Lin Shang(尚琳)and Qing Huang(黃晴)

School of Mathematics,Northwest University,Xi’an 710069,China

Center for Nonlinear Studies,Northwest University,Xi’an 710069,China

1 Introduction

In classical integrable systems,the term complete integrability is well de fined.More precisely inndegrees of freedom,a Hamiltonian system can be integrated up to quadrature if there existninvolutive functions,which are functionally independent.When taking one of these functions as the HamiltonianHand thinking of others as its first integrals,this Hamiltonian is said to be completely integrable in the Liouville sense.Even though the number of independent functions,which are in involution isnat most,maybe additional integrals of the HamiltonianHexist and they will de finitely generate a non-Abelian algebra of integrals ofH.In general,these integrals of motion do not yield finite-dimensional Lie algebras,but more complicated algebraic structures,namely,Possion algebra.These systems,with more integrals,are called super-integrable.The maximal number of further independent integrals isn?1.If there are exactly 2n?1 independent first integrals,we say that the HamiltonianHis maximally superintegrable.The isotropic harmonic oscillator,the Kepler system,and the Calogero-Moser system are the well-known examples.Maximally superintegrable systems have some interesting features.For instance,they can be separable in more than one coordinate system and all bounded classical trajectories are closed.

There are numerous papers on superintegrable systems,both classical and quantum,see Refs.[1–5]and references therein.In most of these work,it was restricted to quadratic integrals of motion.The quadratic integrals have an intimate connection with the separation of variables in the Hamilton-Jacobi and Schr¨odinger equations and generally they produce so-called quadratic algebras.[6]

In contrast,much less attention has been devoted to integrable and superintegrable systems with cubic and higher-order integrals of motion.Ten different potentials in a complex Euclidian plane,which admitted a cubic integral of motion are listed in Ref.[7].Among them,seven are reducible for their cubic integrals can be written as the Poisson commutators of two quadratic integrals.[8]Systematic studies for superintegrable systems with at least one cubic integral was initiated in Ref.[9].There the coexistence of first-and third-order integrals of motion in two-dimensional classical and quantum mechanics are considered.Corresponding potentials and integrals are identi fied.The above mentioned paper was followed by a series of publications.In Refs.[10–11],superintegrable systems that are separable in Cartesian coordinates and admit a cubic integral in classical and quantum mechanics are presented.Polynomial Poisson algebras for classical superintegrable systems with a cubic integral are given in Ref.[12].The complete classi fication of quantum and classical superintegrable systems allowing the separation of variables in polar coordinates and admitting an additional integral of motion of order three in the momentum is performed.[13]In Ref.[14],the conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied.

In Ref.[15],families of Hamiltonians of the form

is considered.There differential Galois theory was utilized to determine necessary conditions for complete integrability.WhenU(φ)=?cosφ,eight integrable systems were isolated,four of which were superintegrable indeed.For each one among the four superintegrable cases,two first integrals,which were quadratic in momenta,were also exhibited.And it means that these systems are separable in two different coordinate systems.Based on this fact,shortly later these superintegrable systems were studied in more detail.[16]These systems were reproduced by change of coordinates.In addition,the identity of these systems was clari fied and the corresponding Poisson algebras were given.Quite recently,superintegrable systems of above form with a position dependent mass were investigated.[17]

Here we deal with the Hamiltonian functions taking the form of

and study the superintegrability of Eq.(1)with one quadratic and one cubic integrals of motion.Unless speci fied otherwise,n=2.In Sec.2,three geometric first integrals of the kinetic energy,related to the Killing vectors,are derived.Starting from the Killing vectors,cubic integrals are built and corresponding integrable systems are determined in Sec.3,since the metric implied by the kinetic energy of Eq.(1)is flat.And in Sec.4,further restriction of quadratic integrals yields superintegrable systems where the potentials are given explicitly.Section 5 is devoted to the integrability of Eq.(1)withn=2.And the last section contains some remarks and conclusion.

2 First Integrals of Kinetic Energy

Now we recall some known facts on Killing vectors.In a Riemannian manifold(M,g),a Killing vector fieldXis the in finitesimal generator of a symmetry of the metricg.NamelyXis a generator of isometries.In geometric termsXsatis fies the conditionLXg=0 with the Lie derivativeLX.For a manifold of dimensionn,the metric possessesn(n+1)/2 linearly independent Killing vectors at most.Note that when the space is either flat or constant curvature,it admits the maximal group of isometries,which isn(n+1)/2 dimension.

For a Riemannian manifold(M,g),which corresponds to a 2n-dimensional con figuration space,with coordinates(q,p),of a system,gdetermines a kinetic Lagrangiansuch that the associated motion is just the geodesic motion with kinetic energywheregijis the inverse of the metric tensorgijwhengijis nonsingular.Note that(q1,q2,...,qn)is generalised position(this can be for example Cartesian coordinates,angles,arc lengths along a curve)and(p1,p2,...,pn)is corresponding generalised momentum.For a metric with isometries,the Killing vectors correspond to functions that are linear in momenta(the so-called Noether integrals)and that Poisson commute with the kinetic energyH0.

Following the standard approach,direct computation shows that the kinetic energy has three first order integrals

which satisfy the following commutative relations

In view of the algebraic structure,the quadratic Casimir of the Euclidean algebra(3)is(1/2)And the corresponding Killing vectors can be written as

andX3=[2/(2?n)]?φ.

Generally speaking,for the kinetic energy,the introduction of a potentialh(r,φ)will destroy its first integrals.Nevertheless,it is well-known that for a flat metric,the leading-order terms of all second-and higher-order integrals of motion are determined by corresponding Killing tensors built from tensor products of Killing vectors.[18]

3 Existence of a Cubic Integral

Applying the canonical transformation governed by the generating function

By straightforward computation,we find that the terms of odd and even orders of momenta can not exist simultaneously since they will commute independently.For the generality of this form,see Ref.[19].Hence,we only need to search for cubic integrals of the form

whereAijkare constants.

And thus,for the Hamiltonian(1),we should consider the cubic integrals

whereKiare de fined in Eq.(3).Here we restrict ourselves to the case

and construct corresponding integrable and superintegrable systems.The condition{H,F1}=0 gives rise to three integrable systems with the following potentials

4 Superintegrable Systems

With integrable systems(4),(5),and(6)allowing one cubic first integrals in hand,we impose a second independent function,which is the second order of motion and Poisson commutes with its Hamiltonians.Here we consider superintegrability of systems(4),(5),and(6)separately.

4.1 Superintegrability of System(4)

(i)Superintegrable Restriction with F2=K2K3+f(r,φ)

Given the second first integrals,the arbitrary functionψin Eq.(4)is speci fied up to finite parameters.In the case of

With these potentials,we have{H,F1}=0 and{H,F2}=0.{F1,F2}is a quartic integral and it may be a polynomial combination ofH,F1,andF2.However,it is not the case.We try to close the algebra at lowest dimension possible.Now we add two elementsK2andF3,with

the integralsH,K2,F1,F2,andF3form a polynomial Poisson algebra.Following the Poisson relations and without using the speci fic representation,the Casimir functionscan be obtained.Inserting the exact forms of the integrals,we find that

which are the algebraic relations among the five first integralsH,K2,F1,F2,andF3.

Remark 1According toF1={F2,F3},the cubic integralF1can be expressed as the commutator of two quadratic integralsF2andF3.Consequently,the cubic integral is reducible.And the superintegrable system coincide with system(14)obtained in Ref.[16].

(ii)Superintegrable Restriction with F2=K1K3+f(r,φ)

Assuming system(4)allows the following quadratic integral

which shows that the functionsH,F1,F2,K2generate a Poisson algebra.And of course,they are dependent and satisfy the relation

(iii)Superintegrable Restriction with F2=K1K2?K1K3+f(r,φ)

Given a quadratic integral of the form

And for this algebra,the constraintholds.

4.2 Superintegrability of System(5)

Here for the system(5)with

we preselect a speci fic quadratic integralF2and verify that the system(5)is superintegrable.

Assume

hold.As a consequence,H,F1,F2bring about a Poisson algebra.

4.3 Superintegrability of System(6)

System(6)is superintegrable if the Hamiltonian there admits another independent first integral.ChoosingF2=K1K2+f(r,φ),we have

Moreover{F1,F2}=(4c1/3)(F2?H).And thusH,F1,F2generate a Poisson algebra.

5 Integrability of Eq.(1)Whenn=2

Providedn=2,the kinetic energy(2)reduces toIts Killing vectors take the forms of

and solving the equations induced by the involutive relation,we have three integrable systems whose potentials are listed as below:

We now look for superintegrable systems(8)with the potential given in Eq.(9)and prove that systems(8)corresponding to Eqs.(10)and(11)are superintegrable themselves.

To setF2=K2K3+f(r,φ)and suppose it is admitted by Eq.(9)give

Following these relations,the first integralsK2,F1,F2,F3,Hform a Poisson algebra with the algebraic relation.In addition,choosingF2=K1K3+f(r,φ)yields another superintegrable system where

With above potentials,H,K2,F1,F2give a Poisson algebra where

For system(10),it also possesses a quadratic first integral

Since{F1,F2}=(4c1/3)(H+F2),now we have a Poisson algebra spanned byH,F1andF2.

Analogously,the quadratic integralF2=K1K2?(c1/3)(φ+lnr)is admitted by the Hamiltonian(8)with(11)and the corresponding Poisson algebra is governed byH,F1,F2,which satisfy{F1,F2}=(4c1/3)(F2?H).

6 Concluding Remarks

In this paper we dealt with the super integrability of some Hamiltonian systems,namely for 2n-dimensional Hamiltonian equation,there exists more thannfirst integrals.Such a property is stronger compared with“complete integrability”,namely the solution structure of such systems has fewer arbitrary parameters since the motion is more restricted by the extra first integrals.Here we constructed some superintegrable systems with one quadratic and one cubic integrals of motion,and build up Poisson algebra for each superintegrable Hamiltonian system.Moreover,the algebraic dependence relations to each Poisson algebra are also given.

A generalization of integrable Hamiltonian equations is the study of the integrability of in finite-dimensional equations,namely integrable partial differential equations(also known as soliton equations)including a number of(1+1)-and(2+1)-dimensional equations?Note that here the dimension refers to the number of independent variables,while in this paper by dimension we mean the number of the components.,in which case an in finite number of independent first integrals being in involution is required in order to guarantee the integrability,see e.g.Adler,[20]Gel’fand and Dikii,[21]Magri,[22]Fuchsteiner and Fokas,[23]and TAH,[24]etc.

The theory of the completely integrability of soliton equations is also applicable to the so-called super soliton equations.We refer the reader to Zhang[25?27]and references therein.

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