One class of solutions with two invariant relations for the problem of motion of the Kowalevski top in double constant field

One class of solutions with t w o in v arian t relations for the problem of motion of the Ko w alevski top in double constan t field ∗ Mikhail P . Kharla mov 05.1 1.200 2 Abstract Consider a rigid b o dy ha ving a fixed p oin t in a su p erp osition of tw o constan t f orce fields (fo r example, gra vitational and ma gnetic). In tro d u cing the condition of K o w alevski t yp e, O.I. Bogo ya vlensky (1984) has found the fi rst int egral generalizi ng that of Ko walevski and p oint ed out the in tegrable case with t wo inv arian t relations, which reduces to the 1st App elrot class w h en one of th e fields v anishes. Th e article presents a new case with tw o in v ariant relations in tegrable in Jacobi sense and ge neralizing the 2nd and 3rd classes of App elrot. In tro duct ion. Inv estigating the problems of classic al mec hanics with n degrees of freedom, one distinguishes the notions of Liouville and Jacobi in tegrability . In the first case w e ha v e n indep enden t first in tegrals in in v olution and the corresp onding Hamiltonian system of 2 n ordinary differen tial equations can b e deriv ed (in theory) to a ”simple” flow on an n -dimensional surface. The second case app ears when the equations hav e 2 n − 2 independent first in tegrals. Then the solution of the problem is reduced t o integrating of t w o differen tial equations ha ving Jacobi’s last m ultiplier (in con temp orary terms—to the system with inv arian t measure on a t w o-dimensional torus). Ob viously if n = 2 these t wo cases are the same. The formal equiv alence of the notions tak es place if among n inv olutive integrals one has n − 2 integrals generated by symmetries o f the p oten tial and kinetic energy . Then b y ignoring the corresp onding features of mo t io n the problem is reduced to a Hamiltonian system with tw o degrees of freedom of natural structure (the phase space is the co ordinate–v elo city space, the Lagrange function is quadratic w.r.t. v elo cit y comp onen ts). The t ypical in tegral manifolds of the reduced system are t w o-dimensional to r i and the t r a jectories are quasi-p erio dic. If w e really w an t to imagine the motion in all details, then, as a result of our three-dimensional w a y o f reasoning, w e prefer the Jacobi in tegrability . Indeed in the case of symmetries only the solutions of the reduced system are double-p erio dic. The motions of the initial system remain essen tially n -dimensional. Let us for brevit y call this situation a reducible problem. Jacobi in tegrability means that the t ra jectories of t he initial mec hanical system fill some tw o-dimensional surfaces, whic h can b e easily represen ted in R 3 . The pro jections o f suc h tra jectories onto the spaces ha ving ph ysical sense ( say , the ho dographs o f the angular v elo cit y or the traces of the v ertical v ector in the mov ing frame) can b e studied in reality . ∗ First published in Russian: Mekh. Tverd. T ela 32(20 02), pp. 32-3 8 1 The general in tegrability cases in the dynamics o f a r ig id b o dy b elong to the r educible prob- lems. They are conditionally in terpreted as tw o- dimensional b y ignoring the precession part of motion. But the tra jectories describing the evolution of the orien tatio n matrix in the general case still fill a three-dimensional tor us. T his is wh y the complete classification of the immo v able ho dographs of the angular v elo city and based on it complete geometrical sim ulation of motion seem to b e extremely complicated pro blems. Moreo v er, considering non-symmetric force fields w e lo ose hop e to find the p ossibilit y of an y natural reduction to t w o-dimensional configuration spaces . Th us, m uc h attention is pa yed to the cases of Jacobi integrabilit y of the full system of equa- tions of a rigid b o dy motion. These are the cases when the in tegral manifolds o f this system are t w o-dimensional. F or the reducible problems t he corresp onding solutions of the Euler–P oisson equations would b e singular p erio dic motions of elliptic or h yp erb olic type. Suc h solutions are called par t ia l. Giv en the bo dy c haracteristics, the partial solution ma y b e the set of closed orbits represen ting, in the reduced system, a family of t w o-p erio dic motions of the b o dy . Man y partia l solutions w ere found lately with the metho d of in v ariant relations [1]. T o describ e a partial solution (one-dimensional in v ar ia n t ma nif o ld in 6- dimensional space of the Euler–P oisson v a riables) one has to find, in addition to three known energy , area, and geometrical in tegrals, t w o indep enden t in v ariant relatio ns. If the f orce field has no symmetries, then the full system of Euler–P oisson equations contain 12 v ariables and admit 7 independen t integrals, namely , the energy inte gra l and 6 geometrical in tegrals. Therefore, to obtain the Jacobi in tegrabilit y one has to find three in v aria n t relatio ns. In this pap er w e presen t suc h a solution for the problem of the rig id b o dy motion in tw o constan t fields. 1. Main equations. Consider the problem of motion o f a rig id b o dy ab out a fixed p oin t in the p oten tial force field with a fo r ce function of the t yp e ( e 1 , α ) + ( e 2 , β ) . (1) Here ( · , · ) denote the scalar pro duct, the vec tors e 1 , e 2 are fixed in the b o dy , a nd α , β are im- mo v able in inertia l space. If β = 0 (or α × β = 0, whic h is the same) w e come to the classical problem of motion of a hea vy rigid b o dy . The force function ( 1) arises, for example, in the prob- lems of motion of a magnetized b o dy with a fixed magnetic momen tum in constan t gra vitational and magnetic fields, o r an electrically charged b o dy (with immo v able c harges in it) in constan t gra vitational a nd electric fields. Giv en that α × β 6 = 0, the orientation mat r ix o f the b o dy and acting forces are completely defined b y the comp onen ts, with respect to the mo ving frame, of the pair α , β . Therefore, the configuration space of the problem (the group of the orthogona l 3 × 3-matrices) cannot b e reduced to a space of less dimension, and this is a principal difference with the case of one field. The corresp onding equations I ˙ ω = I ω × ω + e 1 × α + e 2 × β , (2) ˙ α = α × ω , ˙ β = β × ω (3) can be considered as equations in R 9 with three geometrical in tegrals ( α , α ) = a 2 , ( β , β ) = b 2 , ( α , β ) = c. (4) Eqs. ( 2),(3) ha ve the energy in tegr a l H = 1 2 ( I ω , ω ) − ( e 1 , α ) − ( e 2 , β ) . In g eneral case, there is no linear in tegral of the area integral type. 2 Cho ose the principal axes of the inertia tensor I for the mo ving frame, then I = diag( A 1 , A 2 , A 3 ) . In the sequel, it is conv enien t t o consider the v ectors e 1 , e 2 orthonormal, and include all c harac- teristic m ultipliers into the parameters a, b, c of relations (4). T ak e p A 3 /u 0 as the time unit ( u 0 is some common unit of measuremen t fo r the comp onen ts of α , β ). F ormally it is equiv alen t to the choice A 3 = 1. W e can use u 0 to bring one of the constan t s a, b, c to 1 . How ev er, w e prefer to k eep notation (4) for some natural symmetry in the form ulas b elow. In t he analogue of t he Kow alevski case A 1 = A 2 = 2 A 3 , e 1 = (1 , 0 , 0) , e 2 = (0 , 1 , 0) . the Euler equations tak e the fo rm 2 ˙ ω 1 = ω 2 ω 3 + β 3 , 2 ˙ ω 2 = − ω 1 ω 3 − α 3 , ˙ ω 3 = α 2 − β 1 . (5) They a re closed by the Poiss on equations (3). O.I. Bogoy av lensky [2] show ed that Eqs. (3), (5) ha v e the first in tegral of t he Kow alevski ty p e K = J 2 1 + J 2 2 , (6) where J 1 = ω 2 1 − ω 2 2 + α 1 − β 2 , J 2 = 2 ω 1 ω 2 + α 2 + β 1 , and po in ted out that on the zero lev el of the in tegra l (6), J 1 = 0 , J 2 = 0 , (7) there exists a new pa r tial integral, namely , J 3 = ( ω 2 1 + ω 2 2 ) ω 3 + 2( ω 1 α 3 + ω 2 β 3 ) , the constant of whic h is a rbitrary . Th us, the system of in v arian t relations (7 ) defines the four-dimensional manifo ld M 4 ; this manifold is independen t of integration constan ts. The induced (not reduced) system on it has tw o first in tegrals H = h, J 3 = j (8) with arbitr a ry constan ts h, j . Therefore the initial system has a pa rtial case of Ja cobi in tegrability . The system thus obtained on M 4 can b e represen ted in t he Hamiltonian form [2 ], but M 4 do es not hav e a structure of a phase space of mec hanical system (co ordinate-ve lo cit y structure). The top ology of M 4 and of t wo-dimens ional in tegral manifolds (8) is studied in the w ork [3]. The Bogo ya vlensky solution generalized the classical case of N.B. D elone. 2. New solution with t wo in v arian t relations. Below the term ”deriv ative” will mean differen tiating functions of the v a r ia bles ω i , α j , β k in virtue of Eqs. (3), (5), i.e., the time-deriv ativ e along tra jectories. Note that the deriv ativ e, in the ab o v e sense, of any expression not con taining ω 3 , is linear in ω 3 . Supp ose that some function F is linear in ω 3 and its deriv a tiv e do es not dep end on ω 3 . Then there is a c ha nce that t he second deriv at iv e of F , b eing linear in ω 3 , can app ear to b e prop ort ional to F , th us closing the sequence of differen tiations a nd generating the in v ar ia n t relation in the definition of the w ork [1]. 3 F ollowing the idea of S. Ko w alevski [4] of using complex v ariables, in tro duce the c hange of v ariables ( i 2 = 1): x 1 = ( α 1 − β 2 ) + i ( α 2 + β 1 ) , x 2 = ( α 1 − β 2 ) − i ( α 2 + β 1 ) , y 1 = ( α 1 + β 2 ) + i ( α 2 − β 1 ) , y 2 = ( α 1 + β 2 ) − i ( α 2 − β 1 ) , z 1 = α 3 + iβ 3 , z 2 = α 3 − iβ 3 , w 1 = ω 1 + iω 2 , w 2 = ω 1 − iω 2 . (9) Denoting the deriv at iv e with resp ect to τ = it b y the prime, w e obtain from (3), (5) x ′ 1 = − x 1 w 3 + z 1 w 1 , x ′ 2 = x 2 w 3 − z 2 w 2 , y ′ 1 = − y 1 w 3 + z 2 w 1 , y ′ 2 = y 2 w 3 − z 1 w 2 , 2 z ′ 1 = x 1 w 2 − y 2 w 1 , 2 z ′ 2 = − x 2 w 1 + y 1 w 2 , 2 w ′ 1 = − ( w 1 w 3 + z 1 ) , 2 w ′ 2 = w 2 w 3 + z 2 , 2 ω ′ 3 = y 2 − y 1 . (10) Let θ = x 1 x 2 , Q 1 = x 2 z 1 w 1 + x 1 z 2 w 2 , Q 2 = x 2 z 1 w 1 − x 1 z 2 w 2 . (11) Then θ ′ = Q 2 , Q ′ 1 = 1 2 Q 2 ω 3 + ... , where ” ... ” stands for the terms not dep ending on ω 3 . Construct the combination θ m ω 3 − θ n Q 1 . In it s deriv ativ e, the co efficien t of ω 3 is equal to mθ m − 1 θ ′ − 1 2 θ n Q 2 = ( mθ m − 1 − 1 2 θ n ) Q 2 and v anishes if m = 1 2 , n = − 1 2 . Therefore, t he fo llowing function b ecomes ”suspicious” in the sense of inv ariant relation generating, F 1 = √ x 1 x 2 ω 3 − 1 √ x 1 x 2 ( x 2 z 1 w 1 + x 1 z 2 w 2 ) . Calculate its deriv ative in virtue o f (10), dF 1 dτ = 1 2 √ x 1 x 2 [ x 2 x 1 ( z 2 1 + x 1 y 2 )( w 2 1 + x 1 ) − x 1 x 2 ( z 2 2 + x 2 y 1 )( w 2 2 + x 2 )] . (12) Note that the geometrical in t egr a ls (4 ) imply z 2 1 + x 1 y 2 = ( a 2 − b 2 ) + 2 ic = c 1 = const , z 2 2 + x 2 y 1 = ( a 2 − b 2 ) − 2 ic = c 2 = const . In tro duce the follo wing notation U 1 = x 2 x 1 c 1 ( w 2 1 + x 1 ) , U 2 = x 1 x 2 c 2 ( w 2 2 + x 2 ) , U 2 = U 1 (13) and calculate t he deriv ativ es of (13), dU 1 dτ = c 1 x 2 1 ( w 2 1 + x 1 )[ x 1 x 2 ω 3 − ( x 2 z 1 w 1 + x 1 z 2 w 2 )] , dU 2 dτ = − c 2 x 2 2 ( w 2 2 + x 2 )[ x 1 x 2 ω 3 − ( x 2 z 1 w 1 + x 1 z 2 w 2 )] , 4 whence d dτ ( U 1 − U 2 ) = 1 √ x 1 x 2 [ c 1 x 2 1 ( w 2 1 + x 1 ) + c 2 x 2 2 ( w 2 2 + x 2 )]( √ x 1 x 2 ω 3 − x 2 z 1 w 1 + x 1 z 2 w 2 √ x 1 x 2 ) . (14) Denote F 2 = U 1 − U 2 and rewrite (12) and (14) in t he form dF 1 dτ = 1 2 √ x 1 x 2 F 2 , dF 2 dτ = 1 √ x 1 x 2 ( U 1 + U 2 ) F 1 . Hence t he system of relatio ns F 1 = 0 , F 2 = 0 (15) defines the in v ariant submanifold of the phase space of Eqs. (1 0). G iv en the geometrical iden t i- ties (4) dep ending only on the b o dy parameters and the force fields, this manifold has dimension 4. Denote it by N 4 . Note that the expressions θ a nd Q 1 in (11), (13) are real, while U 1 − U 2 is purely ima g inary . Hence Eqs. (15) can b e written in the form x 1 x 2 ω 3 − 2Re( x 2 z 1 w 1 ) = 0 , Im[ x 2 2 c 1 ( w 2 1 + x 1 )] = 0 . Substitution (9) leads to the inv aria n t relations expressed in the initial v ariables ( ξ 2 1 + ξ 2 2 ) ω 3 − 2[( ξ 1 ω 1 + ξ 2 ω 2 ) α 3 + ( ξ 2 ω 1 − ξ 1 ω 2 ) β 3 ] = 0 , 2[ c ( ξ 2 1 − ξ 2 2 ) − ( a 2 − b 2 ) ξ 1 ξ 2 ]( ω 2 1 − ω 2 2 + ξ 1 )+ +[( a 2 − b 2 )( ξ 2 1 − ξ 2 2 ) + 4 cξ 1 ξ 2 ](2 ω 1 ω 2 + ξ 2 ) = 0 . (16) Here ξ 1 = α 1 − β 2 , ξ 2 = α 2 + β 1 . The equations of motion restricted to the manifold N 4 defined by (16 ) ha v e t w o indep enden t first in tegrals H = ω 2 1 + ω 2 2 + 1 2 ω 2 3 − α 1 − β 2 ≡ h, K = ( ω 2 1 − ω 2 2 + α 1 − β 2 ) 2 + (2 ω 1 ω 2 + α 2 + β 1 ) 2 ≡ k . (17) The singular p oin ts of the initia l system (2), (3) in this problem corresp ond to the b o dy equi- libria; there exis ts o nly four suc h p oin ts. It follo ws from dynamics theorems that, if t he common lev el (17) do es not con tain any singular p oints and the g radien ts of H and K are linearly inde- p enden t, then each connected comp onen t of (17) is a tw o-dimensional torus and the tra jectories on it satisfy the differen tia l equations hav ing the last Jacobi multiplie r. Therefore, b y the time c hange, these tra jectories can b e transformed to quasi-p erio dic ones. As a result w e hav e p o inted out the t w o-parametric family (arbitrary h and k ) of t w o-p erio dic motions of the rigid b o dy in double constan t field under the conditions of Kow alevski t yp e. 3. The classical analogue. Supp ose that in the considered problem β = 0 (the second field v anishes). Then w e obtain the case of S. K ow alevski [4]. The r educed phase space of the Euler–P o isson v ariables has dimension 5. The in v ariant relatio ns (16) tak e the form ( α 2 1 + α 2 2 ) ω 3 − 2( α 1 ω 1 + α 2 ω 2 ) α 3 = 0 , (18) 2 α 1 α 2 ( ω 2 1 − ω 2 2 + α 1 ) − ( α 2 1 − α 2 2 )(2 ω 1 ω 2 + α 2 ) = 0 (19) 5 By means of the appropriate c hoice of the measuremen t unit u 0 mak e ( α , α ) = 1. W rite (19) in the form ω 2 1 − ω 2 2 + α 1 α 2 1 − α 2 2 = 2 ω 1 ω 2 + α 2 2 α 1 α 2 and substitute to the Ko w alevski in tegral to o bta in ω 2 1 − ω 2 2 + α 1 = α 2 1 − α 2 2 α 2 1 + α 2 2 √ k , 2 ω 1 ω 2 + α 2 = 2 α 1 α 2 α 2 1 + α 2 2 √ k . (20) Recall the classical area in tegr a l L = α 1 ω 1 + α 2 ω 2 + 1 2 α 3 ω 3 ≡ ℓ. Calculate the combination Ψ = 2 L 2 − H , Ψ = − 1 2 ( α 2 1 + α 2 2 )[ ω 3 − 2 α 1 ω 1 + α 2 ω 2 α 2 1 + α 2 2 ] 2 + + 1 α 2 1 + α 2 2 [( α 2 1 − α 2 2 )( ω 2 1 − ω 2 2 + α 1 ) + 2 α 1 α 2 (2 ω 1 ω 2 + α 2 )] . (21) Under the conditions (18) and (20) the last expression turns in to the follo wing 2 ℓ 2 − h = √ k . (22) Here the v a lue √ k is algebraic. Eq. (22) defines the 2nd and 3rd classes of motions b y the definition of G.G. Appelrot [5, 6]. Analyzing t he structure of (21) and (22), we conclude that the three-dimensional manifold (18), (19) is exactly the set of critical p oin ts of the com bined first in tegral ( 2 L 2 − H ) 2 − K . In particular, one of the classic in tegrals on this manifold b ecomes redundan t. The other tw o define the closed orbits. The se ar e, nat urally , the solutions on whic h one of the Kow alevski v ariables remains constant. Emphasize that in the full phase space including precession the corresponding motions are t w o-p erio dic for almost all integral constan ts under the condition defined b y (22). Therefore the family of motions (16) found in this pap er generalizes the so-called esp e cial ly r emarkable motions of the 2nd a nd 3rd Appelrot classes. 4. Remarks. After this pap er w as published in Russian journal Mekhanic a tver do go tela, 2002, N 32 , t he author and A.Y. Sa vushkin ha v e receiv ed the separation of v ariables for the class o f motions found here [7]. This separatio n provided the explicit solutions in elliptic Jacobi functions a nd ga v e the p ossibilit y to completely in ves tigate the phase top o logy of the case [8]. In connection with the in v estigation of the set of critical p oints and the arising bifurcation diagra ms of the momen tum mapping of the Ko w alevski top in double constan t field, it w as prov ed by the author in [9] that there exists only one more case of Jacobi integrabilit y , and for this new case the separation of v ariables w as found in [10] leading to h yp erelliptic quadratures. References [1] P .V. Kharlamo v, On in v ariant relations o f a system of differen tial equations, Mekh. tver d. tela , 1974, 6, pp. 15-24. (In Russian) 6 [2] O.I. Bo g o y av lensky , Euler equations on finite-dimension Lie alg ebras arising in ph ysical prob- lems, Commun. Math. Phys. , 1984, 95 , pp. 307-315. [3] D.B. Zotev, F omenk o- Ziesc hang inv arian t in the Bog o y av lenskyi case, R e gular and Cha otic Dynamics , 2000, 5 (4), pp. 437-45 8. [4] S. Kow alevski, Sur le probleme de la r o tation d’un cor ps solide autour d’un p oin t fixe, A cta Mathematic a , 1889, 2 , pp. 177-232. [5] G.G. Appelrot, Non-completely symmetric hea vy gyroscop es, In: Motion of a rigid b o dy ab out a fixed p oin t, Collection of pap ers in memory of S.V.Kov alevsk ay a. Mosco w-Leningrad, 1940, pp. 6 1-156. (In Russian) [6] A.F. Ipato v, Motion of the Kov alevsk ay a gyroscop e on the b o undar y of t he ultra-elliptic region, Sci. Bul letin o f Petr ozavo d sk Univ. , 1970, 18 (2), pp. 6-93. (In R ussian) [7] M.P . K ha r lamo v, A.Y. Savus hkin, Explicit in tegra t ion of one problem of motion of the generalized K ow alewski top, Me chanics R ese ar ch Co mmunic ations , 2005, 3 2 , pp. 547- 552. arXiv:0803.0857v1 [nlin.SI] [8] M.P . K ha r lamo v, A.Y. Sa vushkin, Separation of v ariables and integral manifolds in one prob- lem of motio n of generalized Kow alevski to p, Ukr ainian Mathematic a l Bul letin , 200 4 , 1 (4) , pp. 5 69-586. arXiv:0803.0882v1 [nlin.SI] [9] M.P . K ha r lamo v, Bifurcation diagrams of the Kow alevski top in tw o constan t fields, R e gular and Chaotic Dynamics , 2005 , 10 (4), pp. 381-398. arXiv:0803 .0893v1 [nlin.SI] [10] M.P . Kharlamo v, Separation o f v ar ia bles in the g eneralized 4th App elrot class, R e gular and Chaotic Dynamics , 2007, 12 (3), pp. 267- 280. arXiv:0803 .1024v1 [nlin.SI] 7