Explicit integration of one problem of motion of the generalized Kowalevski top

Explicit in tegration of one proble m of motion of the generalize d Ko w a levs ki top M.P . Kharla mov ∗ , A.Y. Sa vushkin Chair of Mathematic al Simulation, V olgo gr ad A c ademy of Public A dministr ation, Gagarin Str e et 8, V olgo gr ad, 400131, Russia Abstract In the problem of motion of the Ko w alevski top in a dou b le force field the 4-dimen- sional in v ariant submanifold of the ph ase space w as p ointe d out by M.P . Kharlamov (Mekh. Tv erd. T ela, 32 , 2002). W e sh o w that the equations of motion on this manifold can b e separated by the appr opriate c hange of v ariables, the new v ariables s 1 , s 2 b eing elliptic functions of time. The natural phase v ariables (comp onent s of the a ngular v elo cit y and the direction vecto rs of the forces w ith resp ect to the mo v able basis) are expressed via s 1 , s 2 explicitly i n ele mentary algebraic functions. Key wor ds: Ko w alevski top, double force field, separation of v ariables, elliptic functions, explicit solution MSC: 70E17, 70E40, 70H06 1 In tro duct ion The famous solution of S. Ko w alevski [1] for t he motio n of a hea vy rigid b o dy ab out a fixed p oin t w as generalized for the case of double constant fo rce field in [2,3]. This Hamiltonian system essen tially has three degrees of f r eedom, a nd hardly c an receiv e a clear geometrical or mec hanical desc ription of all types of motions. In v arian t subsystems, whic h can b e in terpreted as sy stems with t w o degrees of fr eedom, w ere found in [2,4]. Phase top olo g y of the case [2] w as studied in [5]. The presen t pap er deals with the case [4]. W e sho w that b y prop er c hoice o f lo cal co ordinat es it is p ossible to obtain separated differen tial equations of motion, and express all phase v ariables explicitly in terms o f t w o new v ariables, the latter b eing elliptic functions of time. ∗ Corresp ond ing author. Email addr ess: mharlamov @vags.ru (M.P . K harlamo v). Article pub lished in Mec hanics Researc h Comm unications 32 (2005) 547– 552 2 Equations of motion and kno wn first in tegrals Consider a heav y magnetized rigid b o dy with a fixed p oin t placed in gra vita- tional and magnetic constant force fields. Let ~ α, ~ β be the direction v ectors of the force fields and ~ e 1 , ~ e 2 b e the radius v ector of mass cen ter and the v ector of magnetic momen t of the b o dy . The scalar characteris tics (fo r example, the pro duct of w eight and distance from the mass cen ter to the fixed p oint) ma y b e included in the length of either vec tor of the associat ed pair. W e prefer to consider ~ e 1 , ~ e 2 to b e unit v ectors according to the mo del accepted in [3]. The Euler–P oinsot equations of motion hav e the form I d~ ω dt = I ~ ω × ~ ω + ~ e 1 × ~ α + ~ e 2 × ~ β , d ~ α dt = ~ α × ~ ω , d ~ β dt = ~ β × ~ ω , (1) where ~ ω is the angular velocity , I is the inertia tensor. All vec tor or tensor ob jects are referred to the basis attac hed to the bo dy . Supp ose that the bo dy is dynamically symmetric and denote b y π e the equa- torial plane of inertia ellipsoid. Cho osing measure units o ne can alw a ys mak e the momen t of inertia with resp ect to symmetry axis equal 1. Let I = diag { 2 , 2 , 1 } , ~ e 1 , ~ e 2 ∈ π e , ~ e 1 · ~ e 2 = 0 . (2) Then ~ e 1 , ~ e 2 ma y b e tak en as the first v ectors of the mov a ble ba sis. It is kno wn that under conditions (2) the s ystem (1) is completely in tegrable due to the existence of the first integrals [2] H = ω 2 1 + ω 2 2 + 1 2 ω 2 3 − ( α 1 + β 2 ) , K = ( ω 2 1 − ω 2 2 + α 1 − β 2 ) 2 + (2 ω 1 ω 2 + α 2 + β 1 ) 2 , (3) and a new integral G po in ted out in [3], whic h in case ~ β = 0 turns in to the square of the v ertical comp onen t of the angular momen tum. Belo w w e exclude the case | ~ α | = | ~ β | , ~ α · ~ β = 0, for whic h there exists a cyclic co ordinate [6], a nd the problem reduces to the system with t w o degrees o f freedom. First, w e sho w that without loss of generalit y one can alw a ys ta ke ~ α ⊥ ~ β . The conditions (2) hold if w e replace ~ e 1 , ~ e 2 , ~ α, ~ β with ~ e 1 ( θ ) , ~ e 2 ( θ ) , ~ α ( θ ) , ~ β ( θ ), 2 where ( ~ e 1 ( θ ) , ~ e 2 ( θ )) = ( ~ e 1 , ~ e 2 )Θ , ( ~ α ( θ ) , ~ β ( θ )) = ( ~ α, ~ β )Θ , Θ =  cos θ − sin θ sin θ cos θ  , θ = const . (4) Therefore, ~ e 1 ( θ ) , ~ e 2 ( θ ) ma y b e tak en as the first v ectors of a new mov able basis. A t the same time, substitution (4) preserv es the r otating moment ~ e 1 × ~ α + ~ e 2 × ~ β in Euler equations, and new vec tors ~ α ( θ ) , ~ β ( θ ) satisfy P oisson equations. F or the general case ( | ~ α | 2 − | ~ β | 2 ) 2 + ( ~ α · ~ β ) 2 6 = 0 tak e tan 2 θ = 2( ~ α, ~ β ) / ( | ~ α | 2 − | ~ β | 2 ) if | ~ α | 6 = | ~ β | , and cos 2 θ = 0 if | ~ α | = | ~ β | . Then ~ α ( θ ) ⊥ ~ β ( θ ). Th us, b elo w we consider the nat ura l restrictions (a lso called the geometrical in tegrals) in the form | ~ α | 2 = a 2 , | ~ β | 2 = b 2 , ~ α · ~ β = 0 . (5) The first in tegral [3] can then b e written in a simple wa y G = 1 4 ( ω 2 α + ω 2 β ) + 1 2 ω 3 ω γ − b 2 α 1 − a 2 β 2 , (6) where ω α = 2 ω 1 α 1 + 2 ω 2 α 2 + ω 3 α 3 , ω β = 2 ω 1 β 1 + 2 ω 2 β 2 + ω 3 β 3 , ω γ = 2 ω 1 ( α 2 β 3 − α 3 β 2 ) + 2 ω 2 ( α 3 β 1 − α 1 β 3 ) + ω 3 ( α 1 β 2 − α 2 β 1 ) . The conditions (5) mak e the phase space diffeomorphic to M 6 = R 3 × S O (3). In g eneral, the in tegral manifold J h,k ,g = { H = h, K = k , G = g } ⊂ M 6 consists o f 3- dimensional tori b earing quasip erio dic tra jectories dense on eac h torus for almo st all v alues of the in tegral constan ts. Therefore, a 4-dimensional in v ariant submanifold, on whic h the induced system has a structure of the in tegrable system with t w o degrees of freedom, m ust r eside in the set o f critical p oin ts of the global integral ma pping J = H × K × G : M 6 → R 3 . One submanifold of this t yp e w as found in [2]: M 4 = { K = 0 } ⊂ M 6 . The top ological structure of the induced syste m on M 4 w as studied in [5]. Belo w w e deal with the case [4], whic h generalizes the so-called marv ellous motions of the 2nd and 3rd classes o f Appelrot [7]. 3 3 New equations for t he in tegral ma nifolds Changing, if necessary , the or der of ve ctors in the mov a ble basis w e can a ssume that a > b . Denote p 2 = a 2 + b 2 , r 2 = a 2 − b 2 and consider the set N 4 ⊂ M 6 of critical p oints of the function F = (2 G − p 2 H ) 2 − r 4 K b elonging to the lev el F = 0. In order to obtain simple f orm ulae and to establish the corr esp o ndence with [4], in tro duce new phase v ariables ( i 2 = − 1) w 1 = ω 1 + iω 2 , w 2 = ¯ w 1 , w 3 = ω 3 , x 1 = ( α 1 − β 2 ) + i ( α 2 + β 1 ) , x 2 = ¯ x 1 , y 1 = ( α 1 + β 2 ) + i ( α 2 − β 1 ) , y 2 = ¯ y 1 , z 1 = α 3 + iβ 3 , z 2 = ¯ z 1 . (7) The equations (1) can b e then written as follo ws 2 w ′ 1 = − ( w 1 w 3 + z 1 ) , 2 w ′ 2 = w 2 w 3 + z 2 , 2 w ′ 3 = y 2 − y 1 , 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 . (8) Here strok e stands for d/d ( it ). The conditions (5) tak e the form z 2 1 + x 1 y 2 = r 2 , z 2 2 + x 2 y 1 = r 2 , x 1 x 2 + y 1 y 2 + 2 z 1 z 2 = 2 p 2 , (9) and F = 0 , ∇ 6 F = 0 give F 1 = 0 , F 2 = 0 , (10) where F 1 = √ x 1 x 2 w 3 − 1 √ x 1 x 2 ( x 2 z 1 w 1 + x 1 z 2 w 2 ) , F 2 = i 2 [ x 2 x 1 ( w 2 1 + x 1 ) − x 1 x 2 ( w 2 2 + x 2 )] . 4 The equations (10) corresp ond to the system of in v arian t relations found in [4]. This fact, in par ticular, rev eals the top ological nature of strictly analytical results [4]. Moreo v er, it straightforw a r dly prov es that N 4 is a subset of the phase space in v ariant under the flow of the dynamical system (1). It is easy t o se e that a lmo st ev erywhere on N 4 the system o f equations (10 ) has rank 2, so N 4 has a natural structure of 4-dimensional m anifold exce pt, ma yb e, for a thin subset defined by x 1 x 2 = 0. Fix the constan ts h, k , g of the in tegrals (3), (6) and in tro duce new constants m = 1 r 4 (2 g − p 2 h ) , ℓ = q 2 p 2 m 2 + 2 hm + 1 (the s ign of ℓ is arbitra ry). Then from the first integrals, conditions (9) and equations (10) w e obtain on N 4 w 2 1 = x 1 x 2 r 2 m − x 1 , w 2 2 = x 2 x 1 r 2 m − x 2 , w 3 = z 1 w 1 x 1 + z 2 w 2 x 2 , m ( x 2 + z 2 ) − ℓx + q r 4 m 2 − r 2 m ( x 1 + x 2 ) + x 2 = 0 , (11) where x 2 = x 1 x 2 > 0 , z 2 = z 1 z 2 > 0 . (12) The square ro ot in (11) equals w 1 w 2 , and therefore is non- nega t iv e. The equations (1 1) of the in tegral manifold J h,k ,g ⊂ N 4 sho w that in general case for giv en m, ℓ this manifold is tw o-dimensional. 4 Separation of v ariables W e no w in tro duce new v ariables in ( x, z )-plane s 1 = x 2 + z 2 + r 2 2 x , s 2 = x 2 + z 2 − r 2 2 x . (13) Calculating time deriv ative s fr o m (8 ) w e obtain s ′ 1 − s ′ 2 = r 2 2 x 2 [ z 2 s x 1 x 2 w 2 − z 1 s x 2 x 1 w 1 ] , s ′ 1 + s ′ 2 = r 2 2 x 2 [ z 1 s x 1 x 2 w 2 − z 2 s x 2 x 1 w 1 ] . (14) 5 Let Ψ( s 1 , s 2 ) = 4 ms 1 s 2 − 2 ℓ ( s 1 + s 2 ) + 1 m ( ℓ 2 − 1) , Φ( s ) = 4 ms 2 − 4 ℓs + 1 m ( ℓ 2 − 1) . Then, ha ving the ob vious iden tity Ψ 2 ( s 1 , s 2 ) − Φ( s 1 )Φ( s 2 ) = 4( s 1 − s 2 ) 2 , w e find from (9), (11), (12) x 1 = − r 2 2( s 1 − s 2 ) 2 [Ψ( s 1 , s 2 ) + q Φ( s 1 )Φ( s 2 )] , x 2 = − r 2 2( s 1 − s 2 ) 2 [Ψ( s 1 , s 2 ) − q Φ( s 1 )Φ( s 2 )] , y 1 = 2 (2 s 1 s 2 − p 2 ) − 2 q ( s 2 1 − a 2 )( s 2 2 − b 2 ) Ψ( s 1 , s 2 ) − q Φ( s 1 )Φ( s 2 ) , y 2 = 2 (2 s 1 s 2 − p 2 ) + 2 q ( s 2 1 − a 2 )( s 2 2 − b 2 ) Ψ( s 1 , s 2 ) + q Φ( s 1 )Φ( s 2 ) , z 1 = r s 1 − s 2 ( q s 2 1 − a 2 + q s 2 2 − b 2 ) , z 2 = r s 1 − s 2 ( q s 2 1 − a 2 − q s 2 2 − b 2 ) , w 1 = r q Φ( s 2 ) − q Φ( s 1 ) Ψ( s 1 , s 2 ) − q Φ( s 1 )Φ( s 2 ) , w 2 = r q Φ( s 2 ) + q Φ( s 1 ) Ψ( s 1 , s 2 ) + q Φ( s 1 )Φ( s 2 ) , w 3 = 1 s 1 − s 2 [ q ( s 2 2 − b 2 )Φ( s 1 ) − q ( s 2 1 − a 2 )Φ( s 2 )] . (15) Substitution of the latter expressions for x j , z j , w j ( j = 1 , 2) in to (14) allo ws to obtain the differen tial equations for s 1 , s 2 in the real form ds 1 dt = 1 2 q ( a 2 − s 2 1 )Φ( s 1 ) , ds 2 dt = 1 2 q ( b 2 − s 2 2 )Φ( s 2 ) . (16) Th us, s 1 , s 2 are easily f ound as elliptic functions of time. The explicit f o rm ulae for the phase v aria bles ω j , α j , β j ( j = 1 , 2 , 3) immediately follow from (7), (15). 6 5 The t yp es of solutions The conditions (9) define the global a r ea for the v ariables (13): | s 1 | > a, | s 2 | 6 b. So, the domain of oscillations (16) for the fixed v alues of m, ℓ is obtained f r om the inequalities Φ( s 1 ) 6 0 , Φ( s 2 ) > 0 . Bifurcations of solutions (16) with resp ect to the par a meters m, ℓ ta k e place when Φ( ± a ) = 0, or Φ( ± b ) = 0. It leads to a set of lines in ( m, ℓ )-plane ℓ = − 2 am ± 1 , ℓ = 2 am ± 1 , ℓ = − 2 bm ± 1 , ℓ = 2 bm ± 1 . Analyzing the ev olution of ro o ts of the p olynomial Φ( s ), we o btain all differen t t yp es of motions. The critical motions app ear in the c ases when o ne of the v a riables s 1 , s 2 re- mains constant, coinciding with t he double ro ot of p olynomial pro duct in the righ t part of the correspo nding equation (16). It ob viously leads to the motion of the b o dy of p endulum t yp e: either ω 1 = ω 3 ≡ 0, or ω 2 = ω 3 ≡ 0. An inte resting a sp ect of the case considered is that b oth mo v a ble and immo v- able ho dographs of the ang ula r ve lo cit y are explicitly found sim ultaneously without an y further integration. Actually , b ecause fo r almost all constan ts m, ℓ from the do main of existing of real solutions, ho dographs fill some tw o- dimensional surfaces densely , w e use the expressions (15 ) to obtain the para- metric equations of those surfaces, in whic h s 1 , s 2 are indep enden t para meters. Expressions for ~ α, ~ β (and, therefore, for ~ α × ~ β ) via s 1 , s 2 giv e the paramet- ric equations for the orien tation matrix. Con temp orary metho ds of computer graphics pro vide the p ossibilit y to construct a detailed and clear picture of motion a s rolling without slipping of one surface through the other: at an y momen t t hese surfaces hav e a common p o in t with zero absolute velocity and one common tangen t v ector, expressing the fact that absolute and relative deriv ativ es of the a ng ular v elo cit y coincide. References [1] S. Ko wa levski, Acta Math. 12 , 177-232 (18 89). [2] O.I. Bogo y a vlensky , Comm un. Math. Ph ys. 95 , 307-315 (1984). [3] A.I. Bob enk o, A.G. Reyman, M.A. Semeno v-Tian-Sh ansky , Comm un. Math. Ph ys. 122 , 321-354 (1989). 7 [4] M.P . Kharlamo v, Mekh. Tv erd. T ela, 32 , 32-38 (2002) . [5] D.B. Zotev, Regular and Chaotic Dyn amics, 5 (4), 437 -458 (200 0). [6] H.M. Y ehia, Mec h. Res. Commun. 13 (3), 169- 172 (1986 ). [7] G.G. App elrot, Non-completely s ymmetric heavy gyroscop es, in ”Motion of the rigid b o dy ab out a fixed p oin t”, 1940, Mosco w-Leningrad, 61-156 . 8