Journal of the Physical Society of Japan
Pre-final version of J. Phys. Soc. Jpn. 86 (2017) 095002.
A Lax-Moser Pair of Euler’s Top Kiyoshi Sogo
∗
Institute of Computational Fluid Dynamics 1-16-5, Haramachi, Meguro, Tokyo 152-0011, Japan A Lax-Moser pair of Euler’s top is found for both cases of continuous and discrete time variable.
(1)
Euler’s top
The equation of motion of Euler’s top is given by dω1 I2 − I3 = ω2 ω3 , dt I1 dω2 I3 − I1 = ω3 ω1 , (1) dt I2 dω3 I1 − I2 = ω1 ω2 , dt I3 where ω = (ω1 , ω2 , ω3 ) is the angular velocity and I1 , I2 , I3 are the moments of inertia, which can be assumed as I1 < I2 < I3 without loss of generality. It is well known that Euler’s top is a completely integrable system, whose solution is given by Jacobi’s elliptic functions. Many years ago a discrete-time version of (1) is proposed by Hirota and Kimura1) as ( ) h(I2 − I3 ) ωn+1 − ωn1 = δ1 ω2n+1 ωn3 + ωn2 ωn+1 , δ1 = 1 3 2I1 ( ) h(I3 − I1 ) (2) ω2n+1 − ωn2 = δ2 ω3n+1 ωn1 + ωn3 ωn+1 , δ2 = 1 2I2 ( ) h(I1 − I2 ) n n+1 n n n+1 ωn+1 − ω = δ ω ω + ω ω , δ3 = 3 3 3 1 2 1 2 2I3 with h a time increment (t = nh), which also have an exact solution in terms of Jacobi’s elliptic theta functions. Surprisingly enough, Lax-Moser pair of Euler’s top has not been found until now. Recently Kimura2) proposed a Lax pair of discrete Euler’s top (2), which is unfortunately incorrect since his derived equation must satisfy δ1 + δ2 + δ3 = 0 that does not hold generally. The ∗
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purpose of this paper is to give a correct Lax-Moser pair of Euler’s top, which is completely different from Kimura’s, for both cases of continuous and discrete time. (2)
Lax-Moser pair for continuous time
Let us start from the Nahm equation3) dA dB dC = [B, C], = [C, A], = [A, B], dt dt dt with z 0 0 x 0 −y , , B = , C = A = x 0 y 0 0 −z which gives equation of motion dx dy dz = 2yz, = −2zx, = 2xy. dt dt dt It is easy to confirm that by setting x = aω1 , y = bω2 , z = cω3 with (I2 − I1 )(I3 − I1 ) a2 = , 4I2 I3 (I2 − I1 )(I3 − I2 ) , b2 = 4I1 I3 (I3 − I2 )(I3 − I1 ) c2 = , 4I1 I2 where a, c > 0, b < 0, the equation (5) becomes the equation (1). Now the Nahm equation (3) can be rewritten in a Lax-Moser form as dL = [M, L], dt ) ) i( 1( + L = C + λL+ + λ−1 L− , M = λL − λ−1 L− , 2 2 where we have set z 0 0 x ∓ iy . ± C = , L = 0 −z x ± iy 0
(3)
(4)
(5)
(6)
(7) (8)
(9)
The parameter λ in (8) is a constant which can be considered as a spectral parameter. (3)
Lax-Moser pair for discrete time The equation (7) can be discretized as Ln+1 − Ln = h (Mn+1 Ln − Ln+1 Mn )
(10)
which is considered as a candidate for the discrete Lax-Moser pair. The equation (10) can be rewritten as Ln+1 (1 + hMn ) = (1 + hMn+1 ) Ln .
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(11)
J. Phys. Soc. Jpn.
It should be remarked here that if we set An = Ln , Bn = 1 + hMn , the equation (11) becomes An+1 Bn = Bn+1 An ,
(12)
which is a Kimura’s type equation,2) although his A and B are 4 × 4 matrices. Then it is a straightforward calculation that by substituting ) i( Ln = Cn + λLn+ + λ−1 Ln− , 2 ) ( 1 λLn+ − λ−1 Ln− , Mn = 2 with z 0 0 xn ∓ iyn n ± Cn = , Ln = 0 −zn xn ± iyn 0
(13) (14) ,
(15)
into (11), and by comparing coefficients of each power of λ’s in both sides, we obtain xn+1 − xn = +h (yn+1 zn + yn zn+1 ) , yn+1 − yn = −h (zn+1 xn + zn xn+1 ) ,
(16)
zn+1 − zn = +h (xn+1 yn + xn yn+1 ) , which is a discretization of the Nahm equation (5). By setting xn = aωn1 , yn = bωn2 , zn = cωn3 with a, b, c of (6), we obtain the discretized Euler equation (2), which is the goal of the present paper.
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References 1) R. Hirota and K. Kimura, J. Phys. Soc. Jpn. 69, 627 (2000). 2) K. Kimura, arXiv:1611.02271v6 [nlin.SI] 5 Apr 2017. 3) L.J. Mason and N.M.J. Woodhouse, Integrability, Self-Duality, and Twister Theory (Oxford Science Publications, London, 1996).
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