The Kepler problem: polynomial algebra of non-polynomial first integrals

TL;DR

This paper explores the algebraic structure of non-polynomial first integrals in superintegrable systems related to elliptic curves, revealing new insights into their polynomial algebraic properties.

Contribution

It introduces the polynomial algebra of non-polynomial first integrals linked to elliptic curve divisors, expanding understanding of symmetries in superintegrable systems.

Findings

Polynomial algebra of non-polynomial first integrals is characterized.

Connection between elliptic integrals and orbit determination is established.

New algebraic structures for fixed points on elliptic curves are identified.

Abstract

The sum of elliptic integrals simultaneously determines orbits in thr Kepler problem and the addition of divisors on elliptic curves. Periodic motion of a body in physical space is defined by symmetries, whereas periodic motion of divisors is defined by a fixed point on the curve. Algebra of the first integrals associated with symmetries is a well-known mathematical object, whereas algebra of the first integrals associated with coordinates of fixed points is unknown. In this paper, we discuss polynomial algebras of non-polynomial first integrals of superintegrable systems associated with elliptic curves.

Equations

$$R=\frac{\alpha }{r^2}\text{mbox}andQ=\frac{\beta }{q^2}.$$

$$\begin{array}[]{c}\dfrac{ds}{\sqrt{Hs^{4}+(\alpha+\beta)s^{3}+cs^{2}-h^{2}(\alpha+\beta)s-Hh^{4}-(b^{2}+c)h^{2}}}=\\ =\dfrac{du}{\sqrt{Hu^{4}+(\alpha-\beta)u^{3}+cs^{2}-h^{2}(\alpha-\beta)u-Hh^{4}-(b^{2}+c)h^{2}}}\end{array}$$

$$\begin{array}[]{c}dt=\dfrac{s^{2}ds}{4\sqrt{Hs^{4}+(\alpha+\beta)s^{3}+cs^{2}-h^{2}(\alpha+\beta)s-Hh^{4}-(b^{2}+c)h^{2}}}-\\ -\dfrac{u^{2}du}{4\sqrt{Hu^{4}+(\alpha-\beta)u^{3}+cs^{2}-h^{2}(\alpha-\beta)u-Hh^{4}-(b^{2}+c)h^{2}}}\end{array}$$

$$\int \frac{d{x}_1}{\sqrt{f(x_1)}}+\int \frac{d{x}_2}{\sqrt{f(x_2)}}=const,$$

$$\int \frac{x_1^{2}d{x}_1}{\sqrt{f(x_1)}}+\int \frac{x_2^{2}d{x}_2}{\sqrt{f(x_2)}}=4t,$$

$$X:\Phi (x,y)=y^2-f(x)=0,f(x)=a_4{x}^4+a_3{x}^3+a_2{x}^2+a_{1}x+a_0$$

$$P_1(t)+P_2(t)=P_3.$$

$$C=2a_4{x}_3^2+a_3{x}_3+a_2-2\sqrt{a_4}{y}_3$$

$$C=-\bigl{(}p_{1}q_{2}-(q_{1}-\kappa)p_{2}\bigr{)}^{2}\,,$$

$$r+r^{\prime }=2u_1,r-r^{\prime }=2u_2.$$

$$q_1=\frac{u_1{u}_2}{\kappa },\text{mbox}and{q}_2=\frac{\sqrt{(u_1^2-{\kappa }^2)({\kappa }^2-u_2^2)}}{\kappa }.$$

$$u_2<\kappa <u_1,$$

$$\begin{array}[]{rcl}p_{1}&=&\dfrac{u_{1}u_{2}(p_{u_{1}}u_{1}-p_{u_{2}}u_{2})-\kappa^{2}(p_{u_{1}}u_{2}-p_{u_{2}}u_{1})}{\kappa(u_{1}^{2}-u_{2}^{2})}\,,\\ \\ p_{2}&=&\dfrac{(p_{u_{1}}u_{1}-p_{u_{2}}u_{2})\sqrt{u_{1}^{2}-\kappa^{2}}\sqrt{\kappa^{2}-u_{2}^{2}}}{\kappa(u_{1}^{2}-u_{2}^{2})}\,.\end{array}$$

$$2H=I_1=p_1^2+p_2^2+\frac{\alpha }{r},I_2=\frac{\alpha (r^2-r^{\prime 2})}{4r}-({\kappa }^2+q_2^2)p_1^2-2q_1{q}_2{p}_1{p}_2-q_1^2{p}_2^2.$$

$$\begin{array}[]{rcl}I_{1}&=&\dfrac{(u_{1}^{2}-\kappa^{2})p_{u_{1}}^{2}}{u_{1}^{2}-u_{2}^{2}}+\dfrac{(u_{2}^{2}-\kappa^{2})p_{u_{2}}^{2}}{u_{2}^{2}-u_{1}^{2}}+\dfrac{{\alpha}}{u_{1}+u_{2}}\\ \\ I_{2}&=&\dfrac{u_{2}^{2}(u_{1}^{2}-\kappa^{2})p_{u_{1}^{2}}}{u_{2}^{2}-u_{1}^{2}}+\dfrac{u_{1}^{2}(u_{2}^{2}-\kappa^{2})p_{u_{2}}^{2}}{u_{1}^{2}-u_{2}^{2}}+\dfrac{\alpha u_{1}u_{2}}{u_{1}+u_{2}}\,.\end{array}$$

$$\frac{d{u}_1}{dt}=\{u_1,H\}=\frac{(u_1^2-{\kappa }^2)p_{u_1}}{u_1^2-u_2^2},\frac{d{u}_2}{dt}=\{u_2,H\}=\frac{(u_2^2-{\kappa }^2)p_{u_2}}{u_2^2-u_1^2},$$

$$\begin{array}[]{rcl}\dfrac{du_{1}}{\sqrt{(u_{1}^{2}-\kappa^{2})(I_{1}u_{1}^{2}-{\alpha}u_{1}+I_{2})}}&=&\phantom{-}\dfrac{dt}{u_{1}^{2}-u_{2}^{2}}\,,\\ \\ \dfrac{du_{2}}{\sqrt{(u_{2}^{2}-\kappa^{2})(I_{1}u_{2}^{2}-{\alpha}u_{2}+I_{2})}}&=&-\dfrac{dt}{u_{1}^{2}-u_{2}^{2}}\,,\end{array}$$

$$\int \frac{d{u}_1}{\sqrt{(u_1^2-{\kappa }^2)(I_1{u}_1^2-\alpha u_1+I_2)}}+\int \frac{d{u}_2}{\sqrt{(u_2^2-{\kappa }^2)(I_1{u}_2^2-\alpha u_2+I_2)}}=const$$

$$X:\Phi (x,y)=y^2-f(x)=0,f(x)=I_1{x}^4-\alpha x^3+(I_2-I_1{\kappa }^2)x^2+{\kappa }^2\alpha x-I_2{\kappa }^2.$$

$$x_1=u_1,y_1=(u_1^2-{\kappa }^2)p_{u_1}\text{mbox}and{x}_2=u_2,y_2=(u_2^2-{\kappa }^2)p_{u_2}.$$

$$u_2<\kappa <u_1\Rightarrow x_1\ne x_2\Rightarrow (x_3,y_3)\ne (\infty ,\infty ),$$

$$Y:y=\mathcal{P}(x),\mathcal{P}(x)=\sqrt{a_4}{x}^2+b_{1}x+b_0,$$

$$y_1=\sqrt{a_4}{x}_1^2+b_1{x}_1+b_0\text{mbox}and{y}_2=\sqrt{a_4}{x}_2^2+b_1{x}_2+b_0$$

$$\mathcal{P}(x)=\sqrt{a_4}{x}^2+b_{1}x+b_0=\sqrt{a_4}(x_1-x)(x_2-x)+\frac{(x-x_2)y_1}{x_1-x_2}+\frac{(x-x_1)y_2}{x_2-x_1}.$$

$$\begin{array}[]{rcl}\psi&=&f(x)-\mathcal{P}^{2}(x)\\ \\ &=&(a_{3}-2b_{1}\sqrt{a_{4}})x^{3}+(a_{2}-2b_{0}\sqrt{a_{4}}-b_{1}^{2})x^{2}+(a_{1}-2b_{0}b_{1})x+a_{0}-b_{0}^{2}\\ \\ &=&(a_{3}-2b_{1}\sqrt{a_{4}})(x-x_{1})(x-x_{2})(x-x_{3})\,.\end{array}$$

$$x_3=-x_1-x_2-\frac{2b_0\sqrt{a_4}+b_1^2-a_2}{2b_1\sqrt{a_4}-a_3}$$

$$y_3=-\mathcal{P}(x_3)=-\sqrt{a_4}{x}_3^2-b_1{x}_3-b_0,$$

$$a_4=I_1,a_3=-\alpha ,a_2=(I_2-{\kappa }^2{I}_1),a_1={\kappa }^2\alpha ,a_0=-{\kappa }^2{I}_2,$$

$$x_{3}=\scriptstyle\frac{2\bigl{(}\kappa^{2}(p_{u_{1}}u_{2}-p_{u_{2}}u_{1})-u_{1}u_{2}(p_{u_{1}}u_{1}+p_{u_{2}}u_{2})\bigr{)}\bigl{(}\sqrt{I_{1}}(u_{1}^{2}-u_{2}^{2})-\kappa^{2}(p_{u_{1}}-p_{u_{2}})+p_{u_{1}}u_{1}^{2}-p_{u_{2}}u_{2}^{2}\bigr{)}-\alpha(u_{1}-u_{2})^{2}(\kappa^{2}+u_{1}u_{2})}{\left(u_{1}^{2}-u_{2}^{2}\right)\bigl{(}2\sqrt{I_{1}}\left(\kappa^{2}(p_{u_{1}}-p_{u_{2}})-p_{u_{1}}u_{1}^{2}+p_{u_{2}}u_{2}^{2}\right)+2\kappa^{2}(p_{u_{1}}^{2}-p_{u_{2}}^{2})-2p_{u_{1}}^{2}u_{1}^{2}+2p_{u_{2}}^{2}u_{2}^{2}-\alpha(u_{1}-u_{2})\bigr{)}}\,.$$

$$y_3=-\sqrt{I_1}(u_1-x_3)(u_2-x_3)-\frac{(x_3-u_2)({\kappa }^2-u_1^2)p_{u_1}}{u_1-u_2}-\frac{(x_3-u_1)({\kappa }^2-u_2^2)p_{u_2}}{u_2-u_1},$$

Full text

1 Introduction

The main point of interest in integrable systems relies on the fact that they can be integrated by quadratures. For many known integrable systems these quadratures involve various sums of Abelian integrals, which are inextricably entwined with the arithmetic of divisors. In physics, we describe first integrals of dynamical systems in terms of physical variables, and usually these first integrals are related to symmetries, including dynamical ones. For the Kepler problem the corresponding first integrals are well-known polynomials in momenta [3, 8, 19, 20, 25, 26].

In algebraic geometry we describe the evolution of divisors in terms of coordinates of divisors. The corresponding constants of motion are nothing more than coordinates of fixed points, which are algebraic functions on original physical variables. In fact, algebraic first integrals for the Kepler problem have been obtained by Euler as a by-product of his study of the algebraic orbits appearing in two fixed centers problem [8].

Algebras of the polynomial first integrals of superintegrable systems can be associated with orthogonal polynomials, see e.g. [5, 6, 12, 13, 14, 23, 24] and references within. For instance, it could be the Racah-Wilson algebra, Bannai-Ito algebra, Askey-Wilson algebra, etc. We suppose that the polynomial algebra of non-polynomial first integrals arising in divisor arithmetic on elliptic and hyperelliptic curves may be associated with elliptic and hyperelliptic functions. It could be Weierstrass functions, Jacobi functions, Abelian functions, etc.

In 1762 Euler wrote a paper titled ”Task: the body is attracted to two given fixed centers inversely proportional to the square of the distance; find in which case the curve described by the body will be algebraic” [8]. In this paper he separated algebraic orbits from transcendental ones using elliptic coordinates on a plane and an addition law for the corresponding elliptic integrals. Algebraic orbits were interesting because if one of the centers was absent, the body would move on algebraic orbits, as a solution of the Kepler problem.

Indeed, let us consider motion of the body attracted to two fixed centers by forces inversely proportional to the squares of the distance

[TABLE]

In elliptic coordinates $s$ and $u$ equations of motion are reduced to differential equations

[TABLE]

and

[TABLE]

which we copied from page 106 of Lagrange’s textbook [26]. Here $H$ and $h$ are integrals of motion, which are second order polynomials in momenta, $b$ and $c$ are geometric parameters describing positions of fixed centers.

At $\beta=0$ and $x_{1}=s$, $x_{2}=u$ these equations become well-known Abel’s quadratures for the two-body Kepler problem

[TABLE]

and

[TABLE]

on elliptic curve $X$ defined by an equation of the form

[TABLE]

on a projective plane. The first equation (1.1) determines trajectories of motion, whereas the second equation (1.2) defines time [26].

Equations (1.1-1.2) describe motion of a body in the Kepler problem and, simultaneously, evolution of points $P_{1}(t)=(x_{1},y_{1})$, $P_{2}(t)=(x_{2},y_{2})$ around fixed point $P_{3}=(x_{3},y_{3})$ on $X$ governed by arithmetic equation

[TABLE]

According to Abel’s theorem [1, 2, 11, 17, 18] trajectories of points $P_{1,2}(t)$ on $X$ are uniquely determined by Abel’s sum (1.1) in the same way as trajectories of a body in the Kepler problem. Subsequently, periodic motion of points along plane curve $X$ generates periodic motion in phase space of the Kepler system and vice versa.

According to [9] coordinates of the fixed point $x_{3}$ and $y_{3}$ are algebraic functions on the coordinates of movable points $x_{1,2}$ and $y_{1,2}$ which are constants of divisor motion along elliptic curve $X$ (1.4). These algebraic functions on elliptic coordinates $u_{1,2}$ and momenta $p_{u_{1,2}}$ are also first integrals in the Kepler problem. In [8] Euler used these algebraic first integrals and their combination

[TABLE]

in order to separate algebraic orbits from transcendental ones in the problem of two fixed centers.

In the Kepler case first integral $C$ (1.5) is a square of the component of angular momentum

[TABLE]

Of course, this first integral is related to the well-studied rotational symmetry [3, 19, 20, 25]. The Poisson algebras of polynomial first integrals for the Kepler problem and other dynamical systems, separable in elliptic, parabolic and polar coordinates are well studied objects, see e.g. [5, 6, 12, 13, 14, 23, 24].

Our aim is to calculate the algebra of non-polynomial first integrals $x_{3}$ and $y_{3}$ and to discuss various representations of this algebra. This algebra occurs in a standard arithmetic of divisors on elliptic curves and, therefore, it could belong to a family of algebras associated with the arithmetic of divisors on more complicated hyperelliptic curves.

2 The Kepler problem

In the original physical problem configuration space is 6-dimensional, and phase space is 12-dimensional space. Discussion of the traditional topics, such as symmetries, conservation of angular momentum, conservation of Laplace-Runge-Lenz vector, regularization and so on, may be found in [3, 19, 20, 25] and many other papers and textbooks.

Our aim is to come back to Euler’s calculations in order to get a family of superintegrable systems with nonpolynomial first integrals, which cannot be obtained using symmetries. Following Euler [8] we start with the planar two centers problem. Reduction of the original phase space to the orbital plane, which Euler described by using a picture, may be found in the Lagrange textbook [26].

2.1 Motion in orbit

Let us introduce elliptic coordinates on the orbital plane. If $r$ and $r^{\prime}$ are distances from a point on the plane to the fixed centers, then elliptic coordinates $u_{1,2}$ are

[TABLE]

If the centres are taken to be fixed at $-\kappa$ and $\kappa$ on $OX$-axis of the Cartesian coordinate system, then we have standard Euler’s definition of elliptic coordinates on the plane

[TABLE]

Coordinates $u_{1,2}$ are curvilinear orthogonal coordinates, which take values only in the intervals

[TABLE]

i.e. they are locally defined coordinates. The corresponding momenta are given by

[TABLE]

For the planar Kepler problem with one center of attraction at point $(\kappa,0)$, which is a partial case of Euler’s two-centers problem, Hamiltonian and first integral are equal to

[TABLE]

In elliptic coordinates these integrals of motion have the following form

[TABLE]

Substituting solutions of these equations with respect to $p_{u_{1}}$ and $p_{u_{2}}$ into the equations of motion

[TABLE]

we obtain differential equations of the form

[TABLE]

After integration of the sum of these equations one gets a sum of Abelian integrals

[TABLE]

involving holomorphic differentials on elliptic curve $X$ (1.3) defined by equation

[TABLE]

Here $I_{1,2}$ are values of the integrals of motion, see terminology and discussion in Lagrange’s textbook [26] and in comments by Darboux and Serret [4, 31].

2.2 Motion in elliptic curve

Using the sum of Abelian integrals (2.5) we can transfer from classical mechanics to algebraic geometry and, in particular, to divisors arithmetic on elliptic curve. Indeed, coordinates of movable points $P_{1,2}(t)$ in the equation of motion along elliptic curve $X$ (1.4) are

[TABLE]

Because

[TABLE]

abscissa $x_{3}$ and ordinate $y_{3}$ of fixed point $P_{3}$ are well-defined finite functions on $T^{*}\mathbb{R}^{2}$.

In order to calculate affine coordinates of fixed point $P_{3}$ we have to consider intersection of $X$ and parabola $Y$ with a fixed leading coefficient

[TABLE]

see [1, 17, 18] for details. Solving equations

[TABLE]

with respect to $b_{1}$ and $b_{0}$ we calculate standard interpolation by Lagrange for polynomial

[TABLE]

Substituting $y=\mathcal{P}(x)$ into $f(x)-y^{2}=0$ we obtain Abel’s polynomial

[TABLE]

Evaluating coefficients of this polynomial we determine abscissa of the fixed point $P_{3}$ in (1.4)

[TABLE]

and its ordinate

[TABLE]

where $b_{1}$ and $b_{0}$ are functions of coordinates of movable points $x_{1},x_{2}$ and $y_{1},y_{2}$ defined by equation (2.7).

Now we come back from divisor arithmetics to classical mechanics. For the Kepler problem we have

[TABLE]

so abscissa of fixed point $P_{3}$ is equal to

[TABLE]

Ordinate $y_{3}$ (2.7) is equal to

[TABLE]

Here $I_{1}$ is given by (2.4) and, therefore, $x_{3}$ and $y_{3}$ are algebraic functions on $u_{1,2}$ and $p_{u_{1,2}}$.

In [8, 9] Euler introduced algebraic first integral $C$ (1.5) which is nothing more than a square of angular momentum in the Kepler case:

[TABLE]

It is well-known that existence of this first integral $C$ is related to rotational symmetry of the orbital plane a around center of attraction. Algebraic first integrals $x_{3}$ and $y_{3}$ have no obvious physical meaning, but they have a trivial geometric description as affine coordinates of the fixed point on elliptic curve $X$.

2.3 Symmetry breaking

Let us consider non-canonical transformations of momenta preserving symmetries of configuration space, but breaking symmetry between divisors [16, 36, 37, 38, 39, 41, 42].

It is easy to see, that transformation of momenta

[TABLE]

where $m$ and $n$ are rational numbers, preserves the only symmetry of potential part of first integrals and breaks symmetry of whole integrals of motion (2.3), which now have the form

[TABLE]

In Cartesian coordinates on the plane Hamiltonians (2.11) read as

[TABLE]

According to [41, 42] these Hamiltonians (2.11) are superintegrable Hamiltonians because this non-canonical transformation sends the original sum of elliptic integrals (2.5) to the sum

[TABLE]

i.e. to the sum of elliptic integrals with integer coefficients

[TABLE]

Here we present rational numbers $m=m_{1}/m_{2}$ and $n=n_{1}/n_{2}$ as the ratio of integer numbers. The corresponding first integrals of motion on elliptic curve $X$ were obtained in Problem 83 of Euler’s textbook [9].

Without loss of generality below we consider only positive integer numbers $m$ and $n$. In this case sum of elliptic integrals (2.12), which generates the well-studied arithmetic equation for divisors on elliptic curves

[TABLE]

see [9, 27, 32]. Here $[k]P$ means scalar multiplication of point on an elliptic curve on integer number $k\in\mathbb{Z}$, and we denote coordinates of $[k]P=[k](x,y)$ as $([k]x,[k]y)$, whereas notations for coordinates of $P_{3}$ in (2.13) remain the same $x_{3}$ and $y_{3}$.

In order to get coordinates of fixed point $P_{3}$ in (2.13) we have to:

Multiply divisors $P_{1,2}$ by integer numbers $m$ and $n$ using a recursion procedure proposed by Euler [9] or using standard expressions for scalar multiplication on elliptic curves, see [27, 32, 40] and references within. 2. 2.

Add divisors $[m]P_{1}$ and $[n]P_{2}$. Because points $[m]P_{1}$ and $[n]P_{2}$ belong to intersection divisor of $X$ and $Y$, we can use the equation of parabola $Y$

[TABLE]

in order to calculate its coefficients $b_{1}$ and $b_{0}$. After that we substitute $a_{i}$, $b_{i}$ and $[m]x_{1}$, $[m]y_{1}$ and $[n]x_{2}$, $[n]y_{2}$ into (2.8) and (2.9) and obtain coordinates of fixed point $P_{3}$ in (2.13) :

[TABLE]

Following [9] we can also determine Euler’s first integral of equation of motion (2.13) on $X$:

[TABLE] 3. 3.

Identify affine coordinates on the projective plane with elliptic coordinates on phase space

[TABLE]

so that constants of divisor motion on elliptic curve $X$ become first integrals of Hamiltonian vector field in $T^{*}\mathbb{R}^{2}$.

At $m=n$ first integral $C_{mn}$ (2.15) is a square of angular momentum relating with rotational symmetry. At $m\neq n$ all first integrals $x_{3},y_{3}$ (2.14) and $C_{mn}$ (2.15) are algebraic functions in phase space. Some explicit expressions of these first integrals may be found in [42] .

Now we are ready to formulate the main result in this note.

Proposition 1

Functions $I_{1},I_{2}$ (2.11) and $x_{3},y_{3}$ (2.14) in phase space $T^{*}\mathbb{R}^{2}$ can be considered as representation of the following algebra of the first integrals

[TABLE]

labelled by two integer numbers $m$ and $n$. Here

[TABLE]

are derivatives of function $\Phi(x,y)$ from the definition of elliptic curve $X$ (2.6), and $\{.,.\}$ is the standard canonical Poisson bracket

[TABLE]

The Poisson brackets (2.17) are derived from the brackets

[TABLE]

between action variables $I_{1,2}$ (2.11) and angle variables

[TABLE]

Here we use indefinite integrals determined only up to an additive constant following Euler [9], Abel [1], Jacobi [22] and Stäckel [33], see also discussion in [2, 17].

In (2.17) form of the bracket $\{x_{3},y_{3}\}$ coincides with the form of original brackets $\{x_{1},y_{1}\}$ and $\{x_{2},y_{2}\}$. Two remaining non-trivial brackets can be rewritten in the following form

[TABLE]

which is reminiscent of Hamiltonian equations of motion. The first time brackets (2.17) appeared when we studied superintegrable systems associated with elliptic curve in the short Weierstrass form [42, 43]. Below we discuss similar algebras of the first integrals for other superintegrable systems associated with elliptic curve.

3 Harmonic oscillator

Let us consider 2D harmonic oscillator with the following Hamiltonian and additional integral of motion

[TABLE]

which is a shifted square of angular momentum. In elliptic coordinates (2.1) and (2.2) these constants of motion are equal to

[TABLE]

Rewriting equation of motion

[TABLE]

in the following form

[TABLE]

we can eliminate time and obtain equation

[TABLE]

Substituting solutions of equations (3.19) with respect to $p_{u_{1,2}}$ into this expression and integrating we obtain standard equation defining form of the trajectories [26]:

[TABLE]

This equation is reduced to (1.1) using Euler’s substitution $u_{i}^{2}=x_{i}$, which allows us to consider evolution of divisors on elliptic curve $X$

[TABLE]

governed by equation (1.4)

[TABLE]

Coordinates of movable points $P_{1,2}(t)$ are

[TABLE]

The abscissa of fixed point $x_{3}$ (2.8) reads as

[TABLE]

Ordinate $y_{3}$ (2.9) is a more lengthy rational function in elliptic coordinates, which we do not present for brevity, whereas Euler’s integral (1.5) is the following simple polynomial

[TABLE]

Existence of the polynomial first integrals $I_{1},I_{2}$ and $C$ is related to symmetries of equations of motion in the original physical space. Existence of non-polynomial first integrals $x_{3}$ and $y_{3}$ is related to motion along elliptic curve $X$ around fixed point $P_{3}$.

Symmetry breaking transformation (2.10) generates polynomial integrals of motion

[TABLE]

In Cartesian coordinates on the plane superintegrable Hamiltonians in (3.21) read as

[TABLE]

where $r$, $r^{\prime}$ and $\kappa$ enter into the definition of elliptic coordinates. The corresponding first integrals $x_{3},y_{3}$ (2.14) and $C_{mn}$ (2.15) are rational functions at $m\neq n$. Some particular expressions for these first integrals may be found in [42].

Proposition 2

Functions $I_{1},I_{2}$ (3.21) and $x_{3},y_{3}$ (2.14) on $T^{*}\mathbb{R}^{2}$ can be considered as representation of the following algebra of the first integrals

[TABLE]

labelled by two integer numbers $m$ and $n$. Here

[TABLE]

are derivatives of function $\Phi(x,y)$ from the definition of elliptic curve $X$ (3.20), and $\{.,.\}$ is the standard canonical Poisson bracket.

As in Section 2 algebra of the first integrals (3.22) is derived from the Poisson brackets between the corresponding action-angle variables. We also have computer-assisted proof of this Proposition at $m=1,2,3$ and $n=1,2,3$.

This algebra of the first integrals (3.22) slightly differs from (2.17) because in the Kepler problem we take $x_{1,2}=u_{1,2}$, whereas for oscillator we have to put $x_{1,2}=u_{1,2}^{2}$ and, therefore, we have different Poisson brackets between coordinates of movable points.

3.1 Smorodinsky-Winternitz system

In order to obtain the so-called Smorodinsky-Winternitz system [10] we have to start with elliptic curve $X$ defined by equation $\Phi(x,y)=y^{2}-f(x)=0$ with

[TABLE]

instead of (3.20). Equation (2.13)

[TABLE]

determines evolution of two moving points around a third fixed point in the intersection divisor. Coefficients $I_{1,2}$ of polynomial $f(x)$ together with coordinates $(x_{3},y_{3})$ of fixed point $P_{3}$ are constants of the divisor motion.

Constants of the divisor motion give rise to the first integrals on phase space, which can be calculated using standard algorithm:

• •

identify affine coordinates of movable points $P_{1,2}(t)$ on the projective plane with elliptic coordinates and momenta in phase space

[TABLE]

• •

solve a pair of equations $\Phi(x_{1},y_{1})=0$ and $\Phi(x_{2},y_{2})$ with respect to $I_{1},I_{2}$;

• •

calculate first integrals associated with affine coordinates $(x_{3},y_{3})$ (2.14) of the fixed point $P_{3}$ .

After that we can verify that functions $I_{1},I_{2}$ and $x_{3},y_{3}$ on $T^{*}\mathbb{R}^{2}$ satisfy to the Poisson brackets (3.22).

For the curve $X$ (3.23 ) one gets the following Hamiltonian

[TABLE]

Here potential part is independent on integer numbers $m$ and $n$, whereas kinetic energy $T_{mn}$ is equal to

[TABLE]

where $r$, $r^{\prime}$ and $\kappa$ enter into Euler’s definition of elliptic coordinates on the plane. At $m=n=1$ this Hamiltonian coincides with the Hamiltonian of the Smorodinsky-Winternitz system [10].

4 Drach system

In 1935 Jules Drach classified Hamiltonian systems in $T^{*}\mathbb{R}^{2}$ with third order integrals of motion [7]. Below we consider the so-called (h) Drach system associated with elliptic curve, see details of classification in [35, 36, 37]. Possible generalizations of the Drach systems are discussed in [28].

The (h) Drach system is defined by Hamiltonian

[TABLE]

and first integral

[TABLE]

After canonical point transformation of variables

[TABLE]

integrals of motion look like

[TABLE]

Solving these equations with respect to $p_{u_{1}}$ and $p_{u_{2}}$ we obtain separated relations

[TABLE]

Following [33] we determine Stäckel matrix $S$ with entries

[TABLE]

and Stäckel angle variables

[TABLE]

which can be rewritten in a standard form for the Stäckel systems with $n$ degrees of freedom

[TABLE]

using separated relations (4.24), definition of the Stäckel matrix (4.25 ) and definition of points $P_{i}=(x_{i},y_{i})$ on hyperelliptic curve $X$, see [34].

In action-angle variables $I_{1,2}$ and $\omega_{1,2}$ equations of motion and symplectic form look like

[TABLE]

Because $H=I_{1}$, differential equations are trivially reduced to quadratures, for instance

[TABLE]

Relation $\omega_{2}=const$ involves the sum of Abelian integrals with holomorphic differentials on $X$ and, therefore, it defines swing of two points around a third fixed point on elliptic curve (1.4)

[TABLE]

In the Drach case coordinates of moving points are simple function on physical variables

[TABLE]

and, therefore, abscissa of fixed point $P_{3}$ is a quite observable rational function

[TABLE]

similar to Euler’s integral (1.5), which is the following polynomial in momenta

[TABLE]

Let us apply symmetry breaking transformation (2.10) to this superintegrable Stäckel system. Action variables $I_{1,2}$ associated with the equation

[TABLE]

are equal to

[TABLE]

In original Cartesian coordinates Hamiltonians in (4.27) have the form

[TABLE]

Following [29] we can say that these Hamiltonians describe motion of the body with a position dependent mass.

The corresponding Stäckel angle variables

[TABLE]

involve a holomorphic differential on elliptic curve, which allows us to calculate coordinates of the fixed point using arithmetic equation (2.14).

At $m=2$ and $n=1$ abscissa of fixed point $P_{3}$ remains a quite observable rational function if $\beta=\gamma=0$

[TABLE]

This expression was obtained using doubling of point $P_{1}=(u_{1},p_{u_{1}}/2)$ and addition (2.14) of points $[2]P_{1}$ and $P_{2}=(u_{2},p_{u_{2}})$ on elliptic curve $X$.

At $m=3$ and $n=1$ abscissa of fixed point $P_{3}$ is a bulky function even if $\beta=\gamma=0$

[TABLE]

where

[TABLE]

and

[TABLE]

This expression was obtained using tripling of point $P_{1}=(u_{1},p_{u_{1}}/3)$ and addition (2.14) of points $[3]P_{1}$ and $P_{2}=(u_{2},p_{u_{2}})$ on elliptic curve $X$.

Proposition 3

Functions $I_{1},I_{2}$ (4.27) and $x_{3},y_{3}$ (2.14) on $T^{*}\mathbb{R}^{2}$ can be considered as representation of the following algebra of the first integrals

[TABLE]

labelled by two integer numbers $m$ and $n$. Here

[TABLE]

are derivatives of function $\Phi(x,y)$ from the definition of elliptic curve $X$ (4.24), and $\{.,.\}$ is the canonical Poisson bracket.

This algebra is derived from the Poisson bracket between the corresponding action-angle variables. We also have computer-assisted proof of this Proposition at $m=1,2,3$ and $n=1,2,3$.

So, the so-called (h) Drach system belongs to a family of two-dimensional superintegrable systems associated with elliptic curves of the form $X:\Phi(x,y)=y^{2}-f^{(k)}(x)$, where

[TABLE]

and equation of motion

[TABLE]

For all these superintegrable systems algebra of the first integrals has the standard form (4.28) which directly follows from the Poisson brackets between action-angle variables.

5 3D superintegrable Stäckel system on elliptic curve

Let us consider Stäckel system associated with symmetric product $X\times X\times X$ of elliptic curve $X$ defined by equation of the form

[TABLE]

If we identify coordinates of the points on each copy of $X$ in $X\times X\times X$ with canonical coordinates in $T^{*}\mathbb{R}^{3}$

[TABLE]

we obtain action variables $I_{1},I_{2}$ and $I_{3}$ :

[TABLE]

solving separated relations $\Phi(u_{i},p_{u_{i}},I_{1},I_{2},I_{3})=0$ with respect to $I_{1},I_{2}$ and $I_{3}$. Substituting solutions of the same separated relations with respect to $p_{u_{1}},p_{u_{2}}$ and $p_{u_{3}}$ into the Stäckel definition (4.26) we get standard angle variables

[TABLE]

Equation of motion $\omega_{3}=const$ involves a holomorphic differential on elliptic curve and, therefore, it is equivalent to arithmetic equation for divisors on $X$

[TABLE]

This equation describes the swing of parabola $Y$

[TABLE]

around some fixed point $P_{4}$ on $X$. Because four points $P_{1},P_{2},P_{3}$ and $-P_{4}$ form an intersection divisor of $X$ and $Y$ we can calculate three coefficients $b_{2},b_{1}$ and $b_{0}$ by solving three equations

[TABLE]

Substituting $y=\mathcal{P}(x)$ into definition $y^{2}-f(x)=0$ of $X$ we obtain Abel’s polynomial

[TABLE]

Evaluating coefficients of this polynomial we find coordinates of the fixed point

[TABLE]

which are constants of divisor motion (5.31) on elliptic curve $X$.

The corresponding rational functions on phase space $T^{*}\mathbb{R}^{3}$

[TABLE]

where

[TABLE]

are first integrals of the dynamical system determined by Hamiltonian $H(I_{1},I_{2},I_{3})$ and canonical Poisson brackets.

After symmetry breaking transformation (2.10)

[TABLE]

equation of motion (5.31) on $X$ becomes

[TABLE]

Affine coordinates of the constant part of intersection divisor are given by

[TABLE]

where parabola $Y:y=\mathcal{P}(x)$ is now defined by using Lagrange interpolation by movable points $[m]P_{1}$, $[n]P_{2}$ and $[k]P_{3}$ on elliptic curve $X$, where

[TABLE]

Proposition 4

Functions $I_{1},I_{2},I_{3}$ (5.30) and $x_{4},y_{4}$ (5.32) in phase space $T^{*}\mathbb{R}^{3}$ can be considered as representation of the following algebra of the first integrals

[TABLE]

Here

[TABLE]

are derivatives of function $\Phi(x,y)$ from the definition of elliptic curve $X$ (5.29) and $\{.,.\}$ is the canonical Poisson bracket.

This algebra is derived from the Poisson bracket between the corresponding action-angle variables. We also have computer-assisted proof of this Proposition at $m=n=k$.

Algebra of the first integrals (5.33) slightly differs from the algebra (2.17) in the Kepler case. Abel’s subalgebra of (5.33) consists of two elements $I_{1}$ and $I_{2}$, whereas Abel’s subalgebra of (2.17) has only one central element $I_{1}$.

Summing up, we can construct six families of superintegrable systems using elliptic curves of the form $X:\Phi(x,y)=y^{2}-f^{(k)}(x)$, where

[TABLE]

and intersection divisor equation of motion

[TABLE]

For all these superintegrable systems algebra of the first integrals has the standard form (5.33) which directly follows from the Poisson brackets between action-angle variables. Following [29] we can say that Hamiltonians $I_{1},I_{2}$ and $I_{3}$ describe motion of the body in $T^{*}\mathbb{R}^{3}$ with a position dependent mass.

6 Conclusion

Equations of motion

[TABLE]

for the Stäckel systems on a symmetrized product $X\times\cdots\times X$ of hyperelliptic curve $X$ are equivalent to equation of motion

[TABLE]

describing evolution of the intersection divisor of $X$ and axillary curve $Y(t)$. For superintegrable Stäckel systems, intersection divisor can be divided on moving and fixed parts

[TABLE]

according to Abel’s theorem. It is clear, that constants of divisor motion are the coordinates of fixed part $D^{\prime}$ of the intersection divisor, and integrals of motion in phase space are some functions on these constants of divisor motion. So, algebra of the first integrals in phase space can be obtained from the algebra of constants of divisor motion, which is easily obtained from the Poisson brackets between the Stäckel action-angle variables.

In this note we calculate algebra of constants of divisor motion associated with the Kepler problem, harmonic oscillator, Drach system, Stäckel systems with two and three degrees of freedom and some of their deformations associated with symmetry breaking transformations of the Stäckel matrices. All these systems are related to various elliptic curves, but we can rewrite the corresponding algebras of non-polynomial integrals in a common form.

Scalar multiplication of points on elliptic curves

[TABLE]

generates non-canonical transformation on phase space

[TABLE]

which changes the form of Hamiltonian, but preserves its superintegrability property. Because multiplication of points on $X$ is a special case of isogenies between elliptic curves, we suppose that isogeny arithmetics also generates non-canonical transformations of phase space

[TABLE]

preserving superintegrability. If this conjecture is true, than isogeny volcanoes [30] could generate superintegrable system volcanoes. In a forthcoming publication, we will discuss this conjecture and application of Vélu’s formulas [32, 44] to construction superintegrable systems associated with elliptic curves.

For superintegrable Stäckel systems on hyperelliptic curves of genus two we have affine coordinates of divisors, Mumford’s coordinates of divisors, modified Jacobian coordinates, Chudnovski-Jacobian coordinates, mixed coordinates, etc. First integrals associated with these coordinates could be algebraic, rational or polynomial functions in phase space satisfying various polynomial or non-polynomial relations. It is interesting to study these relations associated with various constants of the divisor motion.

Class of superintegrable or degenerate systems is closely related to the class of bi-Hamiltonian systems with equations of motion

[TABLE]

see [15] and references within. So, we have two algebras of divisor motion constants the with respect to compatible Poisson brackets $\{.,.\}$ and $\{.,.\}^{\prime}$. We suppose that both algebras of the first integrals are similar to ”Hamiltonian equation of motion” with respect to ”Hamiltonian” $\Phi(x,y)$ (2.18).

The work was supported by the Russian Science Foundation (project 18-11-00032).