Symmetry group classification of three-dimensional Hamiltonian systems

Keywords-Hami l ton ian systems, Symmetry groups, Classification. 1. I N T R O D U C T I O N In [1], the Lie point symmet ry groups of a Hamiltonian system with two degrees of freedom were completely classified. In tha t case, a maximum dimension of 15 was obtained for a free particle and all dimensions between 1 and 7. We should clarify tha t we are dealing only with point t ransformations. In other words, the generators are functions of the dependent and independent variables; there is no dependence on the derivatives. This paper is a report on some recent work for three-dimensional systems. We will not present the full classification, which is quite extensive, but instead we will give a list of potentials for each dimension tha t appears, together with the corresponding symmet ry group. The classification for a general ordinary differential equation of second order with one dependent and one independent variable goes back to [2]. Lie showed tha t the dimensions of a maximal admit ted algebra take only the values 1, 2, 3, and 8. Lie actually gave a group classification of all arbi t rary order O.D.E.s. In this way, he identified all equations tha t can be reduced to lower-order equations or completely integrated by group theoretical methods [3]. The problem of classifying symmet ry groups for a system of differential equations is open. This is mentioned in [4] where some known facts are presented. Some results for linear systems of second-order ordinary differential equations can be found in [5]. We consider the motion of a particle of unit mass in three-dimensional space (ql, q2, qa) under the influence of apo ten t i a l of the form V(ql, q2, q3). We will assume tha t the Hamiltonian is t ime independent. This is not really a restriction because a t ime-dependent n-dimensional system is 0893-9659/00/$ see front matter ~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by .A~-TEX PII: S0893-9659(99)00166--4 64 P. A. DAMIANOU AND C. SOPHOCLEOUS equivalent to a time-independent (n + 1)-dimensional system by regarding the time variable as the new coordinate. We assume that the Hamiltonian has the form H(ql,q2,q3,Pl,P2,P3): 2P21 A-~p2-4-2P2q V(ql,q2, q3). (1) Hamilton's equations, in Newtonian form, become OV q'~ = Oq---~' i = 1, 2, 3. (2) We search for point symmetries of system (2). That is, we search for the infinitesimal transformations of the form t ' = t + eT ( t , ql, q2, q3) + O (e2) , (3) q~ = qi + eQi( t , ql, q2, q3) + O (e2), i = 1,2, 3. Equations (2) admit Lie transformations of the form (3) if and only if r (2) {~ + vq,} = 0, i = i, 2, 3, (4) where r (2) is the second prolongation of r = T --° + Q~b~q~ + Q~__0 + Q3 0 Ot Oq2 Oq3 " (5) For the reader who is unfamiliar with the definition and properties of Lie point symmetries, there are a number of excellent books on the subject, e.g., [6-9]. Equations (4) give three identities of the form E~(t, q l ,q2 ,q3, qt ,q2, q3) = O, i = 1,2,3, (6) where we have used that c1~ = o v -b'~q~ for i = 1, 2, 3. The functions El , E2, and E3 are explicit polynomials in ql, q2, and q3. We impose the condition that equations (6) are identities in seven variables t, ql, q2, q3, ~1, q'2, and q'3 which are regarded as independent. These three identities enable the infinitesimal transformations to be derived and ultimately impose restrictions on the functional forms of V, T, Q1, Q2, and Q3. After some straightforward calculations one can show, (see e.g., [10]) that the generators necessarily have the following form: 3 T=a(t) + ~bk(t)qk, k-~l 3 3 k----1 k= l i -1,2,3. (7) As in the case of two degrees of freedom, we obtain a maximum dimension of (n + 2) 2 1 = 24 for a free particle and the other dimensions vary between 1 and n 2 q3 = 12. In this short note, we will not present in detail the structure of the Lie algebras that appear. Most of the systems in [1] are extended from two to three dimensions and completion of these two important eases will enable one to generalize in n dimensions. Connections with integrability will also be presented in a future paper. We should point out that there are symmetries other than point symmetries. One may allow the infinitesimals T, Qi to depend on t, qi and the derivatives of q~. Transformations of this type are commonly called Lie-B~icklund or generalized transformations. There is also the notion of Hamiltonian Systems 65 dynamical or contact symmetries (a subset of Lie-B/icklund transformations) where the generators are velocity dependent. In this paper, we have used the classical method of finding point transformations. This method may well overlook discrete symmetries such as simple reflections. Olver [8], cites the example ~ = xy+ tan ~ which has no continuous symmetry yet possesses a discrete reflection symmetry. d a There is also an example due to Englefield of equation ~ x = 1/x-~x + 4(y2/x2) • It admits a discrete symmetry group which is cyclic of order 4. or 2. C A S E 1 As was proved in [1], when bi(t) # 0 for some i, then the potential takes the form 3 v = (s) i=l 3 v = (9) i----1 The symmetry group has maximum dimension. It is a 24-parameter group of transformations isomorphic to sl(5, It). When hi = 0 in (9), the potential energy is zero and we have a free particle moving in R 3. In this case, the generators take the following simple form: a(t) = c 1 + c2t -4c~ 2, d,(t) = ~ + &d, (10) ca~ = ei + ct, cij = ~0, i # j. This dimension generalizes easily in the case of n degrees of freedom to (n + 2) 2 1. This dimension is in agreement with the results in [11], where upper bounds for the dimension of symmetry groups are obtained. The case of the harmonic oscillator has been studied in [12] where it is shown that the symmetry group for a time-dependent harmonic oscillator is SL(n + 2, R). 3. C A S E 2. b l ( t ) = b2(t) = b3(t) = 0 At this point, it is natural to continue with the classification by considering the function a(t) and proceed according to a"(t) # 0 or a"(t) = O. We treat in detail the case a"(t) # O. In the next section, several examples of the case a"(t) = 0 appear. For the case a(t) nonlinear the most general potential is V ~-~ (ql -4-q2 + q 2 ) -4q2 \ q l ~'1_ " (11) The Lie algebra of symmetries is simple, three-dimensional. This potential generalizes to The potential ° = o ( ,qo + . . . . 021 V = 2k=1 q l / A q2 q3] + V = ~ (q2.4_ .4 2X 1 @ (Alq2 A2ql~ (13) ( A l q 3 A Z q l ) 2 \~1q3 A3ql ) gives a six-parameter group of symmetries. 66 P . A . DAMIANOU AND C. SOPHOCLEOUS By specializing the form of @ in (11) or (13), we obtain different symmetry groups. (i) A # v = ~ (q~ + q: + q~) + (ql 9A2q2 9A3q3) 2 gives an 11-parameter group of symmetries. (ii) (iii) V = ~-(q2 9q2 9q3 2) 9].~ eetan-1 q~/ql 2 2 2 ql + q2 gives a seven-parameter group of symmetries. (14)