aa r X i v : . [ m a t h - ph ] J a n Extensions of non-natural Hamiltonians
Claudia Maria Chanu, Giovanni RastelliDipartimento di Matematica, Universit`a di Torino,via Carlo Alberto 10, 10123, Torino, Italiaemail: [email protected]; [email protected]
Abstract
The concept of extended Hamiltonian systems allows the geometri-cal interpretation of several integrable and superintegrable systems withpolynomial first integrals of degree depending on a rational parameter.Until now, the procedure of extension has been applied only in the caseof natural Hamiltonians. In this article, we give several examples of ap-plication to non-natural Hamiltonians, such as the two point-vortices, theLotka-Volterra and some quartic in the momenta Hamiltonians, obtainingeffectively extended Hamiltonians in some cases and failing in others. Webriefly discuss the reasons of these results.
Given a Hamiltonian L with N -degrees of freedom, the procedure of extensionallows the construction of Hamiltonians H with ( N + 1) degrees of freedom ad-mitting as first integrals L itself, with all its possible constants of motion, and acharacteristic first integral dependent on a rational parameter k . We gave sev-eral examples of extension of natural Hamiltonians on Riemannian and pseudo-Riemannian manifolds in [1, 2, 3, 4, 5, 6, 7], including anisotropic, Harmonicoscillators, three-body Calogero and Tremblay-Turbiner-Winternitz systems. Inall these examples, L and H are natural Hamiltonians and the characteristic firstintegral is polynomial in the momenta of some degree depending on k .However, the procedure of extension do not make any assumption on L otherthan it is a regular function on some cotangent bundle T ∗ M . Until now, wealways considered L as a natural Hamiltonian, in such a way that the extendedHamiltonian is itself natural. In this work, we apply the Extension Procedureon functions L which are no longer quadratic in the momenta and, consequently,the extended Hamiltonian is not a natural one. The construction of an extendedHamiltonian requires the determination of a certain function G well defined onall T ∗ M , up to some lower-dimensional subset of singular points. The extendedHamiltonian is a polynomial in p u , L , while its characteristic first integral isa polynomial in p u , L , G and X L G , the derivative of G with respect to theHamiltonian vector field of L , so that its global definition depends ultimatelyon G and X L G . Therefore, our analysis is focused on the determination of thefunction G in the different cases, and on the study of its global behaviour on T ∗ M . 1ince this work is intended as a preliminary study of the possible applicationsof the extension procedure to non-natural Hamiltonians, we do not pretendhere to obtain complete and general results, but we focus on some meaningfulexamples only.In Sec. 2 we recall the fundamentals of the theory of extended Hamiltonians.In Sec. 3 we consider extensions of functions quartic in the momenta and we findexamples of extended Hamiltonians in analogy with the quadratic Hamiltoniancase. The analysis becomes more subtle in Sec. 4, when we try to extendfunctions which are not polynomial in the momenta, as the case of the two point-vortices Hamiltonian. Here the global definition of the extended Hamiltonianand its characteristic first integral becomes an issue in some cases, so that theextension is possible only for some values of the constant of motion L , while inother cases the extension is always possible.We conclude in Sec. 5 with some examples where we are unable to find anyproperly globally defined extended Hamiltonian. Let L ( q i , p i ) be a Hamiltonian with N degrees of freedom, that is defined onthe cotangent bundle T ∗ M of an N -dimensional manifold M .We say that L admits extensions , if there exists ( c, c ) ∈ R − { (0 , } suchthat there exists a non null solution G ( q i , p i ) of X L ( G ) = − cL + c ) G, (2.1)where X L is the Hamiltonian vector field of L .If L admits extensions, then, for any γ ( u ) solution of the ODE γ ′ + cγ + C = 0 , (2.2)depending on the arbitrary constant parameter C , we say that any Hamiltonian H ( u, q i , p u , p i ) with N + 1 degrees of freedom of the form H = 12 p u − k γ ′ L + k c γ + Ω γ , k = mn , m, n ∈ N − { } , Ω ∈ R (2.3)is an extension of L .Extensions of Hamiltonians where introduced in [2] and studied becausethey admit polynomial in the momenta first integrals generated via a recursivealgorithm. Moreover, the degree of the first integrals is related with the choiceof m, n . Indeed, for any m, n ∈ N − { } , let us consider the operator U m,n = p u + mn γX L . (2.4) Proposition 1. [5] For
Ω = 0 , the Hamiltonian (2.3) is in involution with thefunction K m,n = U mm,n ( G n ) = (cid:16) p u + mn γ ( u ) X L (cid:17) m ( G n ) , (2.5) where G n is the n -th term of the recursion G = G, G n +1 = X L ( G ) G n + 1 n G X L ( G n ) , (2.6) starting from any solution G of (2.1). = 0, the recursive construction of a first integral is more complicated:we construct the following function, depending on two strictly positive integers s, r ¯ K s,r = (cid:0) U s,r + 2Ω γ − (cid:1) s ( G r ) , (2.7)where the operator U s,r is defined according to (2.4) as U s,r = (cid:18) p u + 2 sr γ ( u ) X L (cid:19) , and G r is, as in (2.5), the r -th term of the recursion (2.6), with G = G solutionof (2.1). For Ω = 0 the functions (2.7) reduce to (2.5) and thus can be computedalso when the first of the indices is odd. Theorem 2. [6] For any Ω ∈ R , the Hamiltonian (2.3) satisfies, for m = 2 s , { H, ¯ K m,n } = 0 , (2.8) for m = 2 s + 1 , { H, ¯ K m, n } = 0 . (2.9)We call K and ¯ K , of the form (2.5) and (2.7) respectively, characteris-tic first integrals of the corresponding extensions. It is proved in [2, 6] thatthe characteristic first integrals K or ¯ K are functionally independent from H , L , and from any first integral I ( p i , q i ) of L . This means that the extensionsof (maximally) superintegrable Hamiltonians are (maximally) superintegrableHamiltonians with one additional degree of freedom (see also [4]). In particular,any extension of a one-dimensional Hamiltonian is maximally superintegrable.The explicit expression of the characteristic first integrals is given as follows[5, 6]. For r ≤ m , we have U rm,n ( G n ) = P m,n,r G n + D m,n,r X L ( G n ) , (2.10)with P m,n,r = [ r/ X j =0 (cid:18) r j (cid:19) (cid:16) mn γ (cid:17) j p r − ju ( − j ( cL + c ) j ,D m,n,r = 1 n [( r − / X j =0 (cid:18) r j + 1 (cid:19) (cid:16) mn γ (cid:17) j +1 p r − j − u ( − j ( cL + c ) j , m > , where [ · ] denotes the integer part and D ,n, = n γ .The expansion of the first integral (2.7) is¯ K m,n = m X j =0 (cid:18) mj (cid:19) (cid:18) γ (cid:19) j U m − j )2 m,n ( G n ) , with U m,n ( G n ) = G n , and G n = [ n − ] X k =0 (cid:18) n k + 1 (cid:19) ( − cL + c )) k G k +1 ( X L G ) n − k − . (2.11)3 emark 1. In [3] it is proven that the ODE (2.2) defining γ is a necessarycondition in order to get a characteristic first integral of the form (2.5) or (2.7).According to the value of c and C , the explicit form of γ ( u ) is given (up toconstant translations of u ) by γ = ( − Cu c = 0 T κ ( cu ) = C κ ( cu ) S κ ( cu ) c = 0 (2.12)where κ = C/c is the ratio of the constant parameters appearing in (2.2) and T κ , S κ and C κ are the trigonometric tagged functions (see also [5] for a summaryof their properties) S κ ( x ) = sin √ κx √ κ κ > x κ = 0 sinh √ | κ | x √ | κ | κ < C κ ( x ) = cos √ κx κ > κ = 0cosh p | κ | x κ < T κ ( x ) = S κ ( x ) C κ ( x ) . Therefore, we have γ ′ = (cid:26) − C c = 0 − cS κ ( cu ) c = 0 . (2.13) Remark 2.
The global definition of the characteristic first integral of the ex-tended Hamiltonian is ultimately determined by the definition of G and itsderivative X L G . When these objects are globally defined, then also the charac-teristic first integral is.From the brief exposition given above it is clear that the extensions of a func-tion L on T ∗ M are completely determined once a solution G of (2.1) is known,provided it is regular and well defined on T ∗ M . The fundamental step for theapplication of the extension procedure is therefore the determination of G . In allexisting examples of extended Hamiltonians the function L is always a quadraticpolynomial in the momenta. The examples include the anisotropic harmonic os-cillators, the Tremblay-Turbiner-Winternitz and the Post-Winternitz systems.For several of these systems there exist a quantization theory, based on theKuru-Negro factorization in shift and ladder operators, adapted to Hamiltoni-ans which are extended Hamiltonians [8]. In order to generalise the extensionprocedure to non-natural Hamiltonians, we focus our research here on the de-termination of functions G solution of (2.1), leaving for other works a deeperanalysis of the resulting extended systems. We consider below some examplesof non-natural Hamiltonians, or natural Hamiltonians in a non-canonical sym-plectic or Poisson structure. Since the forms of the extended Hamiltonian andof the characteristic first integral are completely determined once known L and G , we are solely concerned with the determination and analysis of G and X L G . Hamiltonians of degree four in the momenta are considered in [9]. These Hamil-tonians are written in Andoyer projective variables and allow a unified rep-resentation of several mechanical systems, such as harmonic oscillator, Kepler4ystem and rigid body dynamics, corresponding to different choices of parame-ters. We consider here some toy model of Hamiltonians of degree up to four inthe momenta.1. Let us assume L = p + f ( q ) p + f ( q ) p + f ( q ) p + V ( q ) . (3.1)The exension of L is possible if global solutions G of X L G = − cL + c ) G, (3.2)are known. If we assume G ( q ), then the function G = C q + C , is a solution of (3.2) if L is in the form L = (cid:0) C p + 8 C f p + 2 cC q + 4 cC q + C f + 8 C C (cid:1) C − c c , (3.3)where C i are real constants and f ( q ) is an arbitrary function.Hence, we have that this system admits the most general extension, with c positive or negative.2. We consider now L = p + f ( q ) p + V ( q ) , (3.4)If we assume G = g ( q ) p and solve the coefficients of the monomials in p in (2.1) equal to zero, we obtain two solutions, one V = 14 ( 116 q c + 18 C cC q − C C ( C q + C ) + C ) − c c ,g = C q + C ,f = 116 q c + 18 C cC q − C C ( C q + C ) + C , (3.5)that, substituted in L , gives, by assuming c = 0, L = 11024 C ( C q + C ) (cid:0) p ( C q + 2 C C q + C C ) + C cq + 4 C C cq + 16 C C q + 5 C C cq + (32 C C C + 2 C c ) q + 16 C C C − C (cid:1) − c c , (3.6)and the other, holding for c = 0 also V = 1( C q + C ) (cid:18) C − cC q ( − C c q − C C c q − C C c q − C C c q + ( − C C c + 1024 c C − C C c ) q + (4096 c C C − C C C c − C C c ) q − C C c + 6144 c C C − C C C c ) q − C c + 4096 c C C − C C C c ) (cid:1) ,g = C q + C ,f = 116 C ( C q + C ) c + C ( C q + C ) , (3.7)where the C i are constants. It is interesting to remark that in the lastcase L is not in general a perfect square plus a constant, as in (3.3) and(3.6).3. We assume now L = (cid:18) p + 1( q ) p + V ( q , q ) (cid:19) , (3.8)that is the square of a natural Hamiltonian on E , and search for func-tions V allowing the existence of non-trivial solutions G of (2.1). Again,by assuming G ( q , q ) and collecting the terms in ( p , p ) in (2.1), the re-quirement that the coefficients of the momenta are identically zero, afterassuming c = 0, gives the following solution G = (cid:0) sin( q ) C + cos( q ) C (cid:1) q + C ,V = − c C sin( q ) − C cos( q )) (2 tan( q ) C C − C + C ) C (tan( q ) C − C ) ( q ) + c C sin( q ) − C cos( q )) C C (tan( q ) C − C ) q + F (cid:0) (sin( q ) C − cos( q ) C ) q (cid:1) , (3.9)where F is an arbitrary function.In all the examples above all the elements of the extension procedure arepolynomial in the momenta, therefore, the extended Hamiltonian and its char-acteristic first integral are globally defined in the same way as for the naturalHamiltonian case. The dynamics of two point-vortices z j = x j + iy j of intensity k j , j = 1 ,
2, in aplane ( x, y ) is described, in canonical coordinates ( Y i = k i y i , X i = x i ), by theHamiltonian L = − αk k ln ( X − X ) + (cid:18) Y k − Y k (cid:19) ! , where α = π and k i are real numbers [10].If k = − k , the functions ( k z + k z , L ) are independent first integrals ofthe system (three real functions). If k = − k , the functions above give onlytwo real independent first integrals.The coordinate transformation˜ X = ( X − X ) / , ˜ X = ( X + X ) / , ˜ Y = Y − Y , ˜ Y = Y + Y ,
6s canonical and transforms L into L = − αk k ln X + ˜ Y + ˜ Y k + ˜ Y − ˜ Y k ! . The exension of L is possible if global solutions G of (2.1) are known. Weconsider below two cases1. If k = k = k > L = − αk ln X + ˜ Y k ! . For c = 0, G in this case can be computed by using Maple, obtaining G = ˜ Y + 2 ik ˜ X √ Q ! Q √ c αk F + ˜ Y + 2 ik ˜ X √ Q ! − Q √ c αk F , (4.1)where Q = k e − Lαk = 4 ˜ X + ˜ Y k and F i are arbitrary functions of L .The function G is not single-valued in general, but it is, for example, when Q √ c αk is an integer.Let us consider now X L G , since Q depends on the canonical coordinatesthrough L only, Q and the exponents in G behave as constants under thedifferential operator X L , hence, the exponents in it remain integer if theyare integer in G and X L G is well defined on T ∗ M . Therefore, both H andits characteristic first integral are globally well defined for integer valuesof Q √ c αk .We have in this case an example in which the possibility of finding anextension depends on the parameters of the system and, in particular, onthe values of the constant of motion L .2. If k = − k = − k , k > L = αk ln X + ˜ Y k ! . For c = 0, the solution G is obtained by Maple as G = F sin √ c Q ˜ X αk ˜ Y ! + F cos √ c Q ˜ X αk ˜ Y ! , (4.2)where Q = k e Lαk and F i are arbitrary first integrals of L .It is evident that the function G above, real or complex, is always globallydefined, as well as X L G , up to lower-dimensional sets, and this makespossible the effective extension of the Hamiltonian L .We observe that in this case the extended Hamiltonian has four indepen-dent constants of motion. 7 Hamiltonians with no known extension
The procedure of extension can be applied in any Poisson manifold, not onlyin symplectic manifolds with canonical symplectic structure, as in the examplesabove. Indeed, if π is the symplectic form or the Poisson bivector determin-ing the Hamiltonian structure of the system of Hamiltonian L in coordinates( x , . . . , x n ), then the symplectic or Poisson structure of the extended manifoldin coordinates ( u, p u , x , . . . , x n ) is given byΠ = − π . We recall that the Hamiltonian vector field of L on Poisson manifolds withPoisson vector π is πdL .We consider below two cases of Hamiltonian systems for which we are unableto find extended Hamiltonians. The obstruction to the extension lies in bothcases in the non-global definition of the known solutions of (2.1). It is well known that the Lotka-Volterra prey-predator system˙ x = ax − bxy (5.1)˙ y = dxy − gy, (5.2)where a, b, d, g are real constants, can be put in Hamiltonian form (see [11]), forexample with Poisson bivector π = (cid:18) A − A (cid:19) , A = − x g y a e − by − dx , (5.3)and Hamiltonian L = x − g y − a e dx + by . (5.4)Since the manifold is symplectic, there is only one degree of freedom, and theexistence of the Hamiltonian itself makes the system superintegrable. The equa-tion (2.1) with c = 0 admits solution G of the form G = F ( L ) e − B + F ( L ) e B , (5.5)where B = − √− c a Z (cid:20) t (cid:18) W (cid:18) − ba t − ga x ga ye d ( t − x ) − bya (cid:19) + 1 (cid:19)(cid:21) − dt, and W is the Lambert W function, defined by z = W ( z ) e W ( z ) , z ∈ C . If we put F = (cid:16) α + βi (cid:17) , F = (cid:16) α − βi (cid:17) , where α and β are real constants,then G = α cos B − β sin B. C − { } , even if its variableis real (in this case, it is defined only for z ≥ − /e and double-valued for − /e < z < G cannot provide a globally defined firstintegral and does not determine an extension of L .By comparison, the Hamiltonian of the one-dimensional harmonic oscillatoradmits iterated extensions with c = 0 and c always equal to the elastic param-eter of the first oscillator. One obtains in this way the n -dimensional, n ∈ N ,anisotropic oscillator with parameters having rational ratios, and therefore al-ways superintegrable [5]. This is not the case for the Lotka-Volterra system,where the periods of the closed trajectories in x, y are not all equal, as happensfor the harmonic oscillators. It is well known that the Euler rigid-body system is described by the Hamilto-nian L = 12 (cid:18) m I + m I + m I (cid:19) , on the Poisson manifold of coordinates ( m , m , m ) and Poisson bivector π = − m m m − m − m m . The ( m i ) are the components of the angular momentum in the moving frameand they are conjugate momenta of the three components of the principal axesalong one fixed direction.A Casimir of π is M = m + m + m . The system has two functionally independent constants of the motion: L andone of the components of the angular momentum in the fixed frame.A solution of equation (2.1) can be found by using the Kuru-Negro [8] ansatz X L G = ± p − cL + c ) G, (5.6)whose solutions are solutions of the equation (2.1) too. A solution of (5.6) is G = f e (cid:20) ∓ I I I √ I I − I q − cL + c X F (cid:18) m q I I − I X , q I I − I X I I − I X (cid:19)(cid:21) , (5.7)where f is an arbitrary function of the first integrals L and M , X = I I ( M − I L ) , (5.8) X = I I (2 I L − M ) , (5.9)and F ( φ, k ) is the incomplete elliptic integral of first kind F ( φ, k ) = Z φ dθ p − k sin θ , which is a multiple valued function, being the inverse of Jacobi’s sinus amplitu-dinis sn function. The function G , possibly complex valued and with singular9ets of lower dimension, depends essentially on m , since all other arguments init are either constants or first integrals.Since our function G is not single-valued, we cannot in this case build anextended Hamiltonian from L . By the examples discussed in this article we see that extended Hamiltonianscan be obtained also from non-natural Hamiltonians L and not only from thenatural ones. The case when L is quartic in the momenta is very much similarto the quadratic cases studied elsewhere, and the extension procedure doesnot encounter new problems. For the two point-vortices Hamiltonian, we havethat global solutions of (2.1) can be obtained in correspondence of particularchoices of parameters in L . In the remaining examples, we are unable to obtainglobally defined solutions of (2.1) and we cannot build extensions in these cases.In future works, the extended Hamiltonians obtained here could be studiedin more details, while the search for global solutions for the cases of Lotka-Volterra and of the rigid body, or for the reasons of their non-existence, couldbe undertaken. Conflict of Interest : the authors declare that they have no conflicts ofinterest.
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