Elliptic curve arithmetic and superintegrable systems
aa r X i v : . [ n li n . S I] F e b Elliptic curve arithmetic and superintegrable systems
A.V. Tsiganov
St.Petersburg State University, St.Petersburg, Russiae–mail: [email protected]
Abstract
Harmonic oscillator and the Kepler problem are superintegrable systems which admit moreintegrals of motion than degrees of freedom and all these integrals are polynomials in momenta. Wepresent superintegrable deformations of the oscillator and the Kepler problem with algebraic andrational first integrals. Also, we discuss a family of superintegrable metrics on the two-dimensionalsphere, which have similar first integrals.
In 1757-1759 Euler created the theory of elliptic integrals which in turn gave birth to the Abel theoryof Abelian integrals, to the Jacobi theory of elliptic functions, to the Riemann theory of algebraicfunctions, etc. This paper considers two themes in algebraic geometry and elliptic curve cryptographydescended from Eulers work: elliptic curve arithmetic and algebraic integrals of Abel’s equations, seeProblems 81-84 in Euler’s textbook [12].In 1760-1767 Euler applied this mathematical theory to searching of algebraic trajectories in thetwo fixed centers problem. In [9, 10, 11] he reduced equations of motion to one equation definingtrajectory dr √ R + ds √ S = 0 , identified algebraic integral of this equation with a partial first integral in the phase space and sepa-rated partial algebraic trajectories from transcendental trajectories. In particular, Euler obtained anadditional first integral for the superintegrable Kepler problem, which is a partial case of two fixedcenters problem, in terms of elliptic coordinates on the plane. In [18] Lagrange proved that equationsof motion for the two fixed centers problem with three degrees of freedom are separable in prolatespheroidal coordinates, considered generalized two fixed centers problem and then used algebraic inte-gral of Abel’s equation for searching algebraic trajectories in this generalized two fixed centers problem,see [18, 19] and comments by Serret [23] and Darboux [7]. Modern description of algebraic trajectoriesin the two centers problem may be found in [8].Thus, if generic or partial equations of motion are reduced to Abel’s equation on the ellipticcurve, then we have additional partial or complete first integral obtained by Euler in his solution ofProblems 81-84 in [12]. In [13, 24, 26, 27] we used Euler’s construction in order to classify knownsuperintegrable systems with additional first integrals which are polynomials in momenta. This paperconsiders superintegrable deformations of the Kepler problem, harmonic oscillators on the plane andgeodesics on the sphere which have algebraic and rational additional integrals of motion. Let us consider smooth nonsingular elliptic curve X on the projective plane defined by an equation ofthe form X : y = f ( x ) , f ( x ) = a x + a x + a x + a x + a . (1.1)The prime divisors are points on X , denoted P i = ( x i , y i ), including point at infinity P ∞ , which playsthe role of neutral element 0 in arithmetic of elliptic curves.1n 1757 Euler proved an addition formulae for elliptic integrals, in modern terms he proved thatby adding two points on X ( x , y ) + ( x , y ) = ( x , y )one gets the third point with the following abscissa and ordinate x = − x − x − b b + b − a b b − a , and y = −P ( x ) , (1.2)where P ( x ) = b x + b x + b = √ a ( x − x )( x − x ) + ( x − x ) y x − x + ( x − x ) y x − x . Then Euler explicitly defined doubling of divisor[2] P = ( x , y ) + ( x , y ) = ([2] x , y ]) , i.e. point of X with coordinates[2] x = − x − b b + b − a b b − a , [2] y = −P (cid:0) [2] x (cid:1) , P ( x ) = b x + b x + b = √ a ( x − x ) + ( x − x )(4 a x + 3 a x + 2 a x + a )2 y + y , (1.3)and tripling of divisor [3] P = ([2] x , y ]) + ( x , y ) = ([3] x , y ]) , i.e. point of X with coordinates[3] x = − x − a − b b a − b , [3] y = −P (cid:0) [3] x (cid:1) , P ( x ) = b x + b x + b = − ( x − x ) (4 a x + 3 a x + 2 a x + a ) y + ( x − x ) (cid:16) x (cid:0) a x + 3 a x + a (cid:1) − a x + a x + a (cid:17) y + y , (1.4)and described an algorithm for multiplication on any integer m , see Problem 83 in [12]. Later Abelused elliptic curve point multiplication in proving his theorem on m -division points of the lemniscatewhen he introduced some pre-image of the division polynomials. Modern computer algorithms forperforming addition and multiplication on elliptic curve are discussed in [3, 15, 34].Lagrange proves Euler’s addition equation introducing time t and motion of two points P ( t ) and P ( t ) on X governed by the following Newton equations d x dt = 2 y , d x dt = 2 y , see details in [14], p.144. In this terms Eulers solutions of Problems 81-84 [12] can be reformulated inthe following form: If two points P = ( x , y ) and P ( x , y ) move along a curve y = f ( x ), there isan algebraic constraint on their motion with the property that Z ω ( x , y ) dx + Z ω ( x , y ) dx can be expressed, for any differential ω ( x, y ), in terms of elementary functions in coordinates x , y and x , y when these coordinates satisfy the algebraic constraint. In his famous Paris memoir [1],Abel states Euler’s conclusion in almost exactly this form as a preamble to his famous theorem.2n Problem 81 Euler calculates algebraic constraint associated with addition of points and provesthat C = (cid:18) y − y x − x (cid:19) − a ( x + x ) − a ( x + x ) (1.5)is the general integral of the differential relation dx y + dx y = 0 (1.6)when C is a constant and particular integral of dx y + dx y + dx y = 0when C = 2 a x + a x + a − √ a y . (1.7)In 1863 Clebsch proposed geometric approach to construction of algebraic constraints, closely interwo-ven with the intersection theory, which was continued by Brill and Noether in 1857 and formalized byPoincar´e in 1901 and Severy in 1914, see classical textbooks [14, 16], review [4] and modern discussionin [6].Then in Problem 83 Euler proves that C mn = (cid:18) [ m ] y − [ n ] y [ m ] x − [ n ] x (cid:19) − a (cid:0) [ m ] x + [ n ] x (cid:1) − a (cid:0) [ m ] x + [ n ] x (cid:1) (1.8)is the general integral of the differential relation m dx y + n dx y = 0 , (1.9)associated with scalar multiplication of points on integer numbers m, n and addition of the obtainedresults. Here and below we write the coordinates of [ m ]( x, y ) as ([ m ] x, [ m [ y ]).In fact Euler proposed only an algorithm of computations, because explicit expression for C mn isa cumbersome formula. For instance, we have C = AB = 16 A y + 32 A y + 16 A y + 16 f ′ A y + 4 f ′ A y + 4 f ′ A y + f ′ A ( B y + B y + B y + B ) (1.10)where f ′ = df ( x ) /dx is a derivative of the polynomial f ( x ) from (1.1) at point x = x , B = 8 √ a y − (cid:0) a x ( x + 2 x ) + a (2 x + x ) + a (cid:1) y − √ a ( x − x ) f ′ y + f ′ and A = a x + a x + a − √ a y , A = − a / x x − a / (2 a x x + a x +2 a x + a )+(4 a x + a ) y , A = − x + x ) a − (cid:0) a ( x + x )+ a ( x + x ) − a x (cid:1) a + a a − a − a (2 x + x ) a − x (4 x − x ) a +4 a y , A = x ( x + x ) a / + (cid:0) a ( x + x )(5 x +4 x x + x )+8 a x (2 x + x )+3 a (2 x + x ) (cid:1) a / + (cid:0) a (2 x +7 x x + x x +2 x )+4 a x + a a (3 x + x )+ a a (9 x +4 x x + x )+2 a a (cid:1) a / − (cid:0) a x +2(6 a x − a ) a + a (3 a x + a ) (cid:1) y − a / (4 a x + a ) y , = − x x (3 x +2 x ) a − x (cid:0) a (2 x + x )+ a x (3 x +2 x )+ a x (3 x +6 x x +4 x x +2 x ) (cid:1) a − (cid:0) a +8 a a x + a a (18 x +4 x x +2 x ) − a x +4 a a x x + a x (4 x +8 x x +8 x x + x ) (cid:1) a − a (3 a x + a a )+ a + a a (4 x + x )+ a a (3 x +4 x x − x ) − a ( x − x )( x − x x − x )+2 a / (cid:0) a x +4 x (5 a x + a x − a ) a +6 a x +4 a a x − a a + a (cid:1) y +(16 a x +8 a a x + a ) y , A = − a / x (2 x + x )+4 a / (4 a x + a x x + a x x +3 a x + a x )+ a / (cid:0) a x +4 a x x − a x +6 a a x +2 a a x +2 a (cid:1) − (cid:0) a x +4 a ( a x + a ) − a (cid:1) y , A = x + x ) a +4(2 a x + a x + a ) a + a +8 a / y . If m = 3 and n = 1, then one gets an algebraic integral with similar structure C = AB = A y + A y + . . . + A ( B y + B y + B y + B ) , (1.11)where f ′′ = d f ( x ) /dx and B = 64(3 a x + a x + a ) y − f ′′ (cid:0) a x ( x + 3 x ) + 3 a x ( x + x ) + a (3 x + x ) + 2 a (cid:1) y +8 f ′ (cid:0) a x (3 x − x ) + 3 a x x + a ( x + x ) + a (cid:1) y + f ′ ( x − x ) . For brevity we omit expressions for functions A k , which are polynomials in coordinates x , and y .They can be obtained using any computer algebra system.In the next Section we apply these Euler’s results to construction of rational and algebraic functionscommuting with Hamilton functions of superintegrable systems with two degrees of freedom. The traditional way of writing an elliptic curve equation of is to use its short or long Weierstrass form.In elliptic curve cryptography we can find also other forms of the elliptic curve such as Edwards curves,Jacobi intersections and Jacobi quartics, Hessian curves, Huff curves, etc.Following [20] we begin with nonsingular elliptic curve X defined by a short Weierstrass equation y = f ( x ) , f ( x ) = x + ax + b , (2.12)and arithmetic equation on X P + P + P = 0 . Here P , P and P are intersection points of X with a straight line, see standard picture in Figure 1and in textbooks [3, 15, 34].Figure 1: Addition of points P + P + P = 0 on the elliptic curve.4sing coordinates of points P = ( x , y ) and P ( x , y ) we can easily define coordinates of the thirdpoint x = λ − ( x + x ) , y = y + λ ( x − x ) , λ = y − y x − x . In this case Euler’s integral (1.5) coincides with abscissa of P , i.e. C = x and it is the requiredalgebraic constraint for motion of two points P ( t ) and P ( t ) along the elliptic curve X .Let us also present a well-known expression for the elliptic curve point multiplication on anypositive integer m : [ m ]( x, y ) ≡ (cid:2) m ] x, [ m ] y ) = (cid:18) x − ψ m − ψ m +1 ψ m , ψ m ψ m (cid:19) where ψ m are the so-called division polynomials in Z [ x, y, a, b ], which are the ratio of two Weierstrass σ -functions, see [3, 20, 34]. It is easy to see that abscissa of [ m ] P is a rational function strictly interms of x whereas ordinate has the form yR ( x ), where R ( x ) is a rational function.If we identify abscissas and ordinates of points P and P with canonical coordinates on the phasespace x , = u , , y , = p u , , where { u , u } = 0 , { p u , p u } = 0 , { u i , p u j } = δ ij , and solve a pair of equations y i = f ( x i ) , i = 1 , a, b , we obtain two functions on thephase space a = p u u − u + p u u − u − u − u u − u , b = u p u u − u + u p u u − u + ( u + u ) u u , (2.13)which are in involution with respect to canonical Poisson brackets.Taking H = a as a Hamiltonian, one gets integrable system on the phase space T ∗ R withquadratures ω = Z u du p u + au + b + Z u du p u + au + b = − t and ω = Z du p u + au + b + Z du p u + au + b = const . In physical terms a, b and ω , ω are action-angle variables associated with this motion, whereas Euler’salgebraic constraint (1.5) is an additional first integral [33].Second quadrature in the differential form coincides with (1.6) and, therefore, we have superinte-grable system with additional first integrals which are abscissa and ordinate of the third point P ona projective plane x = (cid:18) p u − p u u − u (cid:19) − ( u + u ) , y = p u + (cid:18) p u − p u u − u (cid:19) ( x − u ) . Functions a, b from (2.13) and functions x , y on the phase space T ∗ R form an algebra of integrals { a, b } = 0 , { a, x } = 0 , { a, y } = 0 , { b, x } = 2 y , { b, y } = 3 x + a , { x , y } = − , in which Weierstrass equation (2.12) plays the role of syzygy y = x + ax + b . In this case two points P ( t ) and P ( t ) move along a curve y = f ( x ) with fixed third point P becauseits abscissa x and ordinate y are additional first integrals, see the picture in Figure 2.5igure 2: Rotation of the straight line with points P ( t ) and P ( t ) around fixed point P .Periodic motion of the points P ( t ) and P ( t ) on two bound pieces of the curve X on the projectiveplane generates motion by algebraic curves in the phase space, similar to algebraic trajectories in thetwo fixed centers problem [9, 10, 11].After canonical transformation of variables u = q − √ q , u = q + √ q , p u = p u − u ) p , p u = p − ( u − u ) p a = p p − q − q , b = p − q p p + q p + 2 q ( q − q ) ,x = p − q , y = p p − p q . Superintegrable Hamiltonian H = a (2.13) belongs to a family of superintegrable Hamiltonianson the plane depending on two integer numbers k , . Indeed, let us consider Hamiltonian H = A k k = (cid:0) k − p u (cid:1) u − u + (cid:0) k − p u (cid:1) u − u − u − u u − u , (2.14)commuting with the following integral of motion B k k = u (cid:0) k − p u (cid:1) u − u + u (cid:0) k − p u (cid:1) u − u + ( u + u ) u u . These functions can be obtained from (2.13) by using non-canonical transformation p u i → k − i p u i , seediscussion in [33]. The corresponding quadratures k Z u du p u + au + b + k Z u du p u + au + b = − t and k Z du p u + au + b + k Z du p u + au + b = const , are related to arithmetic equation on the elliptic curve X [ k ] P + [ k ] P + P = 0 . Here two points [ k ] P ( t ) and [ k ] P ( t ) of degree k and k move along curve X , whereas P is afixed point similar to Figure 2, but instead of line we have to take curve Y defined by equation y = λx k + k − + µ k + k − · · · . 6ccording to Euler Hamiltonians H = A k k are in involution with the additional first integral ofthe form (1.8) x = C k ,k = (cid:18) [ k ] y − [ k ] y [ k ] x − [ k ] x (cid:19) − (cid:0) [ k ] x + [ k ] x (cid:1) , which is abscissa x of fixed point P that is a well-defined rational function on x , y , x , y on theprojective plane.At k = 2 and k = 1 this additional first integral is a rational function of the form C = 4 u − u − u − u ) (cid:0) u − u u + 6 u u + p u p u − p u (cid:1)(cid:0) u − u u + 4 u − p u + 4 p u p u − p u (cid:1) + 64 ( u − u ) (cid:0) u − u u + u + p u p u − p u (cid:1)(cid:0) u − u u + 4 u − p u + 4 p u p u − p u (cid:1) . At k = 3 and k = 1 additional first integral is equal to C = − ( u + u ) + ( p u − p u ) p u + p u ) ( u − u ) + 8 p u A p u + p u ) B − p A p u + p u ) B where B = p u − p u p u + 18 (cid:0) p u − u u + 2 u u − u (cid:1) p u − (cid:0) p u − u + 3 u u − u (cid:1) , and A = (15 u + 19 u ) p u − u + 5 u ) p u p u − u + 19 u )(2 u + u )( u − u ) p u +54 (cid:0) ( u + u ) p u + (2 u + u )( u − u ) (cid:1) p u p u − p u (5 u + u ) (cid:0) p u − (2 u + u )( u − u ) (cid:1) ,A = 2( u + u ) p u − u + 5 u ) p u p u + 54 p u (5 u + u ) (cid:0) p u − (2 u + u )( u − u ) (cid:1) +3 (cid:0) u + u ) p u − (7 u + 16 u u + 13 u )( u − u ) (cid:1) p u − (cid:0) u + u ) p u − (23 u + 38 u u + 11 u )( u − u ) (cid:1) p u p u . Algebraic trajectories for these superintegrable systems are generated by rotation of a parabola anda cubic around fixed point P on the projective plane instead of rotation of the straight line, see thepicture in Figure 2.Similarly we can take superintegrable systems on the plane with Hamiltonians H = p p + V ( q , q )listed in [22, 25, 26] and obtain families of the superintegrable Hamiltonians H k k depending on integernumbers k , , see discussion in [33]. Let us come back to physical systems and introduce elliptic coordinates on the plane following Euler[9] and Lagrange [18, 19]. If r and r ′ are distances from a point on the plane to the two fixed centers,then elliptic coordinates u , are r + r ′ = 2 u , r − r ′ = 2 u .
7f two centres are taken to be fixed at − κ and κ on the first axis of the Cartesian coordinate system,then we have standard Euler’s definition of the elliptic coordinates on the plane q = u u κ , and q = p ( u − κ )( κ − u ) κ . Coordinates u , are curvilinear orthogonal coordinates, which take values only on in the intervals u < κ < u , i.e. they are locally defined coordinates.In terms of elliptic coordinates u , and the corresponding momenta p u , kinetic energy has thefollowing form 2 T = p + p = u − κ u − u p u + u − κ u − u p u , see [7, 19, 23]. By adding separable in elliptic coordinates potentials one gets a Hamiltonian and asecond integral of motion2 H = I = ( u − κ )( p u + V ( u )) u − u + ( u − κ )( p u + V ( u )) u − u , (2.15) I = − u ( u − κ )( p u + V ( u )) u − u − u ( u − κ )( p u + V ( u )) u − u . According to Euler and Lagrange there are an equation defining time u du r ( u − κ ) (cid:16) u I − V ( u )( u − κ ) + I (cid:17) + u du r ( u − κ ) (cid:16) u I − V ( u )( u − κ ) + I (cid:17) = dt . and an equation defining trajectories of motion du r ( u − κ ) (cid:16) u I − V ( u )( u − κ ) + I (cid:17) + du r ( u − κ ) (cid:16) u I − V ( u )( u − κ ) + I (cid:17) = 0 . In these equations I , are the values of integrals of motion, see terminology and discussion in theLagrange textbook [19] and comments by Darboux and Serret [7, 23].The second equation is reduced to Euler’s differential relation (1.6) for the Kepler problem2 H = I = p + p + αr , V i ( u i ) = αu i u i − κ (2.16)when Z du p ( u − κ )( I u + αu i + I ) + Z du p ( u − κ )( I u + αu + I ) = const and for the harmonic oscillator2 H = I = p + p − α ( q + q ) , V i ( u i ) = − α u i (2.17)when Z du q ( u − κ ) (cid:0) I u + α u ( u − κ ) + I (cid:1) + Z du q ( u − κ ) (cid:0) I u + α u ( u − κ ) + I (cid:1) = const . The equation for the harmonic oscillator coincides with equation dp p a p + a p + a p + a ± dq p a q + a q + a q + a = 0 , X (1.1) associated with the Kepler problem is defined by the following polynomialof fourth order in xf ( x ) = I x − αx + ( I − I κ ) x + κ αx − I κ , x , = u , . (2.18)For the harmonic oscillator the corresponding quartic polynomial looks like f ( x ) = α x + ( I − α κ ) x + ( α κ − I κ + I ) x − I κ x , x , = u , , (2.19)when abscissas of divisors x , = u , are equal to the squared elliptic coordinates.Substituting elliptic coordinates and first integrals I , into y = f ( x ) one gets expressions forordinates y , of points P = ( x , y ) and P = ( x , y ). For the Kepler problem we have y = ( u − κ ) p u and y = ( u − κ ) p u , whereas for the harmonic oscillator we obtain y = u ( u − κ ) p u , and y = u ( u − κ ) p u . Substituting coefficients a = I , a = − α and coordinates of divisors x , and y , into the Euleralgebraic relation (1.5) one gets an additional first integral for the Kepler problem C = ( u − κ )( u − κ )( p u − p u ) ( u − u ) , (2.20)which is independent on first integrals I , . This partial integral of motion in the two fixed centersproblem and the corresponding algebraic trajectories were studied by Euler and Lagrange [9, 10, 11,18, 19].Substituting coefficients a = α , a = I − ακ and coordinates of divisors x , and y , intothe Euler algebraic relation (1.5) one gets an additional first integral for the harmonic oscillator C = ( u − κ )( u − κ u ) p u − u u ( u − κ )( u − κ ) p u p u + ( u − κ )( u − κ u ) p u ( u − u ) + α κ ( u + u ) . (2.21)First integrals (2.20) and (2.21) have two different forms in elliptic and especially in Cartesian coordi-nates, but they have a common simple form in terms of coordinates of divisors (1.5).Other superintegrable systems separable in elliptic coordinates on the plane with first integral ofthe form (1.5) are discussed in [5, 13, 26, 27, 28]. Two-dimensional metrics which geodesic flows admit three functionally independent first integrals arecalled superintegrable metrics. Superintegrable metrics with first integrals which are second orderpolynomials in momenta were described by Koenigs [17]. In this Section we consider a well-knownsuperintegrable metric on the two-dimensional sphere with quadratic first integrals, one of which has theEuler form (1.5). In the next Section we present superintegrable metrics on the sphere with algebraicand rational first integrals which are easily constructed using multiplication of points on elliptic curve.We have to underline that our main aim is a search of algebraic trajectories for dynamical systemsfollowing to Euler [9, 10, 11] and Lagrange [18, 19]. Construction of the first integrals is only a toolfor the searching of such trajectories.Let us introduce elliptic coordinates on the two-dimensional sphere S ⊂ R embedded into three-dimensional Euclidean space with Cartesian coordinates q , q and q The elliptic coordinate system u , on the sphere S with parameters α < α < α is defined through equation q z − α + q z − α + q z − α = ( z − u )( z − u ) φ ( z ) , φ ( z ) = ( z − α )( z − α )( z − α ) , z . It implies q + q + q = 1 , which is a standard description of the sphere in R . Similar to elliptic coordinates on the plane, ellipticcoordinates on the sphere are also orthogonal and only locally defined. They take values in the intervals α < u < α < u < α . The coordinates and the parameters can be subjected to a simultaneous linear transformation u i → au i + b and α i → aα i + b , so it is always possible to choose α = 0 and α = 1.Let us consider free motion on the sphere S defined by the Hamiltonian H = p + p + p In elliptic coordinates this Hamiltonian and the corresponding second integral of motion have thefollowing form 2 H = I = φ ( u ) p u u − u + φ ( u ) p u u − u , I = u φ ( u ) p u u − u + u φ ( u ) p u u − u . (2.22)As above, there are two Abel’s equations u du p φ ( u )( u I + I ) + u du p φ ( u )( u I + I ) = 2 dt ,du p φ ( u )( u I + I ) + du p φ ( u )( u I + I ) = 0 . defining time and trajectories according to Euler and Lagrange terminology [19].It is easy to see that second equation du p f ( u ) + du p f ( u ) = 0 , coincides with Euler equation (1.6) on the elliptic curve X defined by y = f ( x ) , f ( x ) = φ ( x )( xI + I ) = a x + a x + a x + a x + a . It allows us to directly obtain an additional integral of motion (1.5) which has the following form inelliptic coordinates C = − (cid:0) α α α − ( α α + α α + α α ) u + ( α + α + α ) u − u u (cid:1) φ p u ( u − u ) − (cid:0) α α α − ( α α + α α + α α ) u + ( α + α + α ) u − u u (cid:1) φ p u ( u − u ) − φ φ p u p u ( u − u ) . (2.23)In modern terms, two-dimensional metric g( u , u ) in (2.22) H = X g ij p u i p u j , g = ( u − α )( u − α )( u − α ) u − u ( u − α )( u − α )( u − α ) u − u ! is a superintegrable metric on the sphere, thus, we have global superintegrable Hamiltonian systemon a compact manifold with closed trajectories. Recall, that problem of finding and describing global10ntegrable Hamiltonian systems on a compact manifold is one of the central topics in the classicalmechanics, see discussion in [2].Summing up, equations of motion for the Kepler system, for the harmonic oscillator on the planeand for the geodesic motion on the sphere are reduced to equation which coincide with Abel’s equationdefining rotation of the parabola around a fixed point on the elliptic curve, see Figure 3.Figure 3: Rotation of the parabola with points P ( t ) and P ( t ) around fixed point P .For all these superintegrable systems motion of two points P ( t ) and P ( t ) on bound pieces of thecurve X generates motion by algebraic trajectories in the phase space T ∗ R or T S .For all these superintegrable systems additional first integrals (2.20,2.21) and (2.23) are definedby coordinates of this fixed point P by equation (1.7). Of course, abscissa x and ordinate y are alsofirst integrals depending on I , I and C . In this Section we consider Abel’s equations (1.9) k dx y + k dx y = 0 , defining motion of the straight line, quadric, cubic, quartic and so on around a fixed point on theelliptic curve when two movable points P ( t ) and P ( t ) of degree k , and one fixed point P form anintersection divisor of elliptic curve X with the straight line, quadric, cubic, quartic, etc.In order to construct superintegrable systems associated with this motion of points on the ellipticcurve we just have to identify Abel’s equations (1.9) on a projective plane with Abel’s equations onsome phase space. Let us consider integrable systems with the following Hamiltonian and second integral of motion2 H = I = ( u − κ ) (cid:18)(cid:16) p u k (cid:17) + V (cid:19) u − u + ( u − κ ) (cid:18)(cid:16) p u k (cid:17) + V (cid:19) u − u , (3.24) I = − u ( u − κ ) (cid:18)(cid:16) p u k (cid:17) + V (cid:19) u − u − u ( u − κ ) (cid:18)(cid:16) p u k (cid:17) + V (cid:19) u − u , where u , and p u , are elliptic coordinates and the corresponding momenta. At k = k these integralsof motion coincide with the integral of motion (2.15) up to the scalar factor.11f potentials V , are given by (2.16) or (2.17), and k , are integer positive numbers, then all thetrajectories of motion are algebraic trajectories because equation defined trajectories coincides withthe Abel’s equation k dx y + k dx y = 0 , k , ∈ Z + , on the elliptic curve y = f ( x ) , f ( x ) = a x + a x + a x + a x + a . For the Kepler potential and the harmonic oscillator potential quartic polynomials f ( x ) are given by(2.18) and (2.19), respectively. Consequently, complete integral C k k (1.8) of the Abel’s equation givesrise to a complete first integral for the corresponding Hamiltonian system. Proposition 1
Hamiltonian H = 2 I (3.24) commutes with integral of motion I { H, I } = 0 . for arbitrary potentials V , and parameters k , .If potentials are equal to V i ( u i ) = αu i u i − κ or V i ( u i ) = − α u i , and k , are positive integers, then Hamiltonian H = 2 I (3.24) commutes { H, C k ,k } = 0 . with additional first integral C k ,k (1.8) which is independent on first integrals I , . The proof of this proposition is completely based on the Euler solution of Problem 83 in [12].For the Kepler potential we have to substitute into C k ,k (1.8) the following coordinates of divisors x i = u i , y i = ( u i − κ ) p u i k i , i = 1 , , and coefficients a = I , a = α, a = I − κ I , a = ακ , a = − κ I , whereas for the harmonic oscillator potential these coordinates and coefficients are equal to x i = u i , y i = u i ( u i − κ ) p u i k i , i = 1 , , and a = α , a = I − α κ , a = α κ − κ I + I , a = − κ I , a = 0 . We calculated integrals C and C in two computer algebra systems Mathematica and Maple anddirectly verified that these integrals are in involution with the corresponding Hamiltonians H = 2 I (3.24).In case of the Kepler potential (2.16) additional first integrals C (1.10) depends on √ a = √ I ,i.e. it is the algebraic function on momenta p u , . Additional first integral C (1.11) is the rationalfunction on elliptic coordinates u , and momenta p u , .In case of the harmonic oscillator potential (2.17) both the additional integrals of motion C and C are rational functions on elliptic coordinates u , and momenta p u , . For instance, we explicitlypresent additional first integral C , which is in involution with the Hamiltonian2 H = I = ( u − κ ) p u u − u ) + ( κ − u ) p u u − u + α ( κ − u − u ) , I = u ( κ − u ) p u u − u ) − u ( κ − u ) p u u − u + α u u . Using various tools in Mathematica and Maple one gets the following observable expression C = − κ H + ( u − κ )(2 αu + p u ) p u + α ( κ − κ u + u ) + p u C D + p u C D , where D = ( κ u + u − u u ) p u κ − u + 4 αu ( u − u ) ( αu + p u ) κ − u − u u p u p u + 4 u p u , and C = u ( κ − u − u ) p u +2 u u (cid:0) αu ( κ − u )+( u +4 u − κ ) p u (cid:1) p u − (cid:16) α u ( u − u )( κ − u − u )+2 α ( κ u − κ u − u +2 u ) u p u + (cid:0) κ − ( u +8 u ) κ + u (2 u +7 u ) (cid:1) p u (cid:17) p u + u ( αu + p u ) (cid:0) α ( u − u ) − p u u (cid:1)(cid:0) αu ( u − u )+( κ − u ) p u (cid:1) C = u (cid:0) u u − κ ( u +2 u ) (cid:1) p u − u u (cid:16) αu ( κ u − u u +2 u ) − (cid:0) κ ( u +4 u ) − u u − u (cid:1) p u (cid:17) p u − u (cid:16) α κ u ( u − u ) − α ( κ ( u + u u − u ) − u u +4 u u ) p u − u ( κ − κ (12 u − u )+24 u u ) p u (cid:17) p u − u u (cid:16) α ( u − u ) ( αu − ( κ − u ) p u ) + αu ( κ (6 u − u ) − u u +11 u ) p u + ( κ − κ ( u − +4 u )+4 u u +7 u ) p u (cid:17) p u + u (cid:16) αu ( u − u )( κ − u )+( κ − κ u +8 u ) p u (cid:17) p u . In our opinion, it is practically impossible to use such sophisticated expressions in the direct search ofadditional integrals of motion or for investigations of algebras of integrals of motion, see also discussionin [5]. Nevertheless, because additional first integrals of motion are easily expressed via coordinates ofthird point P = [ k ] P + [ k ] P at any k , , for instance C k k = 2 a x + a x + a − √ a y , we can derive the algebra of integrals using the well-known syzygies on elliptic curve [14, 16]. Let us consider geodesic motion on the two-dimensional sphere S defined by Hamiltonian and secondintegral of motion H = I = φ ( u ) p u k ( u − u ) + φ ( u ) p u k ( u − u ) , I = u φ ( u ) p u k ( u − u ) + u φ ( u ) p u k ( u − u ) . (3.25)At k = k these first integrals coincide with (2.22) up to the scalar factor k .Trajectories of motion are defined as solutions of equation k du p φ ( u )( u I + I ) + k du p φ ( u )( u I + I ) = 0 , k dx y + k dx y = 0 , x , = u , , on the elliptic curve defined by an equation of the form y = f ( x ) , f ( x ) = φ ( x )( xI + I ) = a x + a x + a x + a x + a . Consequently, we have superintegrable systems with additional first integral C k k (1.8). Proposition 2
The following Hamiltonian H = P g ij p u i p u j on the two-dimensional sphere with di-agonal metric g = 1 u − u ( u − α )( u − α )( u − α ) k
00 ( α − u )( α − u )( α − u ) k , k , k ∈ Z + is in involution with two independent first integrals { H, I } = 0 , { H, C k ,k } = 0 . Thus, this metric is a superintegrable metric.
The proof of this proposition is completely based on the Euler solution of Problem 83 in [12].In Cartesian variables q , q , q and momenta p , p , p in T ∗ R , so that q + q + q = 1 , q p + q p + q p = 0 , this Hamiltonian has the following form H = k + k k k ( p + p + p )+ k − k k k ( u − u ) (cid:0) α q q p p + α q q p p + α q q p p + β p + β p + β p (cid:1) , where β = (cid:0) ( q + q ) α + ( q − q ) α + ( q − q ) α (cid:1) ,β = (cid:0) ( q − q ) α + ( q + q ) α + ( − q + q ) α (cid:1) ,β = (cid:0) ( − q + q ) α + ( − q + q ) α + ( q + q ) α (cid:1) , and u − u = (cid:16) ( α − α ) q + ( α − α ) q + ( α − α ) q + 2( α − α )( α − α ) q q +2( α − α )( α − α ) q q + 2( α − α )( α − α ) q q (cid:17) / is a difference of the elliptic coordinates, which is the globally defined strictly positive function on thesphere.At k = 2 , k = 1 explicit expression for the first integral C k ,k can be obtained substituting x , = u , , y , = ( u , − α )( u , − α )( u , − α ) p u , k , and a = I , a = I − ( α + α + α ) I a = ( α α + α α + α α ) I − ( α + α + α ) I a = ( α α + α α + α α ) I − α α α I , a = − α α α I √ a = √ I , first integral C (1.10) is an algebraic function on the elliptic coordinates u , and the corresponding momenta p u , . Additional first integral C is a rational function onvariables of separation of the form (1.11).As above at k = 2 , k = 1 we directly verified that Hamiltonians H = P g ij p u i p u j andfirst integrals C k ,k are in involution by using computer algebra systems Mathematica and Maple. Consider motion of k points P , . . . , P k around m fixed points P k +1 , . . . , P k + m along a plane curve X ,which is governed by Abel’s equations generated by the addition of points on X ( P + · · · + P k ) + ( P k +1 + · · · + P k + m ) = 0 . If the same Abel’s equations arise when studying motion of an integrable Hamiltonian or non-Hamilto-nian system, then this dynamical system is a superintegrable system with additional partial or completeintegrals of motion which are given by the coordinates of fixed points P k +1 , . . . , P k + m . According toAbel’s theorem these integrals are algebraic functions in coordinates of movable points P , . . . , P m and,therefore, they are well-defined algebraic functions on original physical variables. Evolution of movablepoints around the fixed points gives rise to algebraic trajectories of this superintegrable system similarto algebraic trajectories in the Euler two centers problem.In this note we study motion of two points of degree k and k around one fixed point on the ellipticcurve. The corresponding Abel’s equations also arise when studying the following superintegrablesystems with two degrees of freedom: the Kepler problem, the harmonic oscillator, the geodesic motionon the sphere and their superintegrable deformations. Here we only present the corresponding firstintegrals, which are polynomial functions in momenta at k = k and algebraic/rational functions at k = k . We plan to discuss the corresponding algebraic trajectories in forthcoming publication.Arithmetic on elliptic curves has been an object of study in mathematics for well over a century.Recently arithmetic on elliptic curves has proven useful in applications such as factoring [21], ellipticcurve cryptography [3, 15, 34], and in the proof of Fermats last theorem [35]. In real world ellipticcurve point multiplication is one of the most widely used methods for digital signature schemes in cryp-tocurrencies, which is applied in both Bitcoin and Ethereum for signing transactions. In [29, 30, 32]we apply elliptic and hyperelliptic curve point multiplication to discretization of some known inte-grable systems in Hamiltonian and non-Hamiltonian mechanics and to construction of new integrableHamiltonian systems in [30, 31] . In this note we use this universal mathematical tool to constructnew superintegrable systems with algebraic and rational integrals of motion. It will be interesting todiscuss other possible applications of arithmetic on elliptic and hyperelliptic curves in classical andquantum mechanics.The work was supported by the Russian Science Foundation (project 18-11-00032). References [1] Abel N. H., M´emoire sure une propri´et´e g´en´erale d’une classe tr`es ´entendue de fonctions tran-scendantes, Oeuvres compl´etes, Tom I, Grondahl Son, Christiania (1881), pages 145-211, availablefrom https://archive.org/details/OEuvresCompletesDeNielsHenrikAbel1881 12/page/n167[2] Bolsinov A.V., Kozlov V.V., Fomenko A.T.,
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