Reduction of divisors and Clebsch system
aa r X i v : . [ n li n . S I] J a n Reduction of divisors and Clebsch system
A.V. Tsiganov
St. Petersburg State University, St. Petersburg, Russiaemail: [email protected]
Abstract
In 2015 Magri and Skrypnyk found Abel’s equations for the Clebsch system on a couple of genusthree spectral curves of 2 × × × gl ∗ ( n ) case. In 1760-67 Euler studied motion of the body attracted to two fixed centers by forces inversely pro-portional to the squares of the distance in three-dimensional space and, after reduction by cycliccoordinate, introduced elliptic coordinates r and s on the orbit plane in which equations of motionhave the following form: 0 = drR − dsS , and 4 dt = r drR − s dsS , (1.1)see page 238 in [11] and page 106 of Lagrange’s textbook [20]. In the Clebsch interpretation of theEuler and Abel results, equations (1.1) describe evolution of two points (prime divisors) P = ( r, R ) ∈ E and P = ( s, S ) ∈ E ′ on a couple of two different elliptic curves E × E ′ defined by equations E : R = Hr + ( α − β ) r + cr − h ( α − β ) r − Hh − ( b + c ) h and E ′ : S = Hs + ( α + β ) s + cs − h ( α + β ) s − Hh − ( b + c ) h . Here H and h are integrals of motion, b and c are geometric parameters describing positions of fixedcenters, and α , β are values of the corresponding charges.If β = 0, we have a Kepler problem with E = E ′ , where H is a Hamiltonian and h is a componentof the Laplace-Runge-Lenz vector. In the Kepler case equations (1.1) describe evolution of the semi-reduced divisor D = P + P ∈ E, deg D = 2 > g( E ) = 1 , on a common elliptic curve E of genus g( E ) = 1. After reduction of this semi-reduced divisor toreduced divisor D = ( r, R ) + ( s, S ) → D ′ = ( u, U ) , so that drR − dsS = uU = 0Euler proved that affine coordinates of reduced divisor D ′ are algebraic integrals of (1.1). In the Keplercase, independent integral u is a function of the corresponding entry of angular momentum vector and,using this additional algebraic integral, Euler separated algebraic orbits from transcendental ones [11].1n 2015 equations of the form (1.1) were obtained for the Clebsch top [16]. In the Clebsch caseseparated variables define two prime divisors P and P on a couple of spectral curves associatedwith the 3 × × × c = 0. We also prove that this unique reduced divisor is a function on integralsof motion similar to the Kepler problem and, therefore, it’s affine coordinates cannon be identifiedwith separated variables. Nevertheless, we suppose that arbitrary prime divisor from the support ofthe corresponding semi-reduced divisor of degree two can be considered as candidate for the desiredseparated variable.Reduction of divisors on the second spectral curve Γ ′ for 2 × In the Clebsch case, the Kirchhoff equations for motion of a rigid body in an ideal incompressible fluidare equal to ˙ M = p × A p , ˙ p = p × M . (1.2)Here p and M are three-dimensional vectors, × denotes the vector product and A = diag ( a , a , a ), a i ∈ R , is a diagonal matrix. Vector M is the total angular momentum vector, whereas p representsthe total linear momentum of system [7]. There are two geometric integrals of motion c = p + p + p , c = p M + p M + p M and two Hamiltonians H = M + M + M + a ( p + p ) + a ( p + p ) + a ( p + p ) ,K = a M + a M + a M + a a p + a a p + a a p , (1.3)which Poisson commutes with respect to Lie-Poisson bracket on the Lie algebra e ∗ (3) (cid:8) M i , M j (cid:9) = ε ijk M k , (cid:8) M i , p j (cid:9) = ε ijk p k , (cid:8) p i , p j (cid:9) = 0 , (1.4)where ε ijk is the totally skew-symmetric tensor. All these definitions are invariant under cyclic per-mutation of indexes.We know a few separated variables for the Clebsch system [19, 21, 16, 30], bi-Hamiltonian structureof the Kirchhoff equations (1.2) [32], various Lax pair representations [2, 3, 24], algebro-geometriclinearization of the flows [4, 9, 10, 13, 40], relations with the Kowalevski gyrostat [18] and the Frahm-Schottky system [2, 4], Hirota-Kimura type discretization [25], numerical solutions [26], etc.Solving equations of motion (1.2) in the framework of the Jacoby separation of variables methodwe try to find invertible transformations of variables p ( t ) , p ( t ) , p ( t ) , M ( t ) , M ( t ) , M ( t ) → c , c , H, K, x ( t ) , x ( t ) (1.5)which reduce Kirchhoff equations to:1. a pair of linear in velocities ordinary differential equations˙ x = Φ ( x , c , c , H, K ) , ˙ x = Φ ( x , c , c , H, K ) , i.e. x , are angle type variables; 2. a pair of linear in velocities Abel’s differential equations˙ x φ ( x , c , c , H, K ) + ˙ x φ ( x , c , c , H, K ) = β , ˙ x φ ( x , c , c , H, K ) + ˙ x φ ( x , c , c , H, K ) = β , (1.6)where β , ∈ C and x , ( t ) are separated variables commuting to each other;3. other equations on x ( t ) and x ( t ) solvable in quadratures.If we know Lax representation of original equations of motion (1.2) ddt L ( x ) = [ L ( x ) , A ( x )] , (1.7)for two N × N matrix functions L ( x ) and A ( x ) on phase space depending on the auxiliary spectralparameter x , then we can directly integrate equations (1.7) in terms of theta-functions by using finite-gap integration theory. Indeed, the time-independent spectral equation L ( x ) ψ ( x, y ) = y ψ ( x, y )allows us to represent the Baker-Akhiezer vector function ψ in terms of the Riemann theta functionon a nonsingular compactification of the spectral curve defined by equationΓ : f ( x, y ) = det( L ( x ) − y ) = 0 . The second equation defines time ddt ψ ( x, y ) = −A ( x ) ψ ( x, y ) , (1.8)similar (1.1), and evolution of original variables p i ( t ) and M i ( t ) with respect to this time, see detailsin [4, 40].According to Krichever and Sklyanin separated variables are abscissas of poles P i = ( x i , y i ) of theproperly normalized Baker-Akhiezer function ψ and, therefore, four algebraic functions φ i in Abel’sequations (1.6) are associated with a common spectral curve Γ [29].For instance, separated variables x , x for the Clebsch system, which were proposed by K¨otter in[19], are poles of the Baker-Akhiezer function for 2 × n = 2of the n -site elliptic Gaudin model describing integrable completely anisotropic spin systems withlong-range interactions. Separated variables for the n -site elliptic magnets were obtained in [28, 30]by using the generic Sklyanin approach [29].We want to learn how to get fewer separated variables than in [29], that is, find one separationvariable for each of the spectral curves associated with different Lax matrices for the Clebsch system. According to [24], the following 3 × L ( λ ) = A + λ M + λ p ⊗ p = λ p + a λ p p + M λ λ p p − M λλ p p − M λ λ p + a λ p p + M λλ p p + M λ λ p p − M λ λ p + a , (2.1)has a natural generalization to gl ∗ ( n, R ) case. Here M ∈ so ∗ (3) is a skew-symmetric matrix associatedwith vector M ∈ R and second Lax matrix is equal to A = − λ ( A + λ M) , or A = 2 λ p ⊗ p . ψ is the eigenvector of first Lax matrix L ( λ ) ψ ( λ, µ ) = µ ψ ( λ, µ ) (2.2)considered as a function on a spectral curve defined by equationΓ : f ( µ, λ ) = det( L ( λ ) − µ ) = 0 . In our case spectral curve ΓΓ : f ( µ, λ ) = c λ + ( c µ − Hµ + K ) λ + det( A − µ ) = 0 (2.3)is a 2-fold covering of elliptic curve E at λ = yE : f ( µ, y ) = c y + ( c µ − Hµ + K ) y + det( A − µ ) = 0 . (2.4)Partial separation of variables associated with this curve is discussed in [21].According to Horozov and van Moerbeke [14] there are exactly six different genus 2 curves relatedto variety P rym (Γ; E ) via 2:1 isogenies, see [10, 27] and references within. Affine coordinates of thereduced divisors of degree two on these genus two curves satisfy Abel’s equations (1.6) involving tworegular differentials on these curves, see K¨otter [19].We want to discuss the algebra-geometric nature of prime divisor or divisor of degree one on thisspectral curve appearing in [16]. As a first step in this direction we want to study a class of linearlyequivalent divisors D = P P i of poles P i = ( µ i , λ i ) of the vector Baker-Akhiezer function ψ (2.2) withsome fixed normalization ~α ~α · ψ = N X i =1 α i ψ i = 1 , Because ψ j = ( L ( λ ) − µ ) ∧ jk ( ~α · ( L ( λ ) − µ ) ∧ ) k , ∀ k = 1 , , , where the wedge denotes the adjoint or co-factor matrix, poles of the Baker-Akhiezer function ψ ( λ, µ )are common zeroes of the four algebraic equationsdet( L ( λ ) − µ ) = 0 , and ~α · ( L ( λ ) − µ ) ∧ ) = 0 . (2.5)For general normalization ~α algebraic equations (2.5) have five solutions P i = ( µ i , λ i ), i = 1 ..
5, whereasat ~α = ( p , p , p )there are only three solutions of equations (2.5). We chose this normalization because at c = 0 poles ofthe Baker-Akhiezer function are sphero-conical or elliptic coordinates on a sphere, which are separatedvariables for Neumann’s system.Three solutions of equations (2.5) define three points P i = ( µ i , λ i ) on spectral curve Γ (2.3) whichform positive divisor D = P + P + P , deg D = 3on the genus three algebraic curve Γ, g (Γ) = 3. According to the Riemann-Roch theorem, dimensionof linear system | D | , which is the set of all the nonnegative divisors which are linearly equivalent to D | D | = { D ′ ∈ Div(Γ) | D ′ ∼ D and D ′ > } , is equal to dim | D | = deg D − g(Γ) = 3 − , see definitions and other details in textbook [22].In our case, dim | D | = 0 and, therefore, divisor D is a unique reduced divisor in the correspondingclass of equivalent divisors. Thus, we can not reduce this divisor to prime divisor of degree one on a4enus three curve Γ, which is discussed in [16]. According to [16] to avoid this problem we have totake arbitrary point P i from the support of semi-reduced divisor D of order three.On elliptic curve E , reduced divisor consists of one point and we first assumed that semi-reduceddivisor D on E can be reduced to divisor defining one separated variable from [16]. Indeed, threepoints P i = ( µ i , y i ), where λ i = y i , define the positive semi-reduced divisor on the genus one ellipticcurve E (2.4) D = P + P + P , deg D = 3so that dim | D | = deg D − g( E ) = 3 − . According to the Riemann-Roch theorem, we can reduce this semi-reduced divisor D on E , to equivalentdivisors D ′ and D ′′ : D → D ′ → D ′′ , dim | D | = 2 , dim | D ′ | = 1 , dim | D ′′ | = 0 . Below we study these semi-reduced D ′ and reduced D ′′ divisors on the elliptic curve E (2.4) . For generic normalization ~α the last three equations in (2.5) are cubic polynomials in λe i ( µ, λ ) = ( ~α × p ) i λ + e (2) i λ + e (1) i λ + e (0) i = 0 , and solutions of these equations for λ and µ are roots of fifths order polynomials A ( λ ) = 0 , B ( µ ) = 0 , which can be easily obtained using modern computer algebra systems.If ~α = ( p , p , p ), then vector ( ~α × p ) = 0 is equal to zero and algebraic equations (2.5) have thefollowing form f ( µ, λ ) = c λ + ( c µ − Hµ + K ) λ + det( A − µ ) = 0 ,e ( µ, λ ) = M c λ + (cid:0) ( M p − M p ) µ + a M p − a M p (cid:1) λ + p (cid:0) µ − ( a + a ) µ + a a (cid:1) = 0 ,e ( µ, λ ) = M c λ + (cid:0) ( M p − M p ) µ + a M p − a M p (cid:1) λ + p (cid:0) µ − ( a + a ) µ + a a (cid:1) = 0 ,e ( µ, λ ) = M c λ + (cid:0) ( M p − M p ) µ + a M p − a M p (cid:1) λ + p (cid:0) µ − ( a + a ) µ + a a (cid:1) = 0 . Solutions of these equations for λ and µ are roots of the cubic polynomials A ( λ ) = − b c λ + (cid:16) a ( p M − p M )( c M + c p )+ a ( p M − p M )( c M + c p ) + a ( p M − p M )( c M + c p ) (cid:17) λ − (cid:16) ( p + p )( a − a )( a − a ) p M + ( p + p )( a − a )( a − a ) p M + ( p + p )( a − a )( a − a ) p M (cid:17) λ − p p p ( a − a )( a − a )( a − a ) (2.6)and B ( µ ) = b µ + b µ + b µ + b , (2.7)5here b = c ( M + M + M ) − c ,b = 2 c ( a p M + a p M + a p M ) − c ( a M + a M + a M ) − a ( M p − M p ) − a ( M p − M p ) − a ( M p − M p ) ,b = ( a a + a a + a a ) b − p ( a + a ) (cid:16) ( a − a ) M + ( a − a ) M (cid:17) − p ( a + a ) (cid:16) ( a − a ) M + ( a − a ) M (cid:17) − p ( a + a ) (cid:16) ( a − a ) M + ( a − a ) M (cid:17) ,b = a a a c − ( a M + a M + a M )( a a p + a a p + a a p ) . The roots of polynomials A ( λ ) and B ( µ ) determine poles P i = ( µ i , λ i ) of the Baker-Akhiezer functionon spectral curve Γ (2.3).To determine the corresponding poles P i = ( µ i , y i = λ i ) on e elliptic curve E we can replace threeequations depending on µ and λe k ( µ, λ ) = e (2) k λ + e (1) k λ + e (0) k = 0 , k = 1 , , , with three equations depending on µ and y = λ E ( µ, y ) = e (1)2 e − e (1)1 e = 0 , E ( µ, y ) = e (1)3 e − e (1)2 e = 0 , E ( µ, y ) = e (1)1 e − e (1)3 e = 0 . Equations of motion for the Clebsch system (1.2), Hamiltonians H , (1.3) and polynomial B ( µ ) areinvariant under cyclic permutations of the indexes. Thus, we consider a linear combination of E ik g( µ, y ) = X ε ijk p i E jk = p E + p E + p E = c Q ( µ ) y − P ( µ ) = 0 , which is also invariant under permutations. Here Q ( µ ) and P ( µ ) are polynomials of first and secondorder in µ Q ( µ ) = − µb + ( a M + a M + a M ) c − c X , P ( µ ) = P µ + P µ + P , where X = a p M + a p M + a p M . (2.8)and P = ( a p + a p + a p ) c − c X , P = (cid:16) ( a + a + a ) c − a p − a p − a p (cid:17) X − (cid:16) ( a a + a a + a a ) c − ( a a p + a a p + a a p ) (cid:17) c , P = a a a c c − ( a a p + a a p + a a p ) X .
It is easy to prove that three points P i = ( µ i , y i ) with coordinates B ( µ ) = b ( µ − µ )( µ − µ )( µ − µ ) = 0 and y i = P ( µ i ) c Q ( µ i ) , i = 1 , , , lie on elliptic curve E (2.4). Formal sum of these points D = P + P + P , deg D = 36s a semi-reduced divisor of degree three on the elliptic curve E . Abscissas of these three points P , P and P do not commute to each other { µ i , µ k } 6 = 0 , i, k = 1 , , , i = k , with respect to the Lie-Poisson bracket (1.4) and, therefore, Poisson brackets for the first Mumford’scoordinate of semi-reduced divisor DU ( µ ) = MakeMonic B ( µ ) = ( µ − µ )( µ − µ )( µ − µ )have the form { U ( µ ) , U ( µ ′ ) } 6 = 0 , µ, µ ′ ∈ C , similar to the Sklyanin calculations for elliptic magnets [28, 30].Equations of motion for three poles P , P and P can be obtained from equation (1.8) ddt ψ ( µ, λ ) = −A ψ ( µ, λ ) , which determines evolution of the Baker-Akhiezer function ψ . Similar to Abel’s equations for theKowalevski top from [5] and the Clebsch system from [21], these Abel’s equations involve one regulardifferential on elliptic curve E and two Prym differentials associated with 2-fold covering of E .In [28, 30] Sklyanin proposed to use action of the torsion subgroup in generalized Jacobian of Γ D = P + P + P → ˜ D = ˜ P + ˜ P + T , deg ˜ D = 3 , in order to get divisor ˜ D with commuting Mumford’s coordinates { ˜ U ( µ ) , ˜ U ( µ ′ ) } = 0 , µ, µ ′ ∈ C . Here T is an arbitrary 2-torsion point of spectral curve Γ. We briefly discuss this approach in lastSection devoted to Neumann’s system. Following Abel’s idea [1] let us consider variable points of intersection of elliptic curve E with a familyof curves depending on time Υ : g( µ, y ) = c Q ( µ ) y − P ( µ ) = 0 . Substituting y = P ( µ ) c Q ( µ )into the elliptic curve equation (2.4) f ( µ, y ) = c y + ( c µ − Hµ + K ) y + det( A − µ ) = 0we obtain Abel’s polynomial Ψ = θ B ( µ ) B ′ ( µ )which determines an intersection divisor D + D ′ + D ∞ = ( P + P + P ) + ( P + P ) + D ∞ = 0 , (2.9)where D ∞ is a linear combination of the points at infinity.Abscissas of points P and P are the roots of polynomial B ′ ( µ ) = b ′ ( µ − µ )( µ − µ ) = (cid:16) c − c c H + ( a + a + a ) c c − c X (cid:17) µ + (cid:16) c c K − ( a a + a a + a a ) c c + ( c H − c ) X (cid:17) µ + a a a c c − c KX + c X . y i = P ( µ i ) c Q ( µ i ) , i = 4 , . Affine coordinates of points P and P are functions on integrals of motion c , c , H, K and only onevariable x ( t ) = X (2.8) in (1.5) so that { µ , H } { µ , K } − { µ , K } { µ , H } = 0 , { µ , µ } 6 = 0 , and µ − µ = c ( y − y ) , (2.10)i.e. points P and P move along a straight line and slope of this line is equal to c − . In order to directly apply the Euler [12] and Abel [1] formulae we have to rewrite equation (2.4) in thefollowing form z = a µ + a µ + a µ + a µ + a using birational transformation y = z − c µ − µH + K c . Then we consider variable points of intersection of E with a family of curves depending on timeΥ ′ : z = √ a µ + b µ + b , where a = c c . Here b and b are coefficients of the interpolating polynomial which is defined by equations z = √ a µ + b µ + b , z = √ a µ + b µ + b , where ( µ , z ) and ( µ , z ) are abscissas and ordinates of points P and P lying on auxiliary curve Υ ′ .The corresponding intersection divisor of curves E and Υ ′ has the following form D ′ + D ′′ + D ∞ = ( P + P ) + P + D ∞ = 0 . According to [1] abscissa µ of point P is equal to µ = − µ − µ − √ a + b − a √ a − a ≡ νυ . (2.11)In the Clebsch case it is a function on the integrals of motion with a numerator ν = c a a a − c ( a + a + a ) K + c (cid:0) c ( a + a + a ) + H K ) − c (cid:0) a + a + a ) H − K (cid:1) c + c c (cid:0) c ( a + a + a ) + H ) − c c H + c and denominator υ = c (cid:16) c ( a a + a a + a a ) − c (cid:0) ( a + a + a ) H + K (cid:1) + c (cid:0) a + a + a ) c + H (cid:1) − c c H + 2 c (cid:17) . Such constant reduced divisors also appear for various superintegrable systems [36, 37, 38] and for theKowalevski top [39].Points P , P and P lie on parabola Υ ′ of a constant size, which leads to vanishing of determinant (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ z µ z µ z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 08nd to vanishing of the corresponding Abel’s integral relation [1] dµ z + dµ z + dµ z = 0 . Because abscissa µ (2.11) is fixed, i.e. dµ = 0, we immediately obtain the following equation forabscissas of points from the support of semi-reduced divisor D ′ = P + P (2.9) dµ z + dµ z = 0 . (2.12)Because points P , P , P and P , P belong to the intersection divisor (2.9) we also have equation dµ z + dµ z + dµ z = 0involving a regular differential on elliptic curve E . This equation is a consequence of the fact that theunique reduced divisor in this class of equivalent divisors is a constant of motion.Variables µ , depend on integrals of motion c , c , H, K and the one time-dependent variable X (2.8) and, therefore, they satisfy geometric equation (2.10). Thus, only one of the abscissas µ or µ could be candidate for a separated variable from [16] up to the group operations on E .Let us introduce elliptic coordinates u , on e ∗ (3) using standard definition p z − a + p z − a + p z − a = c ( z − u )( z − u )( z − a )( z − a )( z − a ) = 0 , and the corresponding conjugated momenta, see [31]. These coordinates satisfy the ”non-autonomous”Abel’s equations ˙ u √ ( u − a )( u − a )( u − a )( u − Hu + ˜ K ) + ˙ u √ ( u − a )( u − a )( u − a )( c u − Hu + ˜ K ) = 0 , u ˙ u √ ( u − a )( u − a )( u − a )( c u − Hu + ˜ K ) + u ˙ u √ ( u − a )( u − a )( u − a )( c u − Hu + ˜ K ) = 4 , where ˜ K = K − c X ( t ) is a function on time. In the next Section we consider the corresponding”autonomous” Abel’s equations at c = p M + p M + p M = 0, when function ˜ K = K − c X ( t )becomes a constant ˜ K = K . At the special case c = p + p + p = 1 , c = p M + p M + p M = 0the Clebsch system is equivalent to the well-studied Neumann’s problem [17] describing motion of aparticle on the unit sphere in the field of quadratic potential, see algebro-geometric description of thissystem in textbooks [9, 23].Sphero-conical coordinates u and u on the sphere are defined through equation p z − a + p z − a + p z − a = ( z − u )( z − u )( z − a )( z − a )( z − a ) = 0 , which implies that c = P p = 1. Similar to elliptic coordinates in R , these coordinates in S arealso orthogonal and only locally defined. They take values in the intervals a < u < a < u < a ,
9o that p i = s ( u − a i )( u − a i )( a j − a i )( a m − a i ) , i = j = m . If π , are canonically conjugated momenta { u , π } = { u , π } = 1with respect to the Poisson brackets (1.4), then M i = 2 ε ijm p j p m ( a j − a m ) µ − µ (cid:16) ( a i − µ ) π − ( a i − µ ) π (cid:17) . At c = 1 and c = 0 Hamiltonians (1.3) are the second order polynomials in momenta H = 4 ϕ π µ − µ − ϕ π µ − µ + µ + µ , K = 4 µ ϕ π µ − µ − µ ϕ π µ − µ + µ µ , where we for brevity denote ϕ k = det( A − µ k ) = ( a − µ k )( a − µ k )( a − µ k ) , k = 1 , . In this partial case, spectral curve Γ (2.3) is the genus two hyperelliptic curve which can be rewrittenin the following formΓ : (cid:16) det( µ − A ) χ (cid:17) = ( µ − H µ + K ) det( µ − A ) , χ = − λ − . (3.1)Below we will study the evolution of semi-reduced and reduced divisors on this curve. At c = 0 cubic polynomial A ( λ ) (2.6) becomes quadratic polynomial on χ = λ − A ( χ ) = √ ϕ ϕ ( χ − π )( χ − π ) . Two roots of this polynomial define separated variables and reduced divisor of degree two D = P + P , | D | = 0 , P = ( u , π ) , P = ( u , π )on the genus two spectral curve Γ (3.1). Thus, normalization ~α = ( p , p , p ) can be called propernormalization of the Baker-Akhiezer function associated with 3 × c = 0 cubic polynomial B ( µ ) (2.7) remains a cubic polynomial on µB ( µ ) = b ( µ − µ )( µ − µ )( µ − µ ) = 4( µ − u )( µ − u ) ϕ π u − u − µ − u ) ( µ − u ) ϕ π u − u . (3.2)Three roots of this polynomial µ , µ and µ define a semi-reduced divisor of poles on the genus twospectral curve Γ (3.1) D = P + P + P , P = ( u , π ) , P = ( u , π ) , P = ( µ , χ ) , where affine coordinates of the third pole are functions on separated variables u , and , π , µ ≡ K − u u H − u − u , χ = 4 π π H − u − u r ϕ ϕ ϕ (3.3)Thus, the same normalization ~α = ( p , p , p ) is not a proper normalization according to the Sklyaninmagic recipe [29], see also discussion of the excess poles for XY Z magnet [28, 30].Summing up, we can see that normalization ~α of the Baker-Akhiezer function ψ in equation( ~α, ψ ) = 1 could be simultaneously proper and improper normalization.10 .2 Evolution of semi-reduced divisor Evolution of three poles P , P and P along the curve Γ (3.1) is determined by Abel’s equationsΩ ( P ) ˙ µ + Ω ( P ) ˙ µ = 0 Ω ( P ) ˙ µ + Ω ( P ) ˙ µ = 4 , (3.4)and Ω ( P ) ˙ µ = 4 X √ ϕ ( H − u − u ) , where regular differentials on Γ (3.1) have the formΩ = 1det( µ − A ) χ , Ω = µ det( µ − A ) χ , and variable X (2.8) at c = 0 is equal to X = a p M + a p M + a p M = 2 √ ϕ ϕ ( π − π ) u − u . Abscissa and ordinate of the third point P are functions of elliptic coordinates and integrals of motion,and, therefore, two Abel’s equations (3.4) completely determine the evolution of the semi-reduceddivisor of poles D = P + P + P , deg D = 3 , dim | D | = deg D − g = 3 − . (3.5) According to the Riemann-Roch theorem there is a unique reduced divisor D ′ on genus two hyperellipticcurve Γ (3.1) D ′ = P + P , deg D ′ = 2 , dim | D ′ | = deg D ′ − g = 2 − , which is equivalent with semi-reduced divisor D (3.5).Following Abel [1], we can identify reduced divisor D ′ with a part of the intersection divisor D + D ′ + D ∞ = 0of fixed hyperelliptic curve Γ with parabola Υ involving three points P , P and P on a projectiveplane. Here D ∞ is a linear combination of points at infinity and parabola Υ : y = V ( µ ) is defined byinterpolation polynomial V ( µ ) = ( µ − µ )( µ − µ ) µ − µ )( µ − µ ) χ + ( µ − µ )( µ − µ ) µ − µ )( µ − µ ) χ + ( µ − µ )( µ − µ ) µ − µ )( µ − µ ) χ . (3.6)Substituting y = V ( µ ) into (3.1) we obtain Abel’s polynomial Ψ defining abscissas µ , of points P and P Ψ = θB ( µ ) B ′ ( µ ) , B = b ( µ − µ )( µ − µ )( µ − µ ) , B ′ = ( µ − µ )( µ − µ ) , see Abel’s original calculations for genus two hyperelliptic curves [1] or [33] and references within. Inour case B ′ ( µ ) = µ + ( µ + µ + µ − a − a − a − H ) µ + µ + µ + µ − ( a + a + a )( µ + µ + µ )+( a + a + a − µ − µ ) H + a a + a a + a a + 2 µ µ , where µ = u , µ = u , µ = K − u u H − u − u . D ′ = P + P satisfy the same Abel’s equations (3.4)Ω ( P ) ˙ u + Ω ( P ) ˙ u = 0 and Ω ( P ) ˙ u + Ω ( P ) ˙ u = 4 , but { µ , µ } 6 = 0 , similar to one of the Kowalevski separated variables [15]. To get commuting separated variablesKowalevski used translation or shift of the corresponding semi-reduced divisor on elliptic curve byfixed element in Jacobian, see discussion in [39]. Spectral curves Γ (2.3) and Γ (3.1) have three finite 2-torsion points[2] T j = 0 , T = ( a , , T = ( a , , T = ( a , m ] T is a scalar multiplication on m ∈ Z + . These points form 2-torsion subgroups in thegeneralized Jacobian which is a commutative algebraic group associated to a curve with divisors.Torsion subgroups are widely used in post-quantum cryptography based on isogenies [8], and now wewant to apply the corresponding algorithms in classical mechanics.Let us consider translation of the semi-reduced divisor D (3.5), which exchanges point P and oneof the torsion points T i , for instance translation D → ˜ D = D + D T = P + P + P + ( T − P ) = P + P + T . Because the so-called Mumford’s coordinates (
U, V ) can be directly obtained from polynomial B ( µ )(3.2) and interpolation function V ( µ ) (3.6) U = MakeMonic B ( µ ) , V = V ( µ ) mod U , we can easy obtain Mumford’s coordinates ( ˜
U , ˜ V ) of semi-reduced divisor ˜ D using computer imple-mentation of the standard Cantor’s algorithm [6]. The first Mumford’s coordinate is equal to˜ U = ( µ − u )( µ − u )( µ − a )so that { ˜ U ( µ ) , ˜ U ( µ ′ ) } = 0 , µ, µ ′ ∈ C . Sklyanin did a similar translation for the elliptic magnets [28, 30], but at c = 0 it is more a complicatedoperation.Reduction of semi-reduced divisor˜ D = P + P + T = ( P + P ) + ( T + 0) → ˆ D = ˆ P + ˆ P , | ˆ D | = 0 , can be considered as action of the finite torsion subgroup in the generalized Jacobian. It can be per-formed by using second part of the Cantor algorithm [6]. As a result one gets canonical transformation τ on a cotangent bundle to the unit sphere T ∗ S τ : T ∗ S → T ∗ S which maps sphero-conical variables to new separated variables for the Neumann system τ : ( u , u , π , π ) → (ˆ u , ˆ u , ˆ π , ˆ π )so that { ˆ u , ˆ u } = { ˆ π , ˆ π } = 0 , { ˆ u i , ˆ π j } = δ ij . These new variables of separation satisfy the same Abel’s equations (3.4)Ω ( ˆ P ) ˙ˆ u + Ω ( ˆ P ) ˙ˆ u = 0 Ω ( ˆ P ) ˙ˆ u + Ω ( ˆ P ) ˙ˆ u = 4 . All the details can be found in [34, 35], where we prove that affine coordinates of the reduced divisorˆ D = ˆ P + ˆ P are simultaneously separated variables for the Neumann’s problem and the Chaplyginsystem on the sphere with velocity-dependent potential.12 Conclusion
In [16] authors found separated variables ( x , y ) and ( x , y ) which form two prime divisors of degreeone or divisors with deficiency on the two different genus three spectral curves D = ( x , y ) ∈ Γ and D ′ = ( x , y ) ∈ Γ ′ , g(Γ) = g(Γ ′ ) = 3 , deg D = deg D ′ = 1 . Both spectral curves Γ , Γ ′ are the twofold coverings of elliptic curves E, E ′ and, therefore, separatedvariables form two unique reduced divisors on these elliptic curves similar to well-known Euler’s twofixed centers problem.In this note, we try to reproduce one of these reduced divisors using the Euler-Abel constructionof reduced divisors, symmetries of the Clebsch system, and poles of the Baker-Akhiezer function onthe spectral curve of the 3 × P or P from the support of semi-reduced divisor D ′ = P + P (2.9) as the desiredseparated variable.Construction of a separated variable and the corresponding prime divisor on the spectral curve of2 × References [1] Abel N. H.,
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