Nonabelian {\mathfrak so}_3 Euler top
aa r X i v : . [ n li n . S I] J a n Nonabelian s o Euler top
V. Sokolov ∗ ∗ †
22 December 2020
Abstract
Using the nonabilinization procedure, we find an integrable matrix version of theEuler top on so .Keywords: Euler equation, nonabelian system, Lax representation, first integral, symmetryIn the papers [1, 2] an approach for constructing integrable noncommutative generalizations of agiven polynomial integrable system was proposed. It turns out that requiring the noncommutativegeneralization not only for the system itself but also for all its first integrals (conservation lawsfor PDEs) and infinitesimal symmetries, one can easily construct all such generalizations. In thisnote we apply this way for finding noncommutative generalizations of the Euler top. Some othergeneralizations, where a part of first integrals and/or symmetries “disappear”, are known.Consider the system of ODEs u ′ = z vw, v ′ = z uw, w ′ = z uv, z i ∈ C , z i = 0 , (1)where ′ means the derivative with respect to t . In the case z = B − CA , z = C − AB , z = A − BC , where A, B, C are axis of the ellipsoid of inertia, equations (1) describe the Euler top. System (1) possessesthe first integrals I = z u − z w , I = z v − x w . For any i, j the system u τ = z vw I i I j , v τ = z uw I i I j , w τ = z uv I i I j (2)is an infinitesimal symmetry for (1).It is easy to see that the parameters z i can be always reduced to one by some (complex) scallingof the independent and dependent variables. Everywhere below, we assume that z = z = z = 1 . Our goal is to generalize system (1) to the case when the unknown variables become matrices ofarbitrary size n × n. Assuming that the right hand sides are still homogeneous quadratic polynomialsand that the system coincides with (1) for n = 1 , we arrive at the following ansatz with unknownconstant coefficients: u ′ = k vw + (1 − k ) wv + c [ v, w ] + c [ w, u ] + c [ u, v ] ,v ′ = k wu + (1 − k ) uw + c [ w, u ] + c [ u, v ] + c [ v, w ] w ′ = k uv + (1 − k ) vu + c [ u, v ] + c [ v, w ] + c [ w, u ] . (3)To begin with, we require that for each integral R ij = I i I j of system (1) there exists an integralof system (3) of the form trace P ij ([3, chapter 6.1]), where P ij is a matrix polynomial, that coincideswith R ij if n = 1 . We will call the integral trace P ij nonabelinization of the integral R ij . ∗ L.D. Landau Institute for Theoretical Physics, Chernogolovka, Russian Federation. † Federal University of ABC, Santo Andr´e, Sao Paulo, Brazil. E-mail: [email protected] For some purposes the normalization z = − , z = z = 1 seems to be more reasonable. In particular,in this case the trajectories are compact. roposition 1. If there exist nonabelinizations of all integrals R ij with i + j ≤ , then system(3) has the form u ′ = 12 ( vw + wv ) + X [ u, v ] + Z [ u, w ] ,v ′ = 12 ( wu + uw ) + Y [ v, u ] + Z [ v, w ] ,w ′ = 12 ( uv + vu ) + X [ w, v ] + Y [ w, u ] , (4)where X, Y, Z are arbitrary constant parameters.
Remark 1.
The permutation of the variables u, v, w leads to the corresponding permutation ofthe coefficients
X, Y, Z . In addition, one can change the signs of several coefficients. For example,the transformation u → − u, t → − t replaces Y → − Y .For any parameters X, Y, Z, system (4) possesses a Lax representation L t = [ A, L ] in the Liealgebra G = ( U V − V U ! , U, V ∈ H ) of matrices over the skew-field of quaternions (cf. [4]). It is assumed that quaternions commute withnonabelian variables u, v, w. The matrices L and A have the form L = L L − L L ! , A = A A − A A ! , where L = 2( ν − µ )Det( S ) u, L = h P, Ω i v + h Q, Ω i w,A = ( − Y u − X v − Z w ) + σL , A = µ h P, Ω i v + ν h Q, Ω i w. Here σ, µ, ν are arbitrary pairwise distinct parameters,
Ω = ( i , j , k ) , P и Q are 3–dimension vectorssuch that h P, Q i = 0 , h P, P i = 14( σ − µ )( ν − µ ) , h Q, Q i = 14( σ − ν )( µ − ν ) , and S is the matrix with rows P, Q, Ω . Replacing the quaternion units i , j , k with the Pauli matrices i → i − i ! , j → − ! , k → ii ! , we obtain a Lax pair in × -matrices. This Lax pair depends on one essential parameter κ =( σ − µ )( ν − µ )( σ − ν )( µ − ν ) . Other parameters can be removed by means of a conjugation by a quaternion, by ashift A → A + const L and a scalling t → τ − t, u → τ u, v → τ v, w → τ w . For example, one mayset σ = 0 , µ = 1 . Probably, the obtained Lax pair allows an algebraic R -matrix interpretation or a description interms of a decomposition of the corresponding loop algebra into a sum of two subalgebras not forany set of parameters X, Y, Z . Conjecture.
In the case of system (4) all integrals R ij permit a nonabelinization. The corre-sponding nonabelian integrals are generated by the traces of powers of the operator L .Apart from nonabelinization of the integrals R ij , system (4) has other integrals like trace ( uvw − uwv ) . In the commutative case these integrals vanish.Systems (4) with different parameters
X, Y, Z have non-isomorphic algebras of polynomial sym-metries.
Remark 2.
System (4) with X = Y = Z = 0 can be obtained by a reduction from the Nahmequation u ′ = [ v, w ] , v ′ = [ w, u ] , w ′ = [ u, v ] . his system, like the Nahm equation itself, has first integrals, but does not have nonabelian poly-nomial symmetries. The latter fact was verified for symmetries of orders not higher than six.Now let us additionally require the existence of nonabelinizations for all symmetries (2) ([3,chapter 6.1]). Proposition 2.
If system (4) admits nonabelinization of symmetries (2) with i = 1 , j = 0 , thenit can be reduced to one of the following: X = Y = Z = ; Y = Z = 0 , X = by transformations from Remark 2.Apparently, system (4) with these values of the parameters is “more integrable” than in thegeneric case. It would be interesting to understand how these sets of parameters are distinguishedfrom the point of view of the Lax pair presented above. Acknowledgements
The author is grateful to M. Dunajski and V. Rubtsov for useful discussions. This work was carriedout under the State Assignment 0029-2021-0004 (Quantum field theory) of the Ministry of Scienceand Higher Education of the Russian Federation.
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World Scientific, Singapore , 2020, ISBN978-981-121-966-5, 321 pp.[4] K. Kimura, A Lax pair of the discrete Euler top in terms of quaternions, arXiv:1611.02271. |, 2020, ISBN978-981-121-966-5, 321 pp.[4] K. Kimura, A Lax pair of the discrete Euler top in terms of quaternions, arXiv:1611.02271. |