Separation of variables in one partial integrable case of Goryachev

Separation of v ariables in one partial in tegrable case of Gory ac hev P .E. Ry ab o v 19.12.2010 Abstract W e s ho w that the equations of motion in one partial in tegrable case of Goryac hev in the rigid b o dy dynamics can b e separated b y the appropriate chang e of v ariables, the new v ariables x, y b eing hyperelliptic functions of time. The natural phase v ari- ables (comp onen ts of co ordinates and momen ta) are expresse d v ia x, y explicitly in elemen tary algebraic f unc tions. 1 In tro duction The equations of motion of a rigid b o dy abo ut a fixed p oin t in the integrable case found b y D.N. Goryac hev hav e the form ˙ s 1 = s 2 s 3 + cr 2 r 3 − b r 2 r 3 3 , ˙ r 1 = 2 s 3 r 2 − s 2 r 3 , ˙ s 2 = − s 1 s 3 + cr 1 r 3 + b r 1 r 2 3 , ˙ r 2 = − 2 s 3 r 1 + s 1 r 3 , ˙ s 3 = − 2 cr 1 r 2 , ˙ r 3 = s 2 r 1 − s 1 r 2 . (1) This system can b e written in the Hamiltonian form on the space R 6 ( s , r ) with the Poiss on brac k ets { s i , s j } = − ε ij k s k , { s i , r j } = − ε ij k r k , { r i , r j } = 0 , 1 6 i, j, k 6 3 , ε = 1 2 ( i − j )( j − k )( k − i ) and the Hamilton function H = 1 2 ( s 2 1 + s 2 2 + 2 s 2 3 ) + 1 2 [ c ( r 2 1 − r 2 2 ) + b r 2 3 ] . (2) Here s is the kinetic momen t of the b o dy , r is iden tified with the unit v ector. The force dep end s o n t w o ar bitra r y parameters c and b . The g eome trical inte gra l and the area in tegral of the system (1) Γ = r 2 1 + r 2 2 + r 2 3 , L = s 1 r 1 + s 2 r 2 + s 3 r 3 (3) are the Casimir functions. Hence the v ector field (1) restricted to the 4- manifold P 4 = { ( s , r ) ∈ R 6 : Γ = 1 , L = 0 } , 1 is the Hamiltonian system with t wo degrees of freedom. T o b e Liouville integrable it needs, in addition to the Hamiltonian H , o ne more indep enden t integral. Consider the functions F =  s 2 1 − s 2 2 + cr 2 3 − b ( r 2 1 − r 2 2 ) r 2 3  2 + 4  s 1 s 2 − br 1 r 2 r 2 3  2 (D. N. Gory ache v [2]) and K =  s 2 1 + s 2 2 + b r 2 3  2 + 2 cr 2 3 ( s 2 1 − s 2 2 ) + c 2 r 4 3 (A. V. T sigano v [5]) . (4) Eac h of them is t he additional in tegral of (1) o n the symplectic manifold P 4 . Acc or ding to the Liouville–Arnold theorem an y regular lev el of the first in tegrals { ( s , r ) ∈ P 4 : H = h, K = k } (5) is a union of t w o - dime nsional tori b earing quasi-p eriodic tra jectories . The in tegral K is not new b ecause of the functional relation on P 4 K = F + 4 bH − b 2 . Note that in tro ducing some terms linear in s and r in t he Hamiltonian function also leads to the in tegrable system on P 4 . The additional in tegral in the most general form for this case is p oin ted out in [6]. F or suc h generalization no explicit in tegration is found y et. F or the system (1) and b = 0 the separation of v ariables is found b y S.A. Chaplygin [1] leading to elliptic quadratures. In the w or k [5] using t he ideas of bi-Hamiltonian approach some v a r ia ble s of separation a r e suggested. Nev ertheless, the corresp onding equations of Ab el–Jacobi type giv en in [5] are not written in the explicit f o rm. The dep endencie s of the phase v ariables on the prop osed separation v ar ia bles are not found either. Therefore the results of [5] are still far f r o m b eing complete. In this work we presen t the explicit separation of v ariables in the G ory ac hev case. This solution do es not nee d to in volv e an y far-g oing mathematical theories and is based on the pure ob vious calculation follo wing the w orks of S.A. Chaplygin. W e obtain the simple standard form of the separated equations and express all phase v ariables via the separation v ariables x and y in algebraic w a y . 2 The separation of v ariables Note that b y the appropriate c hoice of the measure ment units and the directions of the mo ving axes one can alw ays get c = 1 . Let u = s 2 1 + s 2 2 + b r 2 3 , z = r 2 3 . (6) 2 These v ariables are in tro duced similar to the w ork [1] where for the case b = 0 they lead to the separation. F ro m (2), (3) and (4 ) we find the parametric equations of the in tegral manifold (5) s 2 1 = ( k − 2 b ) − ( u − z ) 2 4 z , s 2 2 = ( u + z ) 2 − ( k + 2 b ) 4 z , s 2 3 = B + √ B 2 − AC 2 4 A , 2 r 2 1 = 2 h − u − 2 s 2 3 + 1 − z , 2 r 2 2 = 1 − z − 2 h + u + 2 s 2 3 , r 2 3 = z . (7) Here A = b 2 − 2 buz + z 2 k , B = − 2 z 2 k u + 2 z 2 k h + 2 z 2 u 2 h + 3 bu 2 z + k uz + k z b − 8 uz hb − − z 2 b + z 3 b + z 3 u − 2 z 4 h − k b + u 2 b − u 3 z + 4 b 2 h − 2 b 2 u, C = − k + u 2 + z 2 − 4 uz h + k z + 4 bh − 2 bu + u 2 z − z 3 . F or the existing of real solutions it is necessary to sim ultaneously ha ve k > 4 bh − b 2 , k > 2 b. (8) In view of (8), transform the v alue B 2 − AC 2 : D ( u ) = B 2 − AC 2 = z 2 ( z 1 − u )( u − z 2 )( u − z 3 )( u − z 4 )( z 5 − u )( u − z 6 ) . Here the ro ots z k of the p olynomial D ( u ) are defined as z 1 , 2 = z ± √ k − 2 b, z 3 , 4 = − z ± √ k + 2 b, z 5 , 6 = (1 − z )( b ± √ b 2 + k − 4 bh ) + 2 z h. The discriminan t set of D ( u ) is k = ± 2 b, k = 4 hb − b 2 , k = (2 h ∓ 1) 2 ± 2 b. (9) The a ccessible region on the ( z , u ) - plane is then b ounded b y the segmen ts of the straigh t lines u = z k . The quadrangle structure of the motion p ossibilit y regions usually leads to some exact separation of v ariables (see [3], [4 ]). Consider the expression of the co efficien t A . By virtue of (4) a nd (6) w e ha v e A = z 2 [( s 2 − r 3 ) 2 + s 2 1 ][ s 2 1 + ( s 2 + r 3 ) 2 ] > 0 Hence k z 2 − 2 buz + b 2 = ξ 2 . (10) This equation defines in space R 3 ( u, z , ξ ) the second order surface whic h is a one-she et hyp e rb oloid and therefore has tw o families of rectilinear generators. T ak e the parameters 3 of these families ( x, y ) fo r new v aria bles. The parametric equations o f the h yp erbolo id (10) tak e the fo r m z = 2 b x + y , u = xy + k x + y , A = ξ 2 = b 2 ( x − y ) 2 ( x + y ) 2 . (11) Denote the p olynomials w 1 = 2 b + k − x 2 , w 2 = 2 b − k + x 2 , w 3 = 4 bh − k − 2 bx + x 2 , w 4 = 2 b + k − y 2 , w 5 = − 2 b + k − y 2 , w 6 = − 4 bh + k + 2 by − y 2 . (12) A ccording to the ab ov e notation B + ξ C = 2 bw 1 w 2 w 6 ( x + y ) 4 , B − ξ C = 2 bw 3 w 4 w 5 ( x + y ) 4 . (13) W e see then that from (7) and (11)–(13) we easily obtain the algebraic expression for s 3 : s 3 = √ w 1 w 2 w 6 + √ w 3 w 4 w 5 2 √ b ( x 2 − y 2 ) . In a similar w a y w e can simpli fy the square ro ots defi ning other phase v a riables in (7). Finally we come to the follo wing statemen t. Theorem 1. On the c ommon level o f the i nte gr als ( 5 ) al l phase variables ( s , r ) ar e alge- br a ic al ly expr esse d in terms of x and y in the form s 1 = − √ w 2 w 5 2 √ 2 b √ x + y , s 2 = √ w 1 w 4 2 √ 2 b √ x + y , s 3 = √ w 1 w 2 w 6 + √ w 3 w 4 w 5 2 √ b ( x 2 − y 2 ) , r 1 = √ w 2 w 3 w 4 − √ w 1 w 5 w 6 2 √ b ( x 2 − y 2 ) , r 2 = − √ w 2 w 4 w 6 + √ w 1 w 3 w 5 2 √ b ( x 2 − y 2 ) , r 3 = s 2 b x + y . (14) T o complete the separation it is necessary to deriv e the differen tial equations for the v ariables x and y . Theorem 2. The derivatives of the intr o duc e d v ariables x , y by virtue of the system ( 1 ) satisfy the e quations ( x − y ) dx dt = − 1 √ b p − W ( x ) , ( x − y ) dy dt = 1 √ b p − W ( y ) , (15) wher e W ( s ) = ( s 2 − k − 2 b )( s 2 − k + 2 b )( s 2 − 2 bs + 4 bh − k ) . Pr o of. On one hand we ha v e ˙ u = { u, H } = 2 r 3 ( s 2 r 1 + s 1 r 2 ) , ˙ z = { z , H } = 2 r 3 ( s 2 r 1 − s 1 r 2 ) , (16) 4 and on the other hand w e obtain ˙ u = ˙ x ( y 2 − k ) + ˙ y ( x 2 − k ) ( x + y ) 2 , ˙ z = − 2 b ( ˙ x + ˙ y ) ( x + y ) 2 . (17) F ro m (16) a nd (17) w e find the express ions for ˙ x and ˙ y : ˙ x = − r 3 ( x + y ) b ( x − y ) [( s 2 r 1 − s 1 r 2 )( x 2 − k ) + 2 b ( s 2 r 1 + s 1 r 2 )] , ˙ y = r 3 ( x + y ) b ( x − y ) [( s 2 r 1 − s 1 r 2 )( y 2 − k ) + 2 b ( s 2 r 1 + s 1 r 2 )] . (18) Substituting (1 4) in to (18), after some simple transformations w e come to (15). The separated equations can a ls o b e written as t he Ab el–Jacobi equations dx p − W ( x ) + dy p − W ( y ) = 0 , xdx p − W ( x ) + y dy p − W ( y ) = − dt √ b . F ro m algebraic p oin t of view the flo w of the Hamiltonian H linearizes on the Jacobian J (Γ) o f the h yp erelliptic curv e of genera 2 g iven b y the equation Γ : τ 2 = W ( s ) . The discriminan t set of W ( s ) , naturally , coincides with (9). Th us, the formulas (14) a nd (15) give the analytical real separation of v ariables whic h, in particular, pro vides the p ossibilit y to completely in ve stigate the phase t o pology of the system including the description of the Lio uv ille to r i bifurcations. The a uthor is grateful to prof . M. Kharlamo v for v aluable discussions and advices. References [1] Chaplygin, S.A., 190 3 . A new partial solution of the problem o f motion of a rigid b o dy in liquid. T rudy Otdel. Fiz. Nauk Obsh. Liub. Est., 11, no. 2, 7–10. [2] Gory ache v, D.N., 1916. New cases of in tegrabilit y of Euler’s dynamical equations. W arsaw Univ. Izv., 3, 1– 1 3. [3] Kharlamo v, M.P ., Savus hkin A.Y., 20 05. Explicit in tegration of one problem of motion of the generalized K ow a levski to p. Mec h. Res Comm un., 32, 547–5 5 2. [4] Kharlamo v, M.P ., 200 8 . Separation o f v ariables in one problem of motion of the general- ized Ko w alevski top. Mech . Res. Comm un. 35 , 276–281. [5] T siganov , A.V., 201 0 . On t he generalized Chaplygin system, J. of Math. Sciences , v ol. 168, no. 8, 901—911. 5 [6] Y ehia, H.M., 1996. New in tegrable problems in the dynamics of rigid b o dies with the K ov alevsk a ya configuration. I – The case of a xi symmetric fo rces . Mec h. Res. Comm un. 23, 423– 427. 6