Superintegrable St\"ackel Systems on the Plane: Elliptic and Parabolic Coordinates

Recently we proposed a generic construction of the additional integrals of motion for the St\"ackel systems applying addition theorems to the angle variables. In this note we show some trivial examples associated with angle variables for elliptic and parabolic coordinate systems on the plane.


Introduction
In classical mechanics Hamiltonian system on a 2n-dimensional phase space M is called completely integrable in Liouville's sense if it possesses n functionally independent integrals of motion H 1 , . . . , H n in involution: where H = H 1 is the Hamilton function and {·, ·} is the Poisson bracket on M . Superintegrable system is a system that is integrable in the Liouville sense and that possesses more functionally independent integrals of motion than degrees of freedom. The construction of superintegrable Stäckel systems using angle variables ω k has been proposed in [12,13,14,15].
In generic case the action variables ω k are multi-valued functions on the whole phase space M . In fact, we can extract polynomial integrals of motion from angle variables only when we can apply addition theorems to the corresponding Abelian integrals. As there are only few addition theorems for the Abel equations [1,3] we can easily classify the corresponding superintegrable systems, see [12,13,14,15].
The goal of this brief note is to present some trivial examples of applying this generic theory associated with elliptic and parabolic coordinate systems on the plane. Superintegrable systems separable in spherical coordinates can be found in [2,6]. The corresponding addition integrals of motion are related with an addition theorem for the logarithmic angle variables. Of course, there is a trivial generalization of the proposed method for all the orthogonal coordinate systems in R 3 (ellipsoidal, paraboloidal, cylindrical, prolate and oblate spheroidal coordinates etc).
The non-Stäckel superintegrable systems in classical and quantum mechanics have been considered in [7,9]. In contrast with the Stäckel case we do not have a generic theory for constructing such superintegrable systems.

The Stäckel systems
The system associated with the name of Stäckel [11] is a holonomic system on the phase space R 2n , with the canonical variables q = (q 1 , . . . , q n ) and p = (p 1 , . . . , p n ): The nondegenerate n × n Stäckel matrix S, whose j column depends only on coordinate q j , defines n functionally independent integrals of motion the separated relations and the action variables w k In generic case the action variables (2.1) are sums of the multi-valued Abelian integrals. However, if we are able to apply the known addition theorems for these Abelian integrals then we can get additional integrals of motion [12,13,14].
Let us discuss addition theorems for the logarithmic (exponential) and elliptic functions [1,3]. In the first case polynomials are the second-order polynomials and So, we can easily get polynomial or rational function at the special choice of k j and z. In the second case m − 1 angle variables ω k , . . . , ω k+m−1 with 1 < m ≤ n have to satisfy to the Abel equations · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · x m−2 with a common polynomial of fixed degree 2m If this case there are some additional Richelot integrals of motion [10] C Here a k are values of x at the branch points of the corresponding hyperelliptic curve and [10].
At m = 2 we have famous Euler algebraic integral [3]. If A 2m = 0 and A 2m−1 = 0 in (2.3) there is another additional Richelot integrals of motion [10]. The Weierstrass generating function of such integrals for any values of the coefficients A k and other constructions of additional integrals of the Abel equations are discussed in [1]. Some example of the Euler and Richelot superintegrable systems may be found in [4,12,15].
Below we show how these addition theorems could help us to classify superintegrable systems.

Elliptic coordinate system
Let us consider elliptic coordinates on the plane q 1,2 defined by The corresponding momenta reads as

The Stäckel matrix and the separated relations
give rise to the following Hamiltonians in the involution The Hamiltonian H 1 commutes with the second angle variable w 2 , which is equal to standing under square root in these integrals are at least third-order polynomials on λ. So, in this case we can not apply addition theorem for the logarithms.
It is easy to see that we can apply addition theorem for the elliptic functions at Namely, if we put λ = x and λ = y in the first and second integrals (3.1), we could apply the Euler addition theorem to angle variable ω 2 . Here X is an arbitrary quartic and Y, S are the same functions of another variables y and s. In this case, symmetrical biquadratic form of x and y F (x, y) = ax 2 y 2 + 2bxy(x + y) + c x 2 + 4xy + y 2 + 2d(x + y) + e = 0 defines the conic section on the plane (x, y). According to [1,3,10], there is a famous Euler integral For the quartic (3.2) associated with the angle variable (3.1) this Euler integral looks like Here q 1,2 and p 1,2 are elliptic coordinates and momenta. It is a third-order polynomial in momenta which commutes with the Hamiltonian The algebra of the polynomial integrals of motion H 1 , H 2 , H 3 can be closed only after some other polynomial generators are added. Thus, we easily find the additional integrals of motion for the Hamilton function of the oscillator 4H 1 = p 2 x + p 2 y + a x 2 + y 2 using the separation of variables in elliptic coordinate system and the corresponding angle variables. Another result is that there is only one superintegrable system separable in elliptic coordinates and associated with the known addition theorems for Abelian integrals.

Parabolic coordinate system
Let us consider parabolic coordinates on the plane q 1,2 defined by and the corresponding momenta The Stäckel matrix and the separated relations give rise to the Hamiltonians The Hamiltonian H 1 commutes with the second angle variable w 2 , which is equal to In contrast with the elliptic coordinates, polynomials standing under square root in these integrals are at least second-order polynomials on λ. So, we can apply both known addition theorems to these Abelian integrals.
In fact, these integrals are expressed via logarithmic functions iff: The addition theorem for the elliptic function is applicable iff:
Here r = x 2 + y 2 and ϕ = arctan x/y are polar coordinates on the plane. According to [8] these systems remain superintegrable in the quantum case.

Case 1
In the first case the second angle variable equals The application of the addition theorem (2.2) to ω 2 gives rise to the following rational integral of motion In order to calculate polynomial integral of motion let us consider a series expansion of the function by momenta p 1,2 . Here we substitute the variables q 1,2 from the separated relations (4.1) into the rational integral Z and α, β are undefined polynomials in H 1,2 .
Equating first coefficient of this expansion to zero one gets the following expressions for these polynomials At this values of α and β the function f becomes a third-order polynomial in momenta such that In order to close the algebra of the polynomial integrals of motion H 1 , H 2 , H 3 we have to add one more polynomial generator by analogy with exp(ω), sin(ω) and cos(ω) functions.
Remark 1. One of the referees proposed another construction of the polynomial integrals of motion from the angle variable ω 2 . Namely, from the separation equations we can deduce that We can therefore write Consequently is a third-order constant of motion. This method of explanation may be clearer than the method of expansion of the rational in momenta function f = αZ + βZ −1 with indefinite coefficients α and β.
Remark 2. Let us remind, that two-dimensional open Toda lattice defined by the following polynomial integrals of motion has the non-rational in momenta additional integral of motion which can be also obtained from the second angle variable ω 2 [14]. However, it is easy to prove, that we can not apply working above constructions of polynomial integral of motion H 3 in this case.

Case 2
In the second case the angle variable is equal to Changing variables µ = λ 2 one gets second-order polynomials under the square root and desired sum of the logarithms The rational integral of motion (2.2) is equal to As above we consider the expansion of the function by momenta p 1,2 . Equating first coefficient of this expansion to zero one gets polynomials α, β At this values of α and β the function f becomes a third-order polynomial in momenta .
such that In order to close the algebra of the polynomial integrals of motion H 1 , H 2 , H 3 we have to add one more polynomial generator
The algebra of the integrals of motion H 1 , H 2 , H 3 is more complicated then the algebra associated with the addition theorem for logarithms. In fact, in order to close this algebra we have to introduce the counterparts of the Jacobi elliptic functions sn(ω), cn(ω) and dn(ω) instead of the trigonometric functions sin(ω) and cos(ω), which we used for the superintegrable systems associated with the addition theorem for logarithms.

Conclusion
It is known that orthogonal coordinate systems on Riemaniann manifolds can be viewed as an orthogonal sum of certain basic coordinate systems and these basic systems can be obtained from the elliptic coordinate system [5] using a degeneration procedure. This degeneration decreases the degree of polynomials standing under square roots into the angle variables (2.1). Thus, we have only one superintegrable systems separable in elliptic coordinates, whereas for degenerations we have a lot of different superintegrable systems. As usual, the addition theorem for logarithms allows us to get additional integrals of higher order in momenta [12,13].