Abstract
The generalization of (super)integrable Euclidean classical Hamiltonian systems to the twodimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon–Heiles system on the sphere and the hyperbolic plane is introduced. Projective Beltrami coordinates are shown to be helpful in this construction, and further applications of this approach are sketched.^{1}^{1}1To appear in “Integrability, Supersymmetry and Coherent States”, A volume in honour of Professor Véronique Hussin. S. Kuru, J. Negro and L.M. Nieto (Eds.), Special volume of the CRM Series in Mathematical Physics (Berlin: Springer, 2019)
Curvature as an integrable deformation
Angel Ballesteros, Alfonso Blasco and Francisco J. Herranz
Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain
Emails: angelb@ubu.es, ablasco@ubu.es, fjherranz@ubu.es
MSC: 37J35, 70H06, 22E60
PACS: 02.30.Ik, 45.20.Jj, 02.20.Sv, 02.40.Ky
Keywords: Integrable systems, Curvature, Sphere, Hyperbolic plane, Integrable perturbations, Oscillator potential, HénonHeiles
Contents
 1 Introduction
 2 Geodesic dynamics on the sphere and the hyperboloid
 3 Beltrami coordinates and projective dynamics
 4 Anisotropic oscillators on the Euclidean plane
 5 Anisotropic oscillators on and
 6 Integrable Hénon–Heiles systems
 7 An integrable KdV Hénon–Heiles system on and
 8 Remarks and open problems
section*
section*
1 Introduction
The aim of this contribution is to review some new recent results related to a seemingly elementary issue in the theory of finitedimensional integrable systems [1, 2, 3, 4, 5], whose solution presents quite a number of interesting features. The problem can explicitly be stated as follows.
Let us consider a certain Liouville integrable natural Hamiltonian system for a particle with unit mass moving on the twodimensional (2D) Euclidean space endowed with the standard bracket in terms of canonical coordinates and momenta, namely
(1) 
where is the kinetic energy and is the potential. The Liouville integrability of this system will be provided by a constant of the motion given by a globally defined function such that .
The proposed problem consists in finding a oneparameter integrable deformation of of the form
with integral of the motion given by the smooth and globally defined function (therefore ), and such that the following two conditions hold:

The smooth function is the kinetic energy of a particle on a 2D space whose constant curvature is given by the parameter , i.e. the 2D sphere S will arise in the case and the hyperbolic plane when .

The Euclidean system given by (1) has to be smoothly recovered in the zerocurvature limit , namely
If these two conditions are fulfilled, we will say that is an integrable curved version of on the sphere and the hyperbolic space. We stress that within this framework the Gaussian curvature of the space enters as a deformation parameter, and the curved system can be thought of as smooth integrable perturbation of the flat one in terms of the curvature parameter. Therefore, integrable Hamiltonian systems on S (), H () and E () will be simultaneously constructed and analysed.
Moreover, it could happen that the initial Hamiltonian is not only integrable but superintegrable, i.e. another globally defined and functionally independent integral of the motion does exist such that
In that case we could further impose the existence of the curved (and functionally independent) analogue of the second integral such that
If we succeed in finding such second integral fulfilling
we will say that we have obtained a superintegrable curved generalization of the Euclidean superintegrable Hamiltonian .
The explicit curvaturedependent description of S and H is wellknown in the literature and can be found, for instance, in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] (see also references therein) where it has been mainly considered in the classification and description of superintegrable systems on these two spaces. In this contribution we will present several recent works in which this geometric framework has been applied for nonsuperintegrable systems where the lack of additional symmetries forces to make use of a purely integrable perturbation approach. Moreover, this perturbative viewpoint shows that the uniqueness of this construction is not guaranteed, since in general different integrable potentials (and their associated integrals) having the same limit could exist and be found. As an outstanding example of this plurality, we will present the construction of different integrable curved analogues on S () and H () of some anisotropic oscillators.
The second novel technical aspect to be emphasized in the results here presented is that in some cases projective coordinates turn out to be helpful in order to construct the (super)integrable deformations , since when these coordinates are considered on S and H then the curved kinetic energy is expressed as a polynomial in the canonical variables describing the projective phase space. Therefore, some of the examples here presented can be thought of as instances of integrable projective dynamics, in the sense of [29, 30].
The structure of the paper is the following. In the next Section we review the description of the geodesic dynamics on the sphere and the hyperboloid by making use of the above mentioned curvaturedependent formalism. In particular, ambient space coordinates as well as geodesic parallel and geodesic polar coordinates for S and H will be introduced. In Section 3 the projective dynamics on the sphere and the hyperboloid in terms of Beltrami coordinates will also be summarized, thus providing a complete set of geometric possibilities for the description of dynamical systems on these curved spaces. In Section 4 we recall the (super)integrability properties of the 2D anisotropic oscillator with arbitrary frequencies and also with commensurate ones, and in Section 5 the explicit construction of the Hamiltonian defining its curved analogue will be presented. Section 6 will be devoted to recall the three integrable versions of the wellknown (nonintegrable) Hénon–Heiles Hamiltonian. In Section 7 the construction of the curved version on S and H of an integrable Hénon–Heiles system related to the KdV hierarchy will be constructed, thus exemplifying the usefulness of the approach here presented for the obtention of new integrable systems on curved spaces. Furthermore, the full Ramani–Dorizzi–Grammaticos series of integrable polynomial potentials will also be generalized to the curved case. Finally, a Section including some remarks and open problems under investigation closes the paper.
2 Geodesic dynamics on the sphere and the hyperboloid
Let us consider the oneparametric family of 3D real Lie algebras with commutation relations given by (in the sequel we follow the curvaturedependent formalism as presented in [31, 32]):
(2) 
where is a real parameter. The Casimir invariant, coming from the Killing–Cartan form, reads
(3) 
The family comprises three specific Lie algebras: for , for , and for . Note that the value of can be reduced to through a rescaling of the Lie algebra generators; therefore setting in (2) can be shown to be equivalent to applying an Inönü–Wigner contraction [33].
The involutive automorphism defined by
generates a grading of in such a manner that is a graded contraction parameter [34], and gives rise to the following Cartan decomposition of the Lie algebra:
We denote and the Lie groups with Lie algebras and , respectively, and we consider the 2D symmetrical homogeneous space defined by
(4) 
This coset space has constant Gaussian curvature equal to and is endowed with a metric having positive definite signature. The generator leaves a point invariant, the origin, so generating rotations around , while and generate translations which move along two basic orthogonal geodesics and .
Therefore (4) covers the three classical 2D Riemannian spaces of constant curvature:
We recall that these three spaces (and their motion groups ) are contained within the family of the socalled 2D orthogonal Cayley–Klein geometries [6, 35, 36], which are parametrized in terms of two graded contraction parameters and [34].
In what follows we describe the metric structure and the geodesic motion on the above spaces in terms of several sets of coordinates that will be used throughout the paper. We stress that all the resulting expressions will have always a smooth and welldefined flat limit (contraction) reducing to the corresponding Euclidean ones.
2.1 Ambient space coordinates
The vector representation of is provided by the following faithful matrix representation [8, 9]
(5) 
which satisfies
(6) 
The matrix exponentiation of (5) leads to the following oneparametric subgroups of :
(7)  
where we have introduced the dependent cosine and sine functions [6, 8]
The tangent function is defined as
These curvaturedependent trigonometric functions coincide with the circular and hyperbolic ones for , while under the contraction they reduce to the parabolic functions: and . Some trigonometric relations read [8]
and their derivatives are given by [9]
Therefore, under the matrix realization (7), the Lie group becomes a group of isometries of the bilinear form (6),
acting on a 3D linear ambient space through matrix multiplication. The subgroup (7) is the isotropy subgroup of the point , which is taken as the origin in the homogeneous space (4). The orbit of is contained in the “sphere” determined by (6):
(8) 
The connected component of is identified with the space and the action of is transitive on it. The coordinates , satisfying the constraint (8) are called ambient space or Weierstrass coordinates. Notice that for we recover the sphere, if we find the twosheeted hyperboloid, and in the flat case with we get two Euclidean planes with Cartesian coordinates . Since , we identify the hyperbolic space with the connected component corresponding to the sheet of the hyperboloid with , and the Euclidean space with the plane .
The metric on comes from the flat ambient metric in divided by the curvature and restricted to :
(9) 
Isometry vector fields in ambient coordinates for , fulfilling (2), are directly obtained from the vector representation (5):
(10) 
where .
Now we consider the ambient momenta conjugate to fulfilling the canonical Poisson bracket subjected to the constraint (8). The vector fields (10) give rise to a symplectic realization of in terms of ambient variables by setting :
(11) 
which close the Poisson brackets defining the Lie–Poisson algebra
The metric (9) provides the free Lagrangian with ambient velocities for a particle with unit mass, so determining geodesic motion on :
(12) 
Thus the corresponding momenta read
(13) 
The time derivative of the constraint (8) provides the relation
Finally, by introducing (13) in (12) we obtain that the kinetic energy in ambient variables is given by
(14) 
Notice that the contraction is welldefined in the r.h.s. of the equations (9), (12) and (14) yielding the Euclidean expresions
2.2 Geodesic parallel and polar coordinates
The ambient coordinates (8) can also be parametrized in terms of two intrinsic variables of geodesic type. For our purposes let us consider the socalled geodesic parallel and geodesic polar coordinates of a point [7, 9], which are defined through the following action of the oneparametric subgroups (7) on the origin :
which gives
(15)  
In this construction, the variable is the distance between the origin and the point measured along the geodesic that joins both points, while is the angle of with respect to a base geodesic (associated with the translation generator ). Let be the intersection point of with its orthogonal geodesic through . Then is the geodesic distance between and measured along and is the geodesic distance between and measured along . On with , the relations (15) lead to and so reducing to Cartesian and polar coordinates.
These coordinates are shown Figure 1 for and . In these pictures, is the base geodesic orthogonal to through , so related to , and is the intersection point of with its orthogonal geodesic through .
Now, we denote and the conjugate momenta of the coordinates and , respectively, and the free Hamiltonian (kinetic energy) turns out to be
(16) 
3 Beltrami coordinates and projective dynamics
The quotients of the ambient coordinates (8) are just the Beltrami coordinates of projective geometry for the sphere and the hyperbolic plane. They are obtained by applying the central stereographic projection with pole of a point onto the projective plane with and coordinates :
giving rise to the expressions
(18)  
Thus the origin goes to the origin in the projective space .
The domain of depends on the value of the curvature . We write in terms of the radius of the space as and we find that in the sphere with , . The points in the equator in with () go to infinity, so that the projection (18) is welldefined for the hemisphere with . In the hyperbolic or Lobachevski space with and it is satisfied that
which is the Poincaré disk in Beltrami coordinates and
The points at the infinity in are mapped onto to the circle . Finally, in the Euclidean plane , with (), the Beltrami coordinates are just the Cartesian ones .
By introducing (18) in the ambient metric (9) and in the free Lagrangian (12) we obtain that
(19) 
where and hereafter we shall use the following notation for any 2vectors and :
The Beltrami momenta conjugate to the coordinates , such that , come from
(20) 
And by inserting these expressions into (19) we get the free Hamiltonian
(21) 
By introducing (18) and (20) in (13) we obtain the ambient momenta written in terms of the Beltrami variables, , and from this result a symplectic realization of the Lie–Poisson generators (11) in these variables is directly found. These expressions are displayed in Table 1. Notice that the kinetic energy (21) can also be recovered by computing the symplectic realization of the Casimir (3) of in Beltrami variables as . Likewise the ambient momenta and symplectic realization of the Lie–Poisson generators can be computed in the geodesic variables introduced in Section 2.2, and these are also presented in Table 1.
We recall that a similar procedure can be performed with Poincaré coordinates [37] which come from the stereographic projection with pole . The resulting expressions can be found in [32].
Beltrami  Geodesic parallel  Geodesic polar  
4 Anisotropic oscillators on the Euclidean plane
To start with, let us consider the Hamiltonian determining the anisotropic oscillator with unit mass and frequencies and on the Euclidean plane in Cartesian coordinates and conjugate momenta :
(22) 
Clearly, this Hamiltonian is always integrable due to its separability in Cartesian coordinates so that it Poissoncommutes with the (quadratic in the momenta) integrals of motion
which are not independent since
Furthermore, it is also wellknown that for commensurate frequencies the Hamiltonian (22) provides a superintegrable system [38, 39, 40], in such a manner that an “additional” (in general higherorder in the momenta) integral of motion does exist.
The (super)integrability properties of the commensurate oscillator will be sketched by following the approach given in [41, 42], which is based on a classical factorization formalism (see [43, 44, 45, 46, 47] and references therein). If we denote
(23) 
then (22) can be written in terms of the parameter and frequency as
(24) 
Next we introduce new canonical variables
(25) 
giving rise to
(26) 
Therefore we obtain two 1D Hamiltonians and given by
(27) 
which are two integrals of the motion for . The 1D Hamiltonian (27) can then be factorized in terms of “ladder functions” as
(28) 
fulfilling
The remaining 1D Hamiltonian (27) can also be factorized through “shift functions” in the form
(29) 
so that
Notice that the sets of functions and span a Poisson–Lie algebra isomorphic to the harmonic oscillator Lie algebra . Hence, the 2D Hamiltonian (26) can finally be expressed in terms of the above ladder and shift functions as
The remarkable fact now is that if we consider a rational value for ,
(30) 
we obtain two additional complex constants of the motion for (26)