Integrable discretizations of the Euler top - CiteSeerX

Integrable discretizations of the Euler top A.I. Bobenko B. Lorbeery Yu.B. Surisz Fachbereich Mathematik, Technische Universitat Berlin, Str. 17 Juni 135, 10623 Berlin, Germany Abstract

Discretizations of the Euler top sharing the integrals of motion with the continuous time system are studied. Those of them which are also Poisson with respect to the invariant Poisson bracket of the Euler top are characterized. For all these Poisson discretizations a solution in terms of elliptic functions is found, allowing a direct comparison with the continuous time case. We demonstrate that the Veselov{Moser discretization also belongs to our family, and apply our methods to this particular example.

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E{mail: E{mail: E{mail:

[email protected] [email protected] [email protected]

1

1 Introduction The subject of integrable discretizations of integrable dynamical systems is slightly more than twenty years old. During the rst decade of its existence several dierent approaches were proposed for discretizing soliton equations [AL], [H], [DJM], [QNCV], [NCW], but neither of them dealt with integrable systems of the classical mechanics. The rst examples in this important subarea seem to appear about ten years ago [V], [S]. The Veselov's paper contained discrete time versions of such classical integrable systems as the Euler case of the rigid body motion and the Neumann system. The algebraic construction behind these examples was later elaborated in more detail in [MV]. However, the nature of these exceptionally beautiful examples remains somewhat mysterious. They still resist to be included in any of the existing general frameworks for integrable discretizations (cf. [DLT]). The origin of the Veselov{ Moser's discretizations, their place in the corresponding continuous time hierarchies, their relations to other existing examples, and several further points remain to be clari ed. The problem of comparing the explicit solutions found in [V], [MV] with the classical continuous time counterparts was also left open (despite the fact that on the level of equations of motion the continuous limit is easy to perform). The present work may be considered as a sort of an extensive comment on [V], [MV] in the part concerned with the Euler top. We do not close all the open problems mentioned above, but we do hope to bring some light into some of them. In particular, we nd formulas for the solution of the discrete time Euler top in the form which makes the comparison with the continuous time case immediate. Moreover, the Veselov{Moser system turns out to be by no means the only reasonable discretization of the Euler top. We introduce a whole family of discretizations sharing integrals of motion with the continuous time system, and characterize those of them which share also the underlying invariant Poisson structure. In this context the Veselov{Moser system becomes just one particular case, and not the most simple one. It has to be mentioned that, unlike [V], [MV], our procedure for obtaining explicit solutions is rather pedestrian; it does not use such advanced tools as spectral theory of dierence operators [V] or Baker{Akhiezer functions [MV]. Actually, our construction is based on the addition formulas for elliptic functions, and could be invented already by Jacobi. However, as a heuristic tool for nding it we used the Lax representation of the Euler top in su(2) with a spectral parameter on an elliptic curve, which was certainly unknown in the times of Euler and Jacobi.

2

2 The Euler top The famous Euler's equations describing the motion of a rigid body with the xed center of mass (Euler top) read [G]: 1 1 _ M = B?C MM ;   M_ = C1 ? A1 M M ; (1)   M_ = A1 ? B1 M M : Here M = (M ; M ; M ) is the kinetic momentum vector in the coordinate system attached rmly to the body; the axes of this system coincide with the principal axes of inertia, and the numbers A; B; C > 0 are the corresponding moments of inertia. The vector form of the equations (1) is: M_ = M  (M) ; (2) 1

2

3

1

2

3

2

3

1

3

1

2

T

where the vector of the angular velocity is introduced:  

(M) = MA ; MB ; MC : (3) The equations (1) are Hamiltonian with respect to the following Poisson bracket on R (M): fM ; M g = M ; fM ; M g = M ; fM ; M g = M : (4) with the Hamilton function H (M) = E (M)=2, where (5) E (M) = MA + MB + MC : The bracket (4) is degenerate and has one Casimir function 1

2

3

T

3

1

2

3

2

3

1

2 2

2 1

3

1

2

2 3

M (M) = M + M + M : (6) Generic symplectic leaves of the bracket (4) are two{dimensional spheres M (M) = const, hence each system Hamiltonian with respect to this bracket is integrable in the Liouville{ Arnold sense; in particular, the Euler top is integrable. An explicit solution to the equations (1) can be given in terms of the Jacobi elliptic functions [G]. Suppose for the sake of de niteness that A>B>C>0; (7) 2

2 1

2 2

2 3

2

3

then

1 E 1: A M C The formulas for the solution look dierent depending on whether this quantity is greater or smaller than 1=B : 2

(

)

(

)

cn  (t ? t ) ; M (t) = b sn  (t ? t ); M (t) = c dn  (t ? t ) : (8) M (t) = a dn  (t ? t ) cn  (t ? t ) 0

1

2

0

0

0

3

0

Here and below in similar situations the upper expressions in curly brackets refer to the case E E B < M 2  C , while the lower ones refer to the case A  M 2 < B . The module k of the elliptic functions in (8) is given by: 1

1

1

8 > > > < k => > > :

1

9 > > > = ; > (B ? C )(AE ? M ) > > (A ? B )(M ? CE ) ; so that we have always 0 < k < 1. The coecients a; b; c are de ned by: 8 M ? CE 9 > > > > B > > < = B ? C ?M : M ? CE ; c = C AE a =A A?C ; b => > A?C > > > : B AE ? M > ; A?B Finally, the frequency  is given by: 8 9 (B ? C )(AE ? M ) > > > > < = ABC  => : > > > > : (A ? B )(M ? CE ) > ; ABC 2

(A ? B )(M ? CE ) (B ? C )(AE ? M ) 2

2

(9)

2

2

2

2

2

2

2

2

2

(10)

2

2

(11)

2

2

In the case

E M2

= B the elliptic functions degenerate to the hyperbolic ones, and we have: M (t) = cosh a(t ? t ) ; M (t) = b tanh  (t ? t ) ; M (t) = cosh c(t ? t ) ; 1

1

where

0

2

0

3

0

(B ? C ) M ; b = M ; c = C (A ? B ) M ; a =A B (A ? C ) B (A ? C ) 2

2

2

2

4

2

2

and

B )(B ? C ) M :  = (A ? AB C In all three cases the numbers a; b; c;  satisfy the condition abc > 0. In what follows, we shall denote by  (M ; E ) the positive square root of the function given in (11), and hence we must have abc > 0. 2

2

2

2

3 Discretizations Our aim here is to nd integrable discretizations of the Euler top. De nition 1 An integrable discretization of the Euler top (2) is a one{parameter family of dieomorphisms F : R  [0; ) 7! R , 3

3

c = F (M; h) ; M (12) such that each one of them F (
; h) : R 7! R is Poisson with respect to the bracket (4), has 3

3

two integrals of motion M 2 and E , i.e.

M (F (M; h)) = M (M) ; 2

E (F (M; h)) = E (M) ;

2

(13)

and the following asymtotics hold:

F (M; h) = M + hM  (M) + o(h) ; h ! 0 :

(14)

Actually, our maps always will be de ned by implicit equations of motion:

Proposition 1 Let f : R

3

c ? M = hf (M; M c ; h) : M (15)  R  [0; ) 7! R be a C {function in each argument such that f (M; M; 0) = M  (M) : 3

3

1

Then for h small enough the equation (15) de nes a local dieomorphism (12) satisfying (14).

Proof. This follows immediately from the implicit function theorem for all h satisfying c ) 6= 0, i.e. for all h small enough. det(I ? h@f=@ M When speaking about discretizations, it is natural to think about M in (12) as about sequences Mm : Z 7! R approximating solutions M(t) : R 7! R of the Euler top (2) in the 3

3

5

sense that Mm  M(mh). In this context the formula (12) takes the form of the dierence equation Mm = F (Mm; h) ; and, similarly, the formula (15) has to be thought of as Mm ? Mm = hf (Mm; Mm ; h) : The approximation property Mm = M(mh) + O(h) holds on nite time intervals and is assured by (14) (this is a standard fact from the numerical analysis). +1

+1

+1

4 Elliptic coordinates and Poisson discretizations Now we would like to nd a manageable criterium for a map (12) to be Poisson with respect to the bracket (4). The corresponding statement becomes rather transparent in a new coordinate system in R (M). The corresponding change of variables  : (M ; E; ') 7! (M ; M ; M ) is suggested by the formulas (8) for a solution of the Euler top. In the above formula (M ; M ; M ) 2 R , and (M ; E; ') 2 R  R  T, where T = R=(4K Z), and K = K (k ) is the full elliptic integral of the rst kind corresponding to the value of k given in (9). The formulas for  are dierent in two regions of R separated by the two planes E (M) = 1 ,  1 ? 1  M =  1 ? 1  M : (16) M (M) B B A C B n o n o Each one of the sets ME2 MM > B and ME2 MM < B consists of two sectors, and each one of these sectors is bounded by two half-planes. The elliptic coordinates in R (M) are introduced by: ( ) ( ) cn ' dn ' M = a dn ' ; M = b sn '; M = c cn ' : (17) with a; b; c ngiven byo(10), and the modulus k of the elliptic functions given by (9). On the subset ME2 > B the sign of c coincides with the sign of M (which is constant in each one of two sectors), n E Mand theo signs of a; b satisfy the condition sign(ab) = sign(c). Similarly, on the subset M 2 M < B the sign of a coincides with the sign of M , while the signs of b; c satisfy sign(bc) = sign(a). Finally, on the four half-planes described by (16) the elliptic coordinates are de ned by continuity according to M = cosha ' ; M = b tanh ' ; M = coshc ' ; the signs of a; c being the same as the signs of M ; M , respectively, and sign(b) = sign(ac). 3

2

1

2

3

3

2

1

+

2

3

2

+

2

3

2 1

2

(

)

(

(

1

)

)

(

2 3

1

)

3

1

2

3

2

1

(

3

)

(

1

1

)

1

2

3

1

6

3

Figure 1: Dierent energy levels for constant M

2

Proposition 2 The formulas (17) de ne a valid change of variables (a local dieomorphism)

near each point of R3 .

Proof. It can be veri ed by a direct calculation that in the region

nEM M

(

o

M 6= B we have:

2(

)

1

)

@ (M ; M ; M ) = ? 1 @ (M ; E; ') 4 (M ; E ) dn ' 6= 0 ; and that the partial derivatives n o of which this Jacobian is composed allow a continuation to the boundary ME2 MM = B . Now it is obvious that an arbitrary map (12) having two integrals of motion M (M) and E (M), can be cast, in the elliptic coordinates (M ; E; '), in the form c = M ; Eb = E ; 'b = 'b(M ; E; ') : M (18) 1

2

3

2

2

(

)

(

2

1

)

2

2

2

2

2

Proposition 3 The map (12) having two integrals of motion M (M) and E (M) is Poisson 2

with respect to the bracket (4) i in the elliptic coordinates (M 2 ; E; ') it takes the form

c = M ; Eb = E ; 'b = ' + g(M ; E ) M 2

2

2

(19)

with the function g not depending on '.

Proof. To prove this statement, we have to calculate the Poisson bracket (4) in the coordi-

nates (M ; E; '). The corresponding formulas read: 2

fM ; E g = fM ; 'g = 0 ; 2

fE; 'g = ?2 (M ; E ) :

2

2

(20)

Indeed, the function M is a Casimir function, hence it Poisson commutes with both E and '. To calculate the bracket fE; 'g, we substitute (away from the boundary ME2 = B ) the expressions (17) into an arbitrary one of the formulas (4), and after straightforward calculations arrive at the expression given above. 2

1

7

Actually, the concrete expression for fE; 'g is not essential for the proof of our proposition. The only important thing is that this Poisson bracket does not depend on '. Indeed, we have: b 'bg = @ 'b fE; 'g : fE; @' Since the Poisson bracket fE; 'g depends only on M ; E which are integrals of motion, we b = 1, see that the necessary and sucient condition for our map to be Poisson reads @ '=@' which is equivalent to the last equation in (19). 2

5 Special discretizations We derive now a family of discretizations of the Euler top. Our derivation is based on a Lax representation with a spectral parameter for the Euler top. We prefere to work with a su(2) Lax representation, since our experience in discretizing various geometric structures (see, for example, [BP]) conviced us that this procedure may be performed most straightforwardly when applied to su(2) Lax formulations. To nd a suitable Lax representation for the Euler top, we use a stationary version of the Lax representation of the chiral eld model due to Cherednik [Ch]. Set

X M (u) = 21i Mk wk (u)k ; 3

(21)

k=1

where k (k = 1; 2; 3) are the Pauli matrices, and wk (u) are the following elliptic functions: 1 ; w (u) =  dn(u; ) ; w (u) =  cn(u; ) : w (u) =  sn(u; ) sn(u; ) sn(u; ) 1

2

3

Here the parameter  and the module  of the elliptic functions are de ned by ? C ;  = C (A ? B ) :  = AAC B (A ? C ) 2

2

(22)

(23)

Further, for a vector V = (V ; V ; V ) 2 R set: 1

2

3

T

3

X V (u) = 21i Vk wk (u ? u )k ; 3

(24)

0

k=1

where the point u is chosen so that 0

w (u ) = A? = ; 1

0

1 2

w (u ) = B ? = ; 2

1 2

0

8

w (u ) = C ? = : 3

0

1 2

(25)

Consider now the Lax equation

M_ (u) = [M (u); V (u)] :

(26)

With the help of identities

wj (u)wk (u ? u ) = wj (u )wl(u ? u ) ? wk (u )wl(u) ; 0

0

0

0

(27)

where (j; k; l) is a permutation of (1; 2; 3), one sees that the matrix equation (26) is equivalent to the set of the following two equations: M_ = M  = (V) ; (28) = 0 = V  (M) : (29) 1 2

1 2

Here the second equation is equivalent to V = = (M) with some 2 R, which, being substituted in the rst equation, results in M_ = M  (M) : (30) 1 2

Obviously, for a nonvanishing function = (t) the latter equation is nothing but a time reparametrization of the Euler top (2). In this sense (26) is a Lax representation of the Euler top. Discretizing the above construction in time, we introduce the matrix

V (u) = I + 2hi

X 3

k=1

Vk wk (u ? u )k ; 0

(31)

and consider instead of (26) the discrete time Lax equation

c(u) = V ? (u)M (u)V (u) : M 1

(32)

(this equation may be interpreted as a stationary version of the lattice chiral eld model by c(u) = M (u)V (u) and using the identities [NP]). Representing the latter equation as V (u)M (27), we nd that our matrix equation is equivalent to the set of the following three equations: c )  = (V) ; c ? M = h (M + M (33) M 2 c) ; 0 = V  = (M + M (34) 1 2

1 2

and

X c (Mk ? Mk )Vk wk (u)wk(u ? u ) = 0 : 3

0

k=1

9

(35)

Now (34) is equivalent to

c) V = 12 = (M + M

(36)

1 2

with some 2 R, which, being substituted in (33), results in c ? M = 1 h (M + M c )  (M + M c) : M (37) 4 It remains to notice that, plugging (36), (37) into (35), we bring the latter equation to the form   X wk (u)wk(u ? u )wk (u ) wl (u ) ? wj (u ) = 0 ; 3

0

k=1

0

2

2

0

0

which is automatically satis ed due to the identity   wl (u)wl(u ? u )wj (u ) ? wj (u)wj (u ? u )wl (u ) = wk (u ) wl (u ) ? wj (u ) : (38) So, the matrix equation (32) is equivalent to (37) with some 2 R, the vector V being given by (36). However, the equation (37) does not completely de ne the discretization due to arbitrariness of . In what follows we shall consider this equation with being a certain xed c . In other words, the subject of our further investigations will consist of function on M, M discretizations governed by implicit equations of motion of the following special form: c ? M = 1 h (M; M c ; h) (M + M c )  (M + M c) : M (39) 4 We call them special discretizations. It is obvious that if

: R  R  [0; ) 7! R is a C {function of each argument satisfying

(M; M; 0) = 1 ; (40) then the equation (39) ful lls the conditions of Proposition 1, and therefore de nes a map (12). In components, special discretizations may be presented as:   c ; h) 1 ? 1 (M + M c )(M + M c); c ? M = 1 h (M; M (41) M 4 B C 1 1 1 c c )(M + M c); c (42) M ? M = 4 h (M; M; h) C ? A (M + M 1 1 1 c c )(M + M c): c (43) M ? M = 4 h (M; M; h) A ? B (M + M 0

0

0

3

3

0

0

0

0

+

1

1

1

2

2

3

3

2

2

3

3

1

1

3

3

1

1

2

2

10

6 Poisson property of special discretizations We investigate now the integrability properties of special discretizations (which are naturally expected due to the existence of a Lax representation with a spectral parameter). Proposition 4 Maps de ned by equations of motion (39) possess M (M) and E (M) as integrals of motion. Proof. It follows from (39) that 2

c ? M; M + M c i = 0 and hM c ? M; (M + M c )i = 0 ; hM

(44)

c; M c i = hM; Mi and hM c ; (M c )i = hM; (M)i : hM

(45)

or, equivalently,

This proves our statement. Actually, this statement can be almost inverted. Namely, consider an arbitrary discretizac ) satisfy (45), tion having M (M) and E (M) as integrals of motion. Then the pairs (M; M c c which is equivalent to (44). If, in addition, (M + M)  (M + M) 6= 0, then there exists a c ) such that (39) holds. real number = (M; M In what follows we shall need also the folowing technical and obvious statement. Lemma 1 The set of xed points of the maps (12) de ned by (39) coincides with the set of c )  (M + M c ) = 0, and this coincides with the union of the the points M for which (M + M coordinate axes 2

fM = M = 0g [ fM = M = 0g [ fM = M = 0g : c , M +M c , M +M c For all other points M, at least two of the three expressions M + M 1

2

2

3

3

1

1

do not vanish.

1

2

2

3

3

Now we nd conditions for a map end notice that   de ned by (39) to be Poisson. To this c c the pairs (M; M) = M; F (M; h) belong to the subset of R (M)  R (M) singled out by c ) and E (M) = E (M c ). The elements of this subset may be the conditions M (M) = M (M parametrized by the quadruples (M ; E; '; 'b) according to 3

2

3

2

2

( )  ( cn ' ) dn ' M = a dn ' ; b sn '; c cn ' ; ) ( ) (  b b cn ' dn ' c= a b c cn 'b : M dn 'b ; b sn '; 11

(46) (47)

Equivalently, we can use the following coordinates: (M ; E; '; 
'), where b b ' = ' +2 ' ;
' = ' ?2 ' : Let us denote on the above mentioned subset: 2

c ; h) = ?(M ; E; ';

(M; M 
'; h) :

(48) (49)

2

Theorem 1 The equations of motion of the special discretization (39) in the elliptic coordi-

nates (M 2 ; E; ') have the form (18), where the function 'b(M 2 ; E; '; h) is implicitly de ned by the equation 2 sn(
') ?(M 2 ; E; '; 
'; h) = (50) 2 2 2 2 1 ? k sn (')sn (
') h (M ; E ) cn(
')dn(
') The map de ned by (39) is Poisson, i the equation (50) may be solved for
' as

 
' = (M ; E; h) = h (M ; E ) 1 + O(h) 2 with the function  not depending on '. 2

2

Proof. Denoting

(51)

D = D(M ; E; '; 
') = 1 ? k sn (')sn (
') ; (52) we nd from (46), (47) with the help of addition formulae for elliptic functions: ( ) ( ) 2 a 2 a sn( '  )dn( '  )sn(
' )dn(
' ) cn( '  )cn(
' ) c c M ? M = ? D k sn(')cn(')sn(
')cn(
') ; M + M = D dn(')dn(
') ; (53) 2 b 2 b c ? M = cn(')dn(')sn(
') ; M c + M = sn(')cn(
')dn(
') ; (54) M D D ( ) ( ) 2 c 2 c k sn( '  )cn( '  )sn(
' )cn(
' ) dn( '  )dn(
' ) c c M ? M = ? D sn(')dn(')sn(
')dn(
') ; M + M = D cn(')cn(
') : (55) Plugging this into an arbitrary one of the equations of motion (41){ (43), we arrive after some cancellations at the equation (50). Notice that Lemma 1 assures that, away from the xed points, these cancellations are legitime in at least one of the equations of motion. So, we have arrived at the equation (50) of the form 1

1

2

2

2

2

2

2

2

2

1

1

3

3

2

2

3

3

(M ; E; '; 
'; h) = 0 ; 2

12

(56)

which serves for determining the function 'b(M ; E; '; h) for our map. The equation (56) may be rewritten as   e b b b h) = 0 : M ; E; ' + ' ; ' ? ' ; h = ( M ; E; '; '; (57) 2 2 b = 1 for our map to be Poisson, Proposition 3 gives a necessary and sucient condition @ '=@' and this is equivalent to the following condition: 2

def

2

2

@ e + @ e = 0 on the solutions of (57) : @ 'b @'

(58)

Since, obviously,

! ! @ e = 1 @ + @ ; @ e = 1 @ ? @ ; @ 'b 2 @ ' @ (
') @' 2 @ ' @ (
') we nd that the Poisson property is equivalent to the following condition: @ = 0 on the solutions of (56) : (59) @ ' In turn, (59) assures that the solutions of (56) for
' do not depend on '. Corollary. In the conditions of the above theorem, solutions Mm = (M ;m ; M ;m; M ;m ) to the dierence equation (60) Mm ? Mm = 41 h (Mm ; Mm ; h) (Mm + Mm )  (Mm + Mm ) in the integrable ( Poisson) case are given by ( ) ( ) cn(2 m + ' ) dn(2 m + ' ) M ;m = a dn(2m + ' ) ; M ;m = b sn(2m + ' ); M ;m = c cn(2m + ' ) : (61) 1

+1

+1

1

0

+1

2

0

0

2

3

T

+1

3

0

0

7 First examples of integrable discretizations We use Theorem 1 to investigate the Poisson property of several discretizations. We start with the following negative result. c ; h)  1 de nes a non{Poisson map. Proposition 5 The function (M; M 0

13

The simplest way to nd Poisson (and therefore integrable) discretizations is to assure that the left{hand side of the equation (50) does not depend on '. Proposition 6 The functions 4M c ; h) = (62)

(M; M c; M + M ci hM + M and 4E c ; h) = (63)

(M; M c ; (M + M c )i hM + M de ne Poisson maps. Proof. Using the second expressions in (53){(55), it is not dicult to derive the following formulas: 1 hM + M c; M + M c i = a + c ? b sn (
') ; 4 1 ? k sn (')sn (
') 2

1

2

2

2

2

2

2

2

2

a + c ? b sn (
') 1 hM + M c ; (M + M c )i = A C B : 4 1 ? k sn (')sn (
') Hence, the left{hand side of the equation (50) for = ; does not depend on '. Taking into account that, according to (10), a +c =E; a +c =M ; A C we nd the following expressions for this left{hand side: 1 ?  ; sn (
') ; where b :  (M ; E ) = Mb ;  (M ; E ) = BE (64) (see (10) for the expression of b through M and E ). Therefore, the equation (50) in these two cases is equivalent to 1 sn(
') = h : (65) cn(
')dn(
') 2 1 ?  ; sn (
') Obviously, its solutions satisfy  
' =  ; (M ; E; h) = h (M ; E ) 1 + O(h) : 2 2

2

2

2

2

2

2

12

2

2

2

2

12

1

2

2

2

2

2

2

2

2

12

12

2

2

2

2

14

2

8 Veselov{Moser discretization In order to introduce the Veselov{Moser discretization, we need, rst of all, the matrix notationfor the Euler top equation (2).  Using the  well{known isomorphism between the Lie algebra R ;  with the Lie algebra so(3); [
;
] of 3  3 skew{symmetric matrices with the usual commutator, we can rewrite the equations of motion (2) also in the matrix form M_ = [M; (M)] ; (66) 3

where

1 0 0 M ?M M=B @ ?M 0 M CA M ?M 0 1 0 0 M =C ?M =B 0 M =A C

(M) = B A: @ ?M =C M =B ?M =A 0 3

2

3

2

and

(67)

1

1

3

2

3

(68)

1

2

1

The relation between the matrices M and = (M) may be expressed as follows: M = J + J ;

(69)

where the entries of the diagonal matrix J = diag(J ; J ; J ) 1

2

(70)

3

are de ned by the relations

A=J +J ; 2

B =J +J ;

3

3

1

C =J +J : 1

2

(71)

A spectral parameter dependent Lax representation for (66) is:

.

(M + J ) = [M + J ; (M) + J] : 2

2

(72)

Now we can describe the Veselov{Moser construction. The dierential equation (66) is replaced by the dierence one, c = ! M! ; M (73) where ! 2 SO(3) is an orthogonal matrix related to M by means of the following relation, coming to replace, or to approximate, (69): T

hM = !J ? J! : T

15

(74)

It is easy to see that the previous two relations imply also

c = J! ? ! J : hM

(75)

T

Moser and Veselov demonstrated that, in general, the equation (74) does not determine the matrix ! 2 SO(3) uniquely. This non{uniqueness may be described as follows. Notice that (74), (75) are equivalent to the following factorizations of matrix polynomials: I ? hM ?  J = (! + J)(! ? J) 2

and

2

T

c ?  J = (! ? J)(! + J) ; I ? hM respectively. Denote by S the set of 6 roots of the equation det(I ? hM ?  J ) = 0. Then to each decomposition S = S \ S? into two disjoint sets of three roots satisfying S = S = ?S? there corresponds a unique solution ! 2 SO(3) of (74) such that S is the set of the roots of det(! + J) = 0, while S? is the set of the roots of det(! ? J) = 0. However, in the continuous limit, when h is supposed to be small, there exists a unique decomposition, for which the roots from S? are positive real numbers O(h){close to J ? , J ? , J ? . This is the only decomposition for which the corresponding ! has the following asymptotics: ! = I + h (M) + O(h ) : (76) Supposing h small enough, we consider from now on only this choice of !, and demonstrate that the discretization of Veselov and Moser also belongs to the class of special discretizations described by the equation (39). Theorem 2 The Veselov{Moser discretization for h small enough may be represented in the form (39) with 2 c ; h) = q

(M; M (77) c )jj : 1 + 1 ? h jj (M + M Proof. We shall need the following lemma, proof of which is put in the Appendix. Lemma 2 For a matrix ! 2 SO(3), O(h){close to I , there exists a unique matrix W 2 so(3) such that W = O(h), and ! = I + W + 2 W ; where

= q2 : 1 + 1 ? jWj 2

2

T

2

2

+

+

2

+

1

3

T

+

1

1

2

VM

1 4

2

2

2

2

16

1

Here

jWj = W + W + W 2

2 1

2 2

1 0 0 W ?W for W = B @ ?W 0 W CA 2 so(3) : W ?W 0 3

2 3

2

3

1

2

1

With the help of this lemma we derive the following two equations:

! ? ! = 2W ; ! + ! = 2I + W : Now from (74), (75) and the rst equation in (78) we nd: c = (! ? ! )J + J(! ? ! ) = 2(WJ + JW) ; h(M + M) hence c : W = 21 h (M + M) Further, from (74), (75) and the second equation in (78) we nd: c ? M) = J(! + ! ) ? (! + ! )J = [J; W ] = [WJ + JW; W] h(M c; (M + M)] c ; = 1 h [M + M 4 T

T

T

2

T

T

T

(79) (80)

2

2

where

(78)

(81)

2 cj : 1 + 1 ? h j (M + M)     Using the above{mentioned isomorphism between R ;  and so(3); [
;
] , we see that the theorem is proved. It remains to reproduce from our point of view the result of Moser and Veselov concerning the Poisson property of their discretization. To this end, we demonstrate that the equation (50) with ? corresponding to allows a '{independent solution for
'. Theorem 3 For the Veselov{Moser discretization the equation (50) is equivalent to   sn(
')cn(
')dn(
') = h 1 +  sn (
' ) ? sn (
' ) ; (82) 2 where 8 8 9 9 > > AC M ? CE > 2ACE ? (A + C ? B )M > > > > > > > < (B ? C ) AE ? M > < (B ? C )(AE ? M ) > = = ; = : (83) => > > > > > > > 2ACE ? (A + C ? B )M AC AE ? M > > : (A ? B ) M ? CE > : (A ? B )(M ? CE ) > ; ;

q

=

1 4

2

2

3

VM

VM

2

4

2

2

2

2

2

2

2

2

2

17

2

The solution of (82) satis es





(84)
' =  (M ; E; h) = h (M ; E ) 1 + O(h) : 2 Proof. For the Veselov{Moser discretization the left{hand side of the equation (50) can be calculated with the help of (77) and the second expressions in (53){(55): ?(M ; E; '; 
'; h) = r 2 ; (85) 1 ? k sn (')sn (
') 1 ? k sn (')sn (
') + 1 ? k sn (')sn (
') ? h G 2

2

VM

2

2

2

2

2

2

2

2

2

2

2

2

where

( ) ( ) a b c cn ( '  )cn (
' ) dn ( '  )dn (
' ) G = A dn (')dn (
') + B sn (')cn (
')dn (
')+ C cn (')cn (
') : (86) Hence the equation (50) in the present case is equivalent to: r  ')dn(
') 1 ? k sn (')sn (
') + 1 ? k sn (')sn (
') ? h G = h cn(
sn(
') 2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

We leave the radical alone on the left{hand side and square the resulting equation to derive: )dn (
') ? 2h cn(
')dn(
') 1 ? k sn (')sn (
') : (87) ?h G = h  cn (
sn'(
') sn(
') Obviously, we have: 2

2

2

2

2

2

2

2

2

G = G (M ; E;
') ? G (M ; E;
') sn (') ; 2

0

where

2

1

(88)

2

( ) ( ) a c cn (
' ) dn (
' ) G = A dn (
') + C cn (
') ; ) ) ( ( a c b cn (
')dn (
') : cn (
' ) k dn (
' ) G = A k dn (
') + C ? cn (
') B We shall prove that G ? k sn (
')G = k  cn (
')dn (
') : This formula allows to derive from (88):   )dn (
') ; G = k snG(
') 1 ? k sn (')sn (
') ?  cn (
sn'(
') 0

2

1

2

2

2

2

2

2

2

1

1

2

2

2

2

2

2

2

2

0

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

18

2

2

2

(89) (90) (91) (92)

so that (87) is equivalent to

h G = 2h cn(
')dn(
') ; k sn (
') sn(
') 2

2

or

1

2

sn(
')cn(
')dn(
') = h G : 2k This is equivalent to (82), as one easily calculates from the de nition (90) that 2

1

  G = k  1 +  sn (
') ? sn (
') 2

1

2

2

4

with  and as in (83). It remains to prove the relation (91). But it follows from the de nitions (89), (90):

8 > a +k c ? b > >