 On a c lass of Hamiltonians with (Class ical) iso chronous motions and (Quantal) equispaced spectra, J. Ph ys. A: Math. The or. 39 11 8 03-1 1824 [5] Ca logero F and Leyvra z F 2007 On a new technique to man ufacture iso chronous Hamiltonian systems: Clas sical and Q uantal trea tment s, J. Nonline ar Math. Phys. 14 612-636 [6] Ca logero F and Leyvra z F 2007 General technique to pro duce iso chronous Hamiltonian, J. Phys. A: Math. The or. 40 1293 1-44 [7] Ca logero F a nd Leyvraz F 2008 Examples of iso chronous Hamiltonian: classic a l and qua nt al treatments, J. Phys. A: Math. The or. 41 175202 [8] Cha ndr asek ar V.K, Senthilv ela n M, a nd Lakshmana n M Unusual Lienard- t yp e nonlinear oscil- lator, Phys. R ev. E 72 06 6 203 June 16, 2018 10:35 WSPC/INSTRUCTION FILE iso˙jnmp 6 V. K. Chandr asekar, A. Dur ga Devi and M. L akshmanan [9] Cha ndr asek ar V.K, Senthilv ela n M , Anjan Kundu and Lak s hmanan M A nonlo cal connection betw een certa in linear a nd nonlinear ordinary differential eq uation/os c illa tors, J . Phys. A: Math. The or. 39 9743 -9754 [10] Guha P a nd Choudhury A G 2 009 Symplectic rectification and iso chronous Hamiltonian sys- tems, J . Phys. A: Math. The or. 42 1 9200 1