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Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems that are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in...

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Опубліковано в: :Symmetry, Integrability and Geometry: Methods and Applications
Дата:2020
Автори: Marchesiello, Antonella, Šnobl, Libor
Формат: Стаття
Мова:Англійська
Опубліковано: Інститут математики НАН України 2020
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/210595
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Цитувати:Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates. Antonella Marchesiello and Libor Šnobl. SIGMA 16 (2020), 015, 35 pages

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Marchesiello, Antonella
Šnobl, Libor
author_facet Marchesiello, Antonella
Šnobl, Libor
citation_txt Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates. Antonella Marchesiello and Libor Šnobl. SIGMA 16 (2020), 015, 35 pages
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems that are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 015, 35 pages Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates Antonella MARCHESIELLO † and Libor ŠNOBL ‡ † Czech Technical University in Prague, Faculty of Information Technology, Department of Applied Mathematics, Thákurova 9, 160 00 Prague 6, Czech Republic E-mail: marchant@fit.cvut.cz ‡ Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Department of Physics, Břehová 7, 115 19 Prague 1, Czech Republic E-mail: Libor.Snobl@fjfi.cvut.cz Received November 05, 2019, in final form March 06, 2020; Published online March 12, 2020 https://doi.org/10.3842/SIGMA.2020.015 Abstract. We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previ- ous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the addi- tional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian co- ordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals. Key words: integrability; superintegrability; higher-order integrals; magnetic field 2020 Mathematics Subject Classification: 37J35; 78A25 1 Introduction In this paper we investigate superintegrability of three-dimensional systems that separate in Cartesian coordinates in the presence of a magnetic field. We say that a mechanical system is superintegrable if it is Liouville integrable and possesses additional independent integrals of motion. Depending on their number we distinguish minimal superintegrability when only one additional integral is present, and maximal superintegrability when the number of additional integrals is the maximal possible, i.e., equal to the number of degrees of freedom minus one. (In three spatial dimensions there is no other possibility.) The study of superintegrability with magnetic fields was initiated in [5] and subsequently followed in both two spatial dimensions [3, 4, 24, 25] and three spatial dimensions [2, 14, 15, 16, 17]; relativistic version of the problem was recently considered too, cf. [11]. Separability of three-dimensional systems with magnetic fields was considered in the papers [1, 27]. Particular planar two-body systems, e.g., Coulomb, in perpendicular constant magnetic field were also studied from the point of solvability and superintegrability, see, e.g., [8, 28, 29, 30]. It turns out that the presence of magnetic field significantly increases the complexity of both calculations and structure of these systems. E.g., contrary to the case without magnetic field separability in orthogonal coordinates and integrability with integrals at most quadratic in the momenta are no longer equivalent, namely separability is stronger and implies the existence of at least one integral linear in the momenta. Similarly, the explicit construction of superintegrable systems and their classification become much harder when magnetic fields are present. mailto:marchant@fit.cvut.cz mailto:Libor.Snobl@fjfi.cvut.cz https://doi.org/10.3842/SIGMA.2020.015 2 A. Marchesiello and L. Šnobl In the present paper we attempt to approach the problem from a different viewpoint. We exploit the fact that in certain situations the three-dimensional system can be rewritten as effec- tively a two-dimensional one without magnetic field, thus generalizing the principal idea of [15]. In other cases we show that the existence of a quadratic integral necessarily implies the exis- tence of an integral in a particular simpler form, which makes our calculations tractable. When the results of the present paper and [14, 16] are viewed together, they provide an exhaustive list of three-dimensional quadratically minimally and maximally superintegrable systems with magnetic fields separable in Cartesian coordinates. We shall investigate the superintegrability of the system defined on the phase space R6, with the canonical coordinates (~x, ~p), by H(~x, ~p) = 1 2 (( pA1 )2 + ( pA2 )2 + ( pA3 )2) +W (~x), (1.1) where W (~x) denotes the so called electrostatic or effective potential, pAj are the covariant ex- pressions for the momenta pAj = pj +Aj(~x), j = 1, 2, 3, (1.2) and Aj(~x) are the components of the vector potential. The magnetic field ~B(~x) is related to ~A(~x) through ~B(~x) = ∇× ~A(~x). Newtonian equations of motion and thus also the physical dynamics are gauge invariant, i.e., depend only on B(~x) and ∇W (~x). However, in the Hamiltonian formulation gauge transforma- tions can be seen as canonical transformations (cf. [12, Problem 11.25]), namely they alter the Hamiltonian, the corresponding Hamilton’s equations of motion and the Hamilton–Jacobi equa- tion in a prescribed way. Separation of variables in the Hamilton–Jacobi equation is related to a specific choice of the coordinate system and is not preserved under canonical transformations – on the contrary, one looks for a suitable canonical transformation such that the system becomes separable after it. Since we are interested in systems that separate in Cartesian coordinates, we find it preferable to work in a suitably chosen fixed gauge adapted to the separation. Furthermore, we will sometimes use canonical transformations to reduce to cyclic coordinates corresponding to integrals. Also in this perspective, it is helpful to fix an appropriate gauge. However, the final results, in particular the superintegrable systems found shall be given in the gauge covariant form, so to express them in the most general way. In gauge dependent form the Hamiltonian (1.1) reads H(~x, ~p) = 1 2 ( p21 + p22 + p23 ) +A1(~x)p1 +A2(~x)p2 +A3(~x)p3 + V (~x), (1.3) where the gauge dependent “scalar” potential V (~x), i.e., the momentum-free term in (1.3), is related to the gauge invariant electrostatic potential W (~x) via V (~x) = W (~x) + 1 2 ∣∣ ~A(~x) ∣∣2. There are only two cases in which the system (1.3) separates in Cartesian coordinates [1, 27], up to a canonical permutation of the variables. Let us write them in both gauge dependent and gauge covariant form: Case I V (~x) = V1(x1) + V2(x2), ~A(~x) = (0, 0, u1(x2)− u2(x1)), (1.4) Classical Superintegrable Systems in a Magnetic Field 3 therefore ~B(~x) = (u′1(x2), u ′ 2(x1), 0), W (~x) = V1(x1) + V2(x2)− 1 2 (u1(x2)− u2(x1))2. (1.5) Case II V (~x) = V1(x1), ~A(~x) = (0, u3(x1),−u2(x1)), (1.6) thus ~B(~x) = (0, u′2(x1), u ′ 3(x1)), W (~x) = V1(x1)− 1 2 ( u3(x1) 2 + u2(x1) 2 ) . (1.7) In these two cases the system admits two Cartesian-type integrals, related to the separation of variables: X1 = ( pA1 )2 − 2(u2(x1)(p A 3 − u1(x2) + u2(x1))− V1(x1)) = p21 − 2(u2(x1)p3 − V1(x1)), X2 = ( pA2 )2 + 2(u1(x2)(p A 3 − u1(x2) + u2(x1)) + V2(x2)) = p22 + 2(u1(x2)p3 + V2(x2)) (1.8) for (1.4) and X1 = pA2 − u3(x1) = p2, X2 = pA3 − u2(x1) = p3 (1.9) for (1.6). Remark. X0 = pA3 − u1(x2) + u2(x1) = p3 is another integral of (1.5), though dependent on the Hamiltonian and (1.8). Minimal superintegrability due to the existence of another first-order integral has been studied in [14, 16]. Here we investigate the conditions for the existence of an additional integral of order at least two for the systems (1.5), (1.7). We give an exhaustive list of systems for which an additional second-order integral exists, and are able to answer the question on the existence of higher-order integrals in special cases. Sections 2 and 3 present two propositions for finding out whether certain classes of systems are superintegrable by reducing to a two-dimensional (2D) problem without magnetic field. In this way we also construct families of systems with higher-order integrals. Next, in Section 4 we address the problem of second-order superintegrability. The determining equations for second- order integrals are given in gauge covariant form, together with their compatibility conditions. In Section 5 we give a necessary condition for second-order superintegrability, which is used in Sections 6 and 7 to simplify the structure of the integral for the classes (1.5), (1.7), respectively. With these simplifications at hand, the determining equations for the integral can be solved. In Section 9.1 we list the superintegrable systems so found; their explicit derivation is rather technical and tedious and we review it in Appendices A, B and C. The special case in which the magnetic field is constant and the functions Vj in (1.5) and (1.7) are at most quadratic polyno- mials is studied in Section 8. Finally, in Section 9.2 we discuss the approaches to construction of higher-order integrals. 2 Minimal superintegrability for Case I when all the integrals commute with one linear momentum Let us consider the natural Hamiltonian systems on the phase space (x1, x2, p1, p2), for κ ∈ R, κ 6= 0 Hκ0(x1, x2, p1, p2) = 1 2 ( p21 + p22 ) + κ(u1(x2)− u2(x1)) + V1(x1) + V2(x2). (2.1) 4 A. Marchesiello and L. Šnobl For the sake of clarity let us refer here to the Hamiltonian of Case I as to H. Since p3 is an integral of motion for (1.4), by setting p3 = κ, Hκ0 = H(x, y, z, p1, p2, κ) − 1 2κ 2. Both systems have a pair of second-order integrals corresponding to separation: Xj as in (1.8) for H and clearly Iκj = Xj(x1, x2, p1, p2, κ), j = 1, 2 for (2.1). If (1.6) possesses any additional integral X3 independent of the variable x3, then Iκ3 (x1, x2, p1, p2) = X3(x1, x2, p1, p2, κ) would be an integral for (2.1). And vice versa, any additional inte- gral Iκ3 of (2.1), would correspond to an integral X3 of (1.6), obtained by simply replacing κ by p3, i.e., X3(x1, x2, x3, p1, p2, p3) = Ip33 (x1, x2, p1, p2). Indeed, {H, X3} = 2∑ i=1 ( ∂Hp30 ∂xi ∂Ip33 ∂pi − ∂Ip33 ∂xi ∂Hp30 ∂pi ) + 1 2 { p23, X3 } = 0, where { , } is the Poisson bracket on the phase space R6. The right hand side of the equality is zero since both H and X3 do not depend on x3 and Ip33 is an integral of Hp30 . Thus, we arrive at the following immediate conclusion Proposition 2.1. Let us consider the Hamiltonian system defined by (1.1) on the phase space (x1, x2, x3, p1, p2, p3) with magnetic field and effective potential as in (1.5). Such system admits an additional independent integral I3 such that {I3, p3} = 0 if and only if (2.1) is superintegrable on the phase space (x1, x2, p1, p2). Therefore all the systems of the form (1.5) that are minimally superintegrable, with an additional integral independent of Cartesian coordinate, can be deduced from 2D natural super- integrable systems of the form (2.1). And vice versa, every superintegrable system in two degrees of freedom can be extended to a minimally superintegrable system in three degrees of freedom with magnetic field. Superintegrable systems of the form (2.1) have been widely studied. In particular they have been completely classified for integrals up to third order [22]. Concerning higher-order integrals, many examples are known, including the harmonic oscillator and the caged oscillator [10, 26], and a wide class of so called exotic potentials [6, 7, 20]. 2.1 Example: extension of 2D second-order superintegrable systems Table 1 contains all three-dimensional systems that can be proven to be (at least) minimally quadratically superintegrable by applying Proposition 2.1 to 2D superintegrable systems that separate in Cartesian coordinates and have integrals at most quadratic. The list of 2D systems is taken from [22], from which we consider only the systems on real phase space. To obtain the most general family of systems (and recalling that the Hamiltonian must depend linearly on κ), we renamed all the parameters as cj = ajκ + bj , aj not all vanishing, then set p3 = κ and applied Proposition 2.1. The third integral, leading to superintegrability, can then be found from the integral I3 of the 2D system, by substituting cj = ajp3 + bj . Since the dependence on the constants cj is linear, the order of the so obtained integral remains quadratic. 2.2 Example: a family of higher-order superintegrable systems from the 2D caged oscillator Let us consider the two-dimensional caged anisotropic oscillator H0 = 1 2 ( p21 + p22 ) + ω ( `2x21 +m2x22 ) + α x21 + β x22 (2.2) Classical Superintegrable Systems in a Magnetic Field 5 T a b le 1 . 3D (a t le as t) m in im al ly q u ad ra ti ca ll y su p er in te g ra b le ex te n si o n s o f 2 D q u a d ra ti ca ll y su p er in te g ra b le sy st em s th a t se p a ra te in C a rt es ia n co o rd in a te s. F or th e re ad er ’s co n ve n ie n ce , w e gi ve th e H am il to n ia n ex p re ss ed in th e g a u g e ch o ic e (1 .4 ), b u t a ls o th e fu n ct io n s u j a n d V j th a t a ll ow to fi n d th e m a g n et ic fi el d ~ B an d p ot en ti al W as in th e m or e ge n er al ga u g e in va ri a n t fo rm (1 .5 ). In th e in te g ra ls , L 3 d en o te s a n g u la r m o m en tu m o n th e p la n e, L 3 = x 1 p 2 − x 2 p 1 . 2 D sy st em a n d it s th ir d in te gr al 3D sy st em E 1 : H 0 = 1 2 ( p2 1 + p 2 2 ) + c 1 ( x2 1 + x 2 2 ) + c 2 x 2 1 + c 3 x 2 2 I 3 = L 2 3 + 2 ( c 2x 2 2 x 2 1 + c 3 x 2 1 x 2 2 ) H = 1 2 ( p2 1 + p 2 2 + p 2 3 ) + ( a 1( x 2 1 + x 2 2 ) + a 2 x 2 1 + a 3 x 2 2 ) p 3+ b 1 ( x2 1 + x 2 2 ) + b 2 x 2 1 + b 3 x 2 2 u 1 (x 2 ) = a 1 x 2 2 + a 3 x 2 2 , u 2 (x 1 ) = − a 1 x 2 1 − a 2 x 2 1 V 1 (x 1 ) = b 1 x 2 1 + b 2 x 2 1 , V 2 (x 2 ) = b 1 x 2 2 + b 3 x 2 2 E 2 : H 0 = 1 2 ( p2 1 + p 2 2 ) + c 1 ( 4 x 2 1 + x 2 2 ) + c 2 x 1 + c 3 x 2 2 I 3 = p 2 L 3 − x 2 2 ( 2 c 1 x 1 + c 2 2 ) + 2 c 3 x 1 x 2 2 H = 1 2 ( p2 1 + p 2 2 + p 2 3 ) + ( a 1( 4x 2 1 + x 2 2 ) + a 2 x 1 + a 3 x 2 2 ) p 3+ b 1 ( 4x 2 1 + x 2 2 ) + b 2 x 1 + b 3 x 2 2 u 1 (x 2 ) = a 1 x 2 2 + a 3 x 2 2 , u 2 (x 1 ) = − 4a 1 x 2 1 − a 2 x 1 V 1 (x 1 ) = 4b 1 x 2 1 + b 2 x 1 , V 2 (x 2 ) = b 1 x 2 2 + b 3 x 2 2 E 3 : H 0 = 1 2 ( p2 1 + p 2 2 ) + c 1 ( x2 1 + x 2 2 ) + c 2 x 1 + c 3 x 2 I 3 = p 1 p 2 + 2 c 1 x 1 x 2 + c 2 x 2 + c 3 x 1 H = 1 2 ( p2 1 + p 2 2 + p 2 3 ) + ( a 1( x 2 1 + x 2 2 ) + a 2 x 1 + a 3 x 2 ) p 3+ b 1 ( x2 1 + x 2 2 ) + b 2 x 1 + b 3 x 2 u 1 (x 2 ) = a 1 x 2 2 + a 3 x 2 , u 2 (x 1 ) = − a 1 x 2 1 − a 2 x 1 V 1 (x 1 ) = b 1 x 2 1 + b 2 x 1 , V 2 (x 2 ) = b 1 x 2 2 + b 3 x 2 6 A. Marchesiello and L. Šnobl for ω ∈ R \ {0}, `, m nonvanishing integers and α, β ∈ R. The system is well known to be superintegrable if ` m rational [10, 26]. A first straightforward extension to a 3D superintegrable system is given by H = 1 2 ( p21 + p22 + p23 ) + ( `2x21 +m2x22 ) p3 + α x21 + β x22 , (2.3) that can be transformed into (2.2) by simply reducing p3 = ω. A more general extension can be constructed as in the previous example. Let us set `2 = `1κ+ `2, α = α1κ+ α2, m2 = m1κ+m2, β = β1κ+ β2. (2.4) The system (2.2) can then be seen as the 2D reduction of H = 1 2 ( p21 + p22 + p23 ) + ( ω ( `1x 2 1 +m1x 2 2 ) + α1 x21 + β1 x22 ) p3 + ω ( `2x 2 1 +m2x 2 2 ) + α2 x21 + β2 x22 , (2.5) by substituting p3 = κ. We obtain in this way the three-dimensional integrable system (2.5) that becomes superintegrable when the frequency ratio of (2.2) (where (2.4) has to be taken into account) is a rational number, i.e., when `1p3 + `2 m1p3 +m2 = `2 m2 , ` m ∈ Q, (2.6) for every possible value of the phase space variable p3. Equivalently, (2.6) can be written as ( m2`1 − `2m1 ) p3 +m2`2 −m2` 2 = 0, ` m ∈ Q. The above equation contains a polynomial in p3 that must be identically zero. This is possible only when the coefficient of each power of p3 vanishes. Namely, when `1 m1 = `2 m2 = `2 m2 , ` m ∈ Q. (2.7) Thus, the family of systems (2.5) is superintegrable if and only its parameters satisfy (2.7) (and in that case also (2.2) is superintegrable). For `j = mj = 0 for some j (not both j = 1, 2), the previous condition reduces to `j mj = `2 m2 , ` m ∈ Q. For α1 = β1 = `2 = m2 = 0 we have the simpler system (2.3). The case αj = βj = 0, `j = mj = ±1, j = 1, 2 was studied in [14] and it is shown there to be quadratically minimally superintegrable, with the fourth independent integral (besides the two Cartesian ones) inherited from the 2D caged oscillator, of first order. In the more general case (2.5), the order of the fourth integral can be arbitrarily high, depending on the value of ` m . Notice that all the systems in Table 1 are contained in the family (2.5), except the systems E2 and E3 for the special case a1 = b1 = 0 (i.e., c1 = 0), in which the linear terms in the space variables cannot be eliminated by translation, due to the absence of quadratic terms. Classical Superintegrable Systems in a Magnetic Field 7 3 Maximal superintegrable class canonically conjugated to natural 2D systems Let us consider the system whose magnetic field and effective potential read ~B(~x) = (0, γ, 0), γ ∈ R \ {0} (3.1) and W (~x) = V (x2), (3.2) respectively. This system can be written in the form (1.4), with the gauge chosen as ~A(~x) = (0, 0,−γx1). Its Hamiltonian reads H = 1 2 ( p21 + p22 + p23 ) − γx1p3 + γ2 2 x21 + V (x2). (3.3) Actually by a different choice of the gauge and a canonical permutation of the variables x1 and x2 we see that the system belongs also to Case II. The Hamiltonian (3.3) admits three independent first-order integrals [14] I1 = p1 − γx3, I2 = p3, I3 = 2l2 + γ ( x21 − x23 ) . (3.4) Out of them, we can construct two Cartesian-type integrals, X1 = I21 + γI3, X2 = 2H − I21 − I22 − γX3. (3.5) The system can be reduced to two degrees of freedom through the following canonical trans- formation x1 = X + P3 γ , x2 = Y, x3 = Z + 1 γ P1, pj = Pj , j = 1, 2, 3, (3.6) with the second type generating function G(~x, ~P ) = ( x1 − 1 γ P3 ) P1 + x2P2 + x3P3. The Hamiltonian in the new coordinates reads K( ~X, ~P ) = 1 2 ( P 2 1 + P 2 2 ) + 1 2 γ2X2 + V (Y ), (3.7) i.e., it is effectively in two degrees of freedom and without magnetic field. This system (3.7) has two cyclic coordinates in the full phase space ( ~X, ~P ), namely Z and P3, that are therefore both integrals. Expressed in the original variables, these integrals correspond to p3 and I1 γ as in (3.4). Moreover (3.7) separates in the Cartesian coordinates (X,Y ), and the corresponding Cartesian-type integrals, I1, I2, once written in the original coordinates, provide (3.5). Thus we have Proposition 3.1. The system with the magnetic field (3.1) and potential (3.2) is maximally superintegrable if and only if (3.7), seen as a system in two degrees of freedom on the phase space (X,Y, P1, P2) has one additional integral of motion, besides I1, I2, and independent of them. 8 A. Marchesiello and L. Šnobl Therefore the problem of maximal superintegrability of (3.3) has been reduced to the two- dimensional problem of superintegrability of (3.7). In particular, all the potentials V (Y ) that make (3.7) superintegrable give (by simply replacing Y = x2) the effective potentials that render (3.3) superintegrable. The cases V (Y ) = c Y 2 + γ2Y 2 8 , (3.8) and V (Y ) = γ2 2 Y 2, (3.9) that correspond to 3D superintegrable systems with additional second-order integral have already been found in [14] with a different approach. All the potentials V (Y ) that lead to second and third-order superintegrability in 2D have been classified [22]. If we focus on second-order integrals, they are listed in Table 1. The systems that can be obtained from it, after applying the transformation (3.6) and are still quadratically superintegrable are given by (3.8), and V (Y ) = γ2 2 Y 2 + cY, that, since γ 6= 0, can be reduced to (3.9) by translation in Y . However, higher-order superintegrable systems can be generated, e.g., from V (Y ) = c Y 2 + γ2Y 2 2 , c ≥ 0. (3.10) The additional integral of (3.7) is second order and reads (see Table 1) X4 = L23 + 2c X2 Y 2 . Here L3 denotes the third component of the angular momentum with respect to the coordinates (X,Y, Z, P1, P2, P3). Inverting the transformation (3.6), it gives the fourth-order integral X4 = 1 γ2 (( pA2 p A 3 + γpA1 x2 )2 + 2c ( pA3 )2 x22 ) . Actually, by polynomial combinations with the other integrals, it can be reduced to the third order one X5 = 2γpA2 p A 3 l A 3 + γ2 ( x21 ( pA2 )2 + x22 (( pA3 )2 − (pA1 )2))+ 2γ x1 x22 ( γ2x42 + 2c ) pA3 + γ2 x21 x22 ( γ2x42 + 2c ) , that cannot be further reduced to lower order by using any of the integrals (3.4) nor (3.5). A more general 3D infinite family of maximally superintegrable system, including the previous cases (3.8) and (3.10) and the one found in [15], corresponds to the caged oscillator V (Y ) = c Y 2 + m2 `2 γ2Y 2, `,m ∈ N (3.11) If we compare it with (2.5), we see that for γ2 = ωl22, α2 = 0, β2 = c and m2 satisfying (2.7) the two obtained 3D families would have the same scalar potential. However, the magnetic fields differ, rendering (3.11) maximally superintegrable, while (2.5) – as far as we can see – is only minimally superintegrable. Classical Superintegrable Systems in a Magnetic Field 9 4 Second-order integrals Any second-order integral of motion we can write X = 3∑ j=1 hj(~x)pAj p A j + 3∑ j,k,l=1 1 2 |εjkl|nj(~x)pAk p A l + 3∑ j=1 sj(~x)pAj +m(~x), (4.1) where εjkl is the completely antisymmetric tensor with ε123 = 1. The condition that the Poisson bracket {a(~x, ~p), b(~x, ~p)} = 3∑ j=1 ( ∂a ∂xj ∂b ∂pj − ∂b ∂xj ∂a ∂pj ) of the integral (4.1) with the Hamiltonian (1.1) vanishes {H,X} = 0 seen as a polynomial in the momenta leads to the determining equations for the unknown functions hj , nj , sj , j = 1, 2, 3 and m in the integral. Order by order (from the third to the zeroth) they read (cf. [16]): ∂x1h1 = 0, ∂x2h1 = −∂x1n3, ∂x3h1 = −∂x1n2, ∂x1h2 = −∂x2n3, ∂x2h2 = 0, ∂x3h2 = −∂x2n1, ∂x1h3 = −∂x3n2, ∂x2h3 = −∂x3n1, ∂x3h3 = 0, ∇ · ~n = 0, (4.2) ∂x1s1 = n2B2 − n3B3, ∂x2s2 = n3B3 − n1B1, ∂x3s3 = n1B1 − n2B2, ∂x2s1 + ∂x1s2 = n1B2 − n2B1 + 2(h1 − h2)B3, (4.3) ∂x3s1 + ∂x1s3 = n3B1 − n1B3 + 2(h3 − h1)B2, ∂x2s3 + ∂x3s2 = n2B3 − n3B2 + 2(h2 − h3)B1, ∂x1m = 2h1∂x1W + n3∂x2W + n2∂x3W + s3B2 − s2B3, ∂x2m = n3∂x1W + 2h2∂x2W + n1∂x3W + s1B3 − s3B1, (4.4) ∂x3m = n2∂x1W + n1∂x2W + 2h3∂x3W + s2B1 − s1B2, ~s · ∇W = 0. (4.5) The equations (4.2) prescribe that the functions hj , nj are such that the highest-order terms in the integral (4.1) are linear combinations of products of the generators p1, p2, p3, l1, l2, l3 of the Euclidean group, where lj = ∑ k,l εjklxkpl [16]. Explicitly, in terms of the expressions (1.2), we have X = ∑ i,j : i≤j αijl A i l A j + ∑ i,j βijp A i l A j + ∑ i,j: i≤j γijp A i p A j + 3∑ j=1 sj(~x)pAj +m(~x), (4.6) where lAj = ∑ k,l εjklxkp A l . By subtracting the Hamiltonian and the two Cartesian integrals we can a priori set γ11 = γ22 = γ33 = 0. There are compatibility conditions on equations (4.3), consequence of the following conditions on the derivatives of the functions sj , namely, ∂2x2∂x1s1 + ∂2x1∂x2s2 = ∂x1∂x2(∂x2s1 + ∂x1s2), 10 A. Marchesiello and L. Šnobl ∂2x3∂x1s1 + ∂2x1∂x3s3 = ∂x1∂x3(∂x3s1 + ∂x1s3), ∂2x3∂x2s2 + ∂2x2∂x3s3 = ∂x2∂x3(∂x3s2 + ∂x2s3), ∂x1∂x3(∂x2s1 + ∂x1s2) = 2∂x2∂x3(∂x1s1)− ∂x1∂x2(∂x3s1 + ∂x1s3) + ∂2x1(∂x3s2 + ∂x2s3), ∂x2∂x3(∂x2s1 + ∂x1s2) = 2∂x1∂x3(∂x2s2)− ∂x1∂x2(∂x3s2 + ∂x2s3) + ∂2x2(∂x3s1 + ∂x1s3), ∂x2∂x3(∂x3s1 + ∂x1s3) = 2∂x1∂x2(∂x3s3)− ∂x1∂x3(∂x3s2 + ∂x2s3) + ∂2x3(∂x2s1 + ∂x1s2).(4.7) These translate into compatibility conditions on the magnetic field and the constants in the coefficients of the second-order terms. Further compatibility constraints come from (4.4), con- sequence of ∂xi∂xjm = ∂xj∂xim, i, j = 1, 2, 3, i 6= j. (4.8) 5 A necessary condition for second-order superintegrability Both classes of systems that separate in Cartesian coordinates have at least one first-order integral and it is always possible to choose a gauge so that such integral reads as one of the linear momenta. To fix the ideas, let us work in such a gauge choice and assume that the constant momentum is p3. If a second-order integral X exists, then K1 = {X, p3} is still an integral at most of second order or a constant. Since the highest-order terms in X are as in (4.6), they can be at most quadratic in x3. This means that if K1 is quadratic in the momenta, its second-order terms are at most linear in x3, since K1 = {X, p3} = ∂X ∂x3 . Thus, K2 = {K1, p3} can be, as above, either an integral at most quadratic or a constant. If K2 is again quadratic, K3 = {K2, p3} can be now at most linear in the momenta, since the highest-order terms in K2 do not depend on x3. Therefore, we can conclude that if a second-order independent integral X exists, then necessarily there must exist a second-order integral (which could be X itself) such that {X, p3} is at most linear in the momenta. In general, for a conserved momentum pj , the result is the same, it is enough to replace x3 by xj in the argument above. Thus, we obtain the following Proposition 5.1. Let the system defined by H as in (1.1) separate in Cartesian coordinates and have a quadratic integral I independent of the Cartesian integrals. Then there exists a second- order integral X, not necessarily different from I, such that {X, pj} is a polynomial expression in the momenta of at most first order, for some j. Thus, to answer the question on the existence of an additional second-order integral for the class of systems we are considering here, we can start by answering the simpler question on the existence of the necessary integral X that satisfies the above property. This is done in the following Sections 6 and 7 and Appendices A, B, C. Since we found that the special case in which the magnetic field is constant and the func- tions Vj are second-order polynomials in the respective variables appears several times in the computation therein, we discuss it at once in the separate Section 8. 6 Quadratic superintegrability in Case I We start with the class of systems in (1.5). To fix the ideas, let us choose a gauge as in (1.4) and assume that there exists a quadratic independent integral I. Thus, by Proposition 5.1 there exist another quadratic integral X such that {X, p3} is at most first order as a polynomial in the momenta. Here we consider only the case in which the two Cartesian-type integrals do not reduce to first-order integrals. In case one of them does, then the system is at the intersection of Case I and Case II (up to a permutation of indices) and it is treated at once in Section 7. Classical Superintegrable Systems in a Magnetic Field 11 Moreover, we assume there does not exist a linear integral, other than p3. If it exists, the corresponding systems can be found in [14], where there is a complete study of quadratically superintegrable systems with Cartesian integrals and one independent first-order integral. We can have several cases: (i) {X, p3} is at most linear and not vanishing. Thus the only possibility of finding something new is in assuming that {X, p3} is a dependent integral or a constant (we excluded the case there is an independent first-order integral). We therefore look for a quadratic integral X such that ∂x3X = {X, p3} = c1p3 + c0, cj ∈ R, (6.1) and cj not both vanishing, j = 0, 1. (ii) {X, p3} = 0 and there exist no quadratic integral independent of the Cartesian integrals and commuting with p3. Then X is trivial, in the sense that it depends on the Carte- sian integrals and p3. However, to have a quadratic superintegrable system, a quadratic integral I as in Proposition 5.1 must exist. Without loss of generality, we can assume X = {I, p3} with {I, p3} = a0p 2 3 + a1X1 + a2X2 + c1p3 + c0, (6.2) where X1 and X2 are as in (1.8), a0, a1, a2, c0, c1 ∈ R, not all aj vanishing, otherwise we are in the previous point i). (iii) {X, p3} = 0 and X is independent of the Cartesian integrals. Since X commutes with p3, it satisfies the assumptions of Proposition 2.1. Thus, the corresponding systems can be found in Table 1. If an additional quadratic independent integral exists, then its Poisson bracket with p3 cannot vanish. This is a consequence of the fact that the 2D system (2.1) cannot have more than 3 independent integrals. However, as in the previous point, there could exist a quadratic independent integral I such that {I, p3} depends on the others, namely {I, p3} = a0p 2 3 + a1X1 + a2X2 + a3X3 + c1p3 + c0, (6.3) where a0, a1, a2, a3, c0, c1 ∈ R and not all aj are vanishing (otherwise we are in case (i)), X1, X2 as in (1.8) and X3 = X. Let us investigate the possibilities for X3 in (6.3). Its highest-order terms should come from a Poisson bracket of the quadratic terms of I with p3, i.e., their derivatives with respect to x3. Moreover, by assumption X3 does not depend on x3. Thus, its second-order terms can arise only by taking derivatives of a second-order polynomial that contains terms of the form pi · lj , i = 1, 2, 3, j = 1, 2. By computing their Poisson bracket with p3, we see that the only outcome (for an integral X3 independent of X1 and X2) is in terms of the type pip`, i 6= `. Looking at the integrals of the of 2D systems in Table 1, and the dependent integrals obtained by their Poisson bracket with the Cartesian integrals, we see that the only possibility is (8.4) below. Now that we outlined all the possibilities, we need to solve the determining equations (4.3)– (4.5), for the different cases. For this, it is necessary to work in gauge covariant setting. The conditions (6.1), (6.2) and (6.3) can be written together as (we can now set a3 = 0): ∂x3X = a0 ( pA3 − u1(x2) + u2(x1) )2 + a1X1 + a2X2 + c1 ( pA3 − u1(x2) + u2(x1) ) + c0, (6.4) where with an abuse in the notation we denoted I as X (the unknown independent integral we are looking for), with aj , cj ∈ R and not all vanishing. For aj = 0, j = 0, 1, 2 we are in case (i). 12 A. Marchesiello and L. Šnobl Equation (6.4) implies the following values for the second-order terms of X as in (4.6): α11 = α22 = α12 = α13 = α23 = β31 = β32 = 0, β11 = β22, a0 = 0, a1 = β12, a2 = −β21. (6.5) Moreover, since ~p · ~L = 0 we can set β22 = 0 (and consequently also β11 = 0). Concerning the lower-order terms, by integrating the right-hand side of (6.4), we obtain the following restriction on the structure of X: sj = Sj(x1, x2), j = 1, 2, s3 = S3(x1, x2)− (2β12u2(x2) + 2β21u1(x2)− c1)x3, m3 = c0x3 + (u1(x2)− u2(x1)) ((2β12u2(x1) + 2β21u1(x2)− c1)x3) + (2β12V1(x1)− 2β21V2(x2))x3 +M(x1, x2). (6.6) With this simplifications at hand, we can solve equations (4.3)–(4.5). Let as assume that a1 and a2 in (6.4) are not both zero; e.g., let it be a1 6= 0. Then we can shift both the potential V1(x1) and the third component of the vector potential by a constant, thus absorbing the constants c0 and c1. Similarly, if a2 6= 0 we could use X2. Therefore, by (6.5), we see that if either β12 6= 0 or β21 6= 0, we can proceed in the solution of (4.3)–(4.5) as if c1 = c0 = 0. We obtain that no new superintegrable system can be found in this case. The details of the computation are in Appendix B. For β12 = β21 = 0 we find it convenient to start from (4.3), in which the third equation simplifies to (β33x1 + γ23)u ′ 1(x2) + (β33x2 − γ13)u′2(x1)− c1 = 0. (6.7) The above equation could be trivially satisfied for some of the functions uj or not. This deter- mines a major splitting in the computation. For the details see Appendix A, the resulting list of systems is given in the conclusions, Section 9.1. 7 Quadratic superintegrability in Case II For the class of systems (1.7) we can choose a gauge so that there are two mutually orthogonal conserved linear momenta. Let us assume that they are p2 and p3 as in (1.9). As above, we assume there exists an independent quadratic integral. Thus, by Proposition 5.1 we can have two possibilities: (i) there exists a quadratic integral X such that {X, p2} = {X, p3} = 0. Then X is an integral of the reduced system obtained from (1.7) by setting the conserved momenta to constants, i.e., function of the 1-dimensional Hamiltonian. Thus, it is dependent on the Hamiltonian and the conserved momenta. The only hope to find something interesting is to look for a quadratic integral I such that {I, pj} = X for some j. (ii) There exists a quadratic integral X such that {X, pj} is linear and not vanishing for at least one pj , j = 2, 3. Without loss of generality we can assume that {X, p3} 6= 0, otherwise we permute the coordinates x2 and x3. Let us set j = 3 in both cases and with an abuse of notation let us rename I in case (i) as X. Thus, we look for a quadratic integral X such that {X, p3} = 2a0 ( H −X2 1 −X2 2 ) + a1X 2 1 + a2X 2 2 + a3X1X2 + c0 + c1X1 + c2X2, (7.1) X1, X2 as in (1.9). For aj = 0, j = 1, . . . , 4, we have case (ii). Classical Superintegrable Systems in a Magnetic Field 13 Equation (7.1) implies the following conditions on the coefficients of the higher-order terms of the integral, expressed as in (4.6) (again, we use the condition ~p · ~L = 0) α11 = α22 = α12 = α13 = α23 = β11 = β22 = β32 = 0, a0 = β12, a1 = −β21, a2 = 0, a3 = −β31. (7.2) Moreover, by subtracting X1X2 from X, we can set γ23 = 0. Still as a consequence of (7.1), we have further conditions on the coefficients of the lower-order terms s1 = S1(x1, x2), s2 = S2(x1, x2) + (2(β12 + β21)u3(x1)− β31u2(x1) + c1)x3, s3 = S3(x1, x2) + z(β31u3(x1)− 2β12u2(x1) + c2) and m = M(x1, x2)− ( (2β12 + β21)u3(x1) 2 − β31u2(x1)u3(x1) + c1u3(x1)− c2u2(x1) + 2β12u2(x1) 2 − 2β12V1(x1)− c0 ) x3. With these simplifications at hand, we are able to solve the determining equations (4.2)–(4.5). Let us perform the substitution uj(x1) = U ′j(x1), j = 2, 3. (7.3) Since uj are defined in (1.5) up to addition of arbitrary constants and Uj is defined as in (7.3), in the following we can set to zero all the coefficient of first and zero-order powers of x1 in the solutions for Uj . From (4.3) we find S1(x1, x2) = s1(x2) + β12U2(x1) + (β13 − 2α33x2)U3(x1)− (β12x1 + β33x2 − γ13)U ′2(x1) + (2α33x1x2 − β13x1 + β23x2 − γ12)U ′3(x1), S2(x1, x2) = s2(x1)− ( α33x1x 2 2 − β13x1x2 + 1 2 β23x 2 2 − γ12x2 ) U ′′3 (x1), S3(x1, x2) = s3(x1) + c1x2 + β31x2U ′ 2(x1)− 2(β12 + β21)x2U ′ 3(x1) + ( α33x1x 2 2 − β13x1x2 + 1 2 β23x 2 2 − γ12x2 ) U ′′2 (x1) − ( β12x1x2 + 1 2 β33x 2 2 − γ13x2 ) U ′′3 (x1), where Uj and s` must satisfy the third, fourth and fifth equation of (4.3). Let us continue by considering the third of these equations, namely U ′′2 (x1)(β12x1 + β33x2 − γ13) + 2β12U ′ 2(x1)− β31U ′3(x1)− c2 = 0, (7.4) together with the compatibility conditions (4.7). The first one is trivially satisfied, while the remaining five read β33U ′′′ 2 (x1) = 0, (2α33x1 + β23)U ′′′ 2 (x1) + 6α33U ′′ 2 (x1)− β33U ′′′3 (x1) = 0, (β12x1 + β33x2 − γ13)U (4) 2 (x1) + 4β12U ′′′ 2 (x1)− β31U ′′′3 (x1) = 0, −(2α33x1x2 − β13x1 + β23x2 − γ12)U (4) 3 (x1) + 4(β13 − 2α33x2)U ′′′ 3 (x1)− β21U ′′′2 (x1) = 0, 14 A. Marchesiello and L. Šnobl (8α33x2 − 4β13 − β31)U ′′′2 (x1)− (4β12 + β21)U ′′′ 3 (x1) (7.5) + (2α33x1x2 − β13x1 + β23x2 − γ12)U (4) 2 (x1)− (β12x1 + β33x2 − γ13)U (4) 3 (x1) = 0. We can have different subcases according to whether the equations (7.4), (7.5) are trivially sa- tisfied for some of the functions Uj or not. This determines a major splitting in the computation. The details are given in Appendix C. 8 Constant magnetic field and second-order polynomial potentials Let us consider the particular case in which the magnetic field is constant ~B(~x) = (a1, a2, 0) (8.1) and in (1.5) we have V1(x) = v11x1 + v12x 2 1, V2(x2) = v21x2 + v22x 2 2, u1 = a1x2, u2 = −a2x1. This system appears in various branches of calculation in the appendices; thus we find it practical to discuss it separately here. Notice that since the magnetic field is constant, by rotation around x3-axis we could reduce it to the case in which it is aligned with one of the Cartesian axis. However, the system would no longer separate in the corresponding rotated Cartesian coordinates, therefore we prefer not to perform such a rotation. Let us also point out that if V1(x1) = 0, for constant magnetic field a rotation around x2 brings the system (1.5) into (1.7). Thus, what we will deduce in the following for V1 = 0 applies also for (1.7). 8.1 v12 and v22 both not vanishing Let us assume v12 and v22 are both not vanishing. Then by the translation of the coordinate system we can set v11 = v21 = 0 without loss of generality. Then, similarly to Section 3, we can reduce to a natural Hamiltonian system through canonical transformations. Namely, let us take as the generating function G = ( x1 − a2P3 2v12 ) P1 + ( a1P3 2v22 + x2 ) P2 + x3P3, (8.2) so that pj = Pj , j = 1, 2, 3 and x1 = X + a2P3 2v12 , x2 = Y − a1P3 2v22 , x3 = Z + a2v22P1 − a1v12P2 2v12v22 . After the transformation, with gauge chosen as in (1.4), the Hamiltonian reads H = 1 2 ( P 2 1 + P 2 2 + ( 1− a21 2v22 − a22 2v12 ) P 2 3 ) + v12X 2 + v22Y 2. If a21 2v22 + a22 2v12 6= 1 we can, by a canonical transformation P3 = 1 λ P̃3, Z = λZ̃, λ2 = ∣∣∣∣1− a21 2v22 − a22 2v12 ∣∣∣∣ Classical Superintegrable Systems in a Magnetic Field 15 scale the P 2 3 term to have the Hamiltonian of the form H = 1 2 ( P 2 1 + P 2 2 ± P̃ 2 3 ) + v12X 2 + v22Y 2. The system can therefore be reduced to a system determined by a two-dimensional, possibly inverted, anisotropic harmonic oscillator and free motion along the Z-direction. The original 3D system is minimally superintegrable if and only if the corresponding 2D oscillator is super- integrable as a system in the (X,Y, P1, P2) space. If v12 = v22 we have a special case of the system E3 in Table 1. If v12 v22 ∈ Q, v12 v22 6= 1 we have a higher-order integral when expressed in the variables xj , pj . If a21 2v22 + a22 2v12 = 1, the coordinate P3 becomes cyclic and its conjugated variable Z is an independent constant of motion. In this case the system (1.5) becomes at least minimally su- perintegrable. It is maximally superintegrable if and only if its reduction on the (X,Y, P1, P2) space is superintegrable. Indeed, we have reduced to the system (3.7) for V (Y ) = v22Y 2 and 2v12 = γ2 = a22. Its maximally superintegrable exception is included in the family of sys- tems (3.11). 8.2 v22 = 0 and v12 not vanishing In this case by translation in x1 we can still set v11 = 0. Then by a canonical transformation such that x1 = X + a2 2v12 P3, x2 = Y, x3 = Z + a2 2v12 P1, pj = Pj , j = 1, 2, 3, we obtain the system H = 1 2 ( P 2 1 + P 2 2 + P 2 3 ) + v12X 2 + a1P3Y + v21Y. If a1 = 0 we have reduced to a natural system. By reducing the integral P3 we have a 2D system that, to our knowledge, is not superintegrable. If a1 6= 0 we have reduced to the case with magnetic field aligned along one axis. The effective potential of the so obtained system reads W = v12X 2 + v21Y − ( a21Y 2 ) /2. (8.3) Thus, by the translation Y → Y + v21 a21 we can eliminate the linear term from the effective potential. A shift of the vector potential by a constant, i.e., A3 → A3 − v21 a1 , gives H = 1 2 ( P 2 1 + P 2 2 + P 2 3 ) + v12X 2 + a1P3Y. Thus, without loss of generality, we can set v21 = 0. By plugging (8.3) and (8.1) with these simplifications into the determining equations for a second-order integral as in (4.6), we find that they have no solution. 16 A. Marchesiello and L. Šnobl 8.3 v12 = v22 = 0 We have a subcase of the system E3 in Table 1. Thus, the system admits a second-order integral. With the gauge chosen as in (1.4), that integral reads X3 = p1p2 + (v11 − a2p3)x2 + (v21 + a1p3)x1. Equivalently, in gauge covariant form, we have X3 = pA1 p A 2 + (a1x1 − a2x2)pA3 − ( a21 + a22 ) x1x2 + a1a2 ( x21 + x22 ) + v11x2 + v21x1, corresponding to the fact that the system actually separates in any rotated system of Cartesian coordinates, since the Hamiltonian is linear in the space variables. Without altering the structure of the Cartesian-type integrals, we can therefore by rotation align the magnetic field along one Cartesian axis, let us say the x2-axis. Thus, without lost of generality, let us assume a1 = 0. The determining equations for an additional second-order integral can be solved. We find for v11 = 0 one maximally superintegrable system: ~B(~x) = (0, a2, 0), W (~x) = v21x2 − 1 2 a22x 2 1, (8.4) with the integral X4 = 3pA3 l A 1 − pA1 lA3 − 3v21 a2 lA2 + a2x1x2p A 3 + 3a2x1l A 1 + v21x 2 1 + a22x 2 1x2 = 3p3l1 − p1l3 − 3 a2 ( 3v21l2 + 2a22x1x2p3 + 2a2v21x 2 1 ) . 9 Conclusions Let us summarize our results. We have provided an exhaustive determination of quadratically superintegrable systems which separate in the Cartesian coordinates with magnetic field. In addition, we have found classes of systems minimally and maximally superintegrable with higher- order integrals. We list them below for reader’s convenience. 9.1 Superintegrable systems with second-order integrals We have constructed an exhaustive list of quadratically superintegrable systems with nonvanish- ing magnetic field which separate in Cartesian coordinates. Under the assumption that there is no independent first-order integral other than the Cartesian ones (in that case we refer the reader to our previous work [14, 16]) we have found 8 classes of minimally superintegrable systems, among which one contains a quadratically maximally superintegrable subclass, cf. (8.4). For brevity, we write here the magnetic field, the electrostatic potential and the leading order terms in the integral(s) together with the reference to the equation in which the system was introduced. We refer the reader to the relevant formulas therein encoding the complete information about the integral(s). Case I, i.e., the magnetic field and potential are of the form (1.5) and the Cartesian integrals as in (1.8). The superintegrable systems read (a) ~B(~x) = ( aebx2 , c, 0 ) , W (~x) = a ( w + c b x1 ) ebx2 − a2 2b2 e2bx2 , X3 = pA1 p A 3 + · · · , cf. (A.11). Classical Superintegrable Systems in a Magnetic Field 17 (b) ~B(~x) = 2 ( a1x2 − a3 x32 ,−a1x1 + a2 x31 , 0 ) , W (~x) = −1 2 a21 ( x21 + x22 )2 − a22 2x41 − a23 2x42 − a1 ( a2 x22 x21 + a3 x21 x22 ) − a2a3 x22x 2 1 + b3 x22 + b1 ( x21 + x22 ) + b2 x21 , X3 = ( lA3 )2 + · · · , cf. Table 1. (c) ~B(~x) = ( 2 ( a1x2 − a3 x32 ) ,−8a1x1 − a2, 0 ) , W (~x) = −a 2 1 2 ( 4x21 + x22 )2 − a22 2 x21 − a23 2x42 − a2a3 x1 x22 − a1a2x1 ( 4x21 + x22 ) − 4a1a3 x21 x22 + b3 x22 + b1 ( 4x21 + x22 ) + b2x1, X3 = pA2 l A 3 + · · · , cf. Table 1. (d) ~B(~x) = (2a1x2 + a3,−2a1x1 − a2, 0), W (~x) = −a 2 1 2 ( x21 + x22 )2 − a23 2 x22 − a22 2 x21 − a2a1x1 ( x21 + x22 ) − a2a3x1x2 − a1a3x2 ( x21 + x22 ) + b1 ( x21 + x22 ) + b2x1 + b3x2, X3 = pA1 p A 2 + · · · , cf. Table 1. When a1 = a3 = 0 and b1 = b2 = 0 the system becomes maximally superintegrable, with the additional integral of the form X4 = pA1 l A 3 − 3pA3 l A 1 + · · · , cf. (8.4). Case II, i.e., the magnetic field and potential are of the form (1.7) and the Cartesian integrals as in (1.9). The superintegrable systems read (a) ~B(~x) = ( 0, aebx1 , 0 ) , W (~x) = wx1 + cebx1 − 1 2 a2 b2 e2bx1 , X3 = pA1 p A 2 − bpA3 lA1 + · · · , cf. (C.7), (b) ~B(~x) = ( 0, a(b− 2)xb−31 , 0 ) , W (~x) = −a 2x 2(b−2) 1 2 + a(b− 2)cxb−21 + w x21 , X3 = pA1 l A 3 − bpA3 lA1 + · · · , cf. (C.9), (c) ~B(~x) = ( 0, a x1 , 0 ) , W (~x) = −1 2 a2 (ln |x1|)2 + b ln |x1|+ w x21 , X3 = 2pA1 l A 3 − pA3 lA1 + · · · , cf. (C.10), 18 A. Marchesiello and L. Šnobl (d) ~B(~x) = ( 0, 0, a x31 ) , W (~x) = −ab ln |x1| x21 − a2 8x41 + w x21 , X3 = pA1 l A 2 + · · · , cf. (C.18). Our approach also demonstrates that any quadratically maximally superintegrable system with magnetic field which separates in Cartesian coordinates would necessarily appear at the intersection of the presented classes. Given the different structure of the magnetic field in each of the cases we find only few potential candidates. One is the intersection of Case I.d and Case II.b which, as we already observed, leads to the maximally superintegrable system (8.4). Another is the system Case I.a which for c = 0 reduces to the system Case II.a (upon interchange of the x1 and x2 coordinates and momenta). However, the integral X3 of Case I.a when c = 0 becomes a function of the two first-order Cartesian integrals, i.e., it is not independent anymore. Last but not least, the systems Case I.b, Case I.c, Case II.b and Case II.d (after a permutation of coordinates) overlap for a1 = a2 = 0 (Case I.b/c) and b = 0 (Case II.b/d) but the integrals again turn our to be dependent. Thus we conclude that no other quadratically maximally superintegrable systems separating in Cartesian coordinates other than (8.4) and the ones found in [14] exist. 9.2 Superintegrable systems with higher-order integrals Above we have provided a complete answer to the problem of quadratic superintegrability for the considered classes of systems (1.5) and (1.7). As we have seen, maximal superintegrability via at most quadratic integrals is very rare in the presence of magnetic field, as opposed to numerous purely scalar maximally superintegrable systems discussed, e.g., in [9, 13]. Thus one should consider also the possible existence of higher-order integrals. However, these are computationally very difficult to find. In this paper we have presented two propositions, namely Propositions 2.1 and 3.1 which can be used to construct three-dimensional maximally superintegrable systems with magnetic field out of two-dimensional scalar ones. In particular, Proposition 3.1 states that a system with ~B(~x) = (0, γ, 0), γ 6= 0, W (~x) = V (x2), is maximally superintegrable if and only if the two-dimensional system with the Hamiltonian K( ~X, ~P ) = 1 2 ( P 2 1 + P 2 2 ) + 1 2 γ2X2 + V (Y ) is superintegrable (where V is the same function of a single variable). Using Proposition 3.1 we have arrived at an explicit example of maximally superintegrable system with ~B(~x) = (0, γ, 0), W (~x) = c x22 + m2 `2 γ2x22, `,m ∈ N, c ∈ R, cf. (3.11), with three first-order integrals (3.4) and an additional integral coming from the integral of two-dimensional caged oscillator through the change of variables (3.6). Similarly, Proposition 2.1 led us to minimally superintegrable systems (2.5) ~B(~x) = 2 ( ωm1x2 − β1 x32 ,−ω`1x1 + α1 x31 , 0 ) , Classical Superintegrable Systems in a Magnetic Field 19 W (~x) = −ω 2 2 ( `1x 2 1 +m1x 2 2 )2 + ω ( `2x 2 1 +m2x 2 2 − α1m1 x22 x21 − β1`1 x21 x22 ) + α2 x21 + β2 x22 − 1 2 ( α1 x21 + β1 x22 )2 , l1 m1 = l2 m2 = l2 m2 , l,m ∈ Z. Systems I.b and I.c of Section 9.1 with a1 6= 0 are special subcases of it when the integral X3 becomes second order one. Another class of minimally superintegrable systems with ~B(~x) = (a1, a2, 0), W (~x) = v12x 2 1 + v22x 2 2 − 1 2 (a2x1 + a1x2) 2 , v12 v22 ∈ Q, v12, v22 6= 0 can be constructed out of anisotropic harmonic oscillator in two dimensions through the canon- ical transformation (8.2). Of course, more efficient and widely applicable tools for construction of higher-order super- integrable systems are needed. Given the recent rapid progress on a similar problem for scalar potentials [7, 18, 19, 21, 23] (see also references in [22]) we hope that in foreseeable future we will be able to report on further development also in the case with magnetic field. A Solution of the determining equations in Case I for β12 = β21 = 0 We consider here β12 = β21 = 0. Thus c0 and c1 cannot both vanish, since by assumption the right-hand side of (6.4) is not identically zero and, by (6.5), aj = 0, j = 1, 2. There are several subcases, according to whether equation (6.7) is trivially satisfied or not. For the reader’s convenience, let us type it here again (β33x1 + γ23)u ′ 1(x2) + (β33x2 − γ13)u′2(x1)− c1 = 0. (A.1) We can have (a) the above equation is not trivially satisfied for u2 nor for u1, thus β33 6= 0 or β33 = 0 and both γ13, γ23 not vanishing; (b) equation (A.1) is trivially satisfied for u2 but not for u1, thus β33 = γ13 = 0 and γ23 6= 0; (c) equation (A.1) is trivially satisfied for u1 but not for u2, thus β33 = γ23 = 0 and γ13 6= 0; (d) equation (A.1) is trivially satisfied for both uj , thus c1 = β33 = γ23 = γ13 = 0. Notice that Case (b) can be reduced to Case (c), and viceversa, by a canonical exchange of p1 with p2. Thus both cases are recovered by Appendix A.3. Case (a) is splitted for convenience into Appendices A.1 and A.2, while Case (d) is treated in Appendix A.4. A.1 β33 6= 0 In this case, by translation in x1 and x2, we can set γ13 = γ23 = 0. By taking second derivatives of (A.1) with respect to x1 and x2 and by equating them to zero, we find that u1(x2) = a12 2 x22 + a11x2, u2(x1) = −a12 2 x21 + a21x1, aij ∈ R. (A.2) 20 A. Marchesiello and L. Šnobl By imposing then that (A.2) solves (A.1), we obtain a polynomial expression in x1 and x2 which must vanish c1 − (a11x1 − a21x2)β33 = 0. We conclude a11 = a21 = c1 = 0. Solving the remaining equations in (4.3) we find the functions Sj . Next we look at the first- order equations (4.4), and in particular at the third one, for mx3 . We proceed as above to solve it for Vj . By considering its second-order derivatives with respect to x1 and x2 and by equating them to zero, we find that also the functions Vj must be second-order polynomials. Then we plug such solution back into the equation for mx3 and find a polynomial in x1 and x2 that must vanish. This implies c0 = 0. Thus {X3, p3} = 0. Nothing new can be found in this case. A.2 β33 = 0, γ13 6= 0, γ23 6= 0 Proceeding as above we find from (A.1) that c1 = (a1γ23 − a2γ13) and u1(x2) = a1x2, u2(x1) = a2x1, aij ∈ R, (A.3) where |a1|2 + |a2|2 6= 0. The solution (A.3) implies that the magnetic field is constant. By substituting (A.3) into the compatibility conditions (4.7) for the magnetic field, they imply α33 = 0. The second-order equations (4.3) can now be solved and give S1(x1, x2) = a2γ13x1 + (S − a1γ13a2γ23)x2 + s11, S2(x1, x2) = −Sx1 − a1γ23x2 + s21, S3(x1, x2) = 1 2 ( a1β13x 2 1 + a2β23x 2 2 ) − a2γ1x2 + (a2β13x2 + a1γ1)x1, S, sij ∈ R, together with the condition β13a2 = β23a1. We now look at the first-order equations (4.4), starting with the equation for mx3 . Its first- order derivatives with respect to x1 and x2 imply that each Vj is a second-order polynomial in its variable, j = 1, 2. This case is discussed in Section 8. A.3 β33 = γ23 = 0, γ13 6= 0 In this case equation (A.1) is trivially satisfied for u1 and as an equation for u2 implies u2(x1) = −c1x1 γ13 . (A.4) By solving the second-order equations (4.3) for sj , j = 1, 2, 3, we find S1(x1, x2) = −c1x1 − Sx2 − γ13u1(x2) + s11, S2(x1, x2) = Sx1 + s21, S3(x1, x2) = c1x2 ( 2α33x1x2 − 2β13x1 − 1 2β23x2 + γ12 ) γ13 + x1 ( x1 ( −α33x2 + 1 2 β13 ) − β23x2 + γ12 ) u′1(x2), Classical Superintegrable Systems in a Magnetic Field 21 where S, sij ∈ R and u1 has to solve the remaining compatibility conditions (4.7) (some are already satisfied by the conditions imposed on the constant αij , βij , γij and (A.4)): 6c1α33 γ13 − β23x2u(3)1 (x2) + γ12u (3) 1 (x2)− 4β23u ′′ 1(x2) = 0, (A.5) (2α33x2 − β13)u′′1(x2) + 6α33u ′ 1(x2) = 0. (A.6) The second derivative with respect to x1 of the third first-order equation in (4.4) implies V1(x1) = v11x1 + v12x 2 1, vij ∈ R. (A.7) To proceed, we must solve the equations (A.5)–(A.6) for u1. There can be several subcases. A.3.1 α33 = β13 = β23 = γ12 = 0 In this case (A.5)–(A.6) are trivially satisfied. Let us consider the second set of compatibility conditions (4.8) coming from the first-order equations, which simplify to c1 2 γ13 − 2v12γ13 − Su′1(x2) = 0, (A.8) c1 ( S γ13 + u′1(x2) ) − s21u′′1(x2) = 0. (A.9) Let us first assume u′′1 6= 0. If c1 = 0, then S = v12 = s21 = 0 and u2(x1) = 0. Equation (4.5) implies that then u1 is a constant, in contradiction with the assumption u′′1 6= 0. Thus, let us consider the case u′′1 and c1 both not zero. From equation (A.9) we obtain S = 0 and u1(x2) = a1s21 c1 e c1x2 s21 , a1 ∈ R together with v12 = c21 2γ213 that we substitute into (A.7). The solution of the compatibility constraints (A.8)–(A.9) assures that a solution of (4.4) for m exists. Thus, let us look at the zero-order equation (4.5), which in this case reduces to c1v11x1 − a1s21e c1x2 s21 ( 4v11γ 2 13 + c1s11 ) c1γ13 + s11v11 + s21V ′ 2(x2) = 0. (A.10) Therefore v11 = 0 and (A.7) simplifies to V1(x1) = c21x 2 1 2γ213 . Let us notice that if s21 = 0 we have constant magnetic field along the x2 direction and arbit- rary V2. Thus, we obtain the class of systems already studied in Section 3. Otherwise, if s21 is not zero, by solving (A.10) we find V2(x2) = a1s11s21e c1x2 s21 c1γ13 . 22 A. Marchesiello and L. Šnobl Thus, we arrive at the system determined by ~B(~x) = ( aebx2 , c, 0 ) , W (~x) = a ( w + c b x1 ) ebx2 − a2 2b2 e2bx2 , (A.11) where we relabelled the integration constants as a1 = a, c1 = −γ13c, s11 = γ13bw, s21 = − cγ13 b , with a, b, c, w ∈ R such that a, b, and c are not vanishing. The system (A.11) is (at least) minimally superintegrable, with the integral X3 as in (4.6), where all the coefficients of the second-order terms are zero except γ13, and Sj and m3 of equation (6.6) are given by S1(x1, x2) = γ13 ( wb− ebx2 a b + cx1 ) , S2(x1, x2) = −γ13 c b , S3(x1, x2) = 0, m3(~x) = γ13c ( ebx2 a b − cx1 − wb ) x3. It remains to be considered the case in which u′′1 = 0, therefore u1(x2) = a1x2, the magnetic field is constant. If c1 = 0, from the zero-order equation (4.5) we get that either V2 is a second- order polynomial (not of interest here since V1 is already a second-order polynomial, too), or S = s21 = v11 = 0 and v12 = 0 from (A.8). This gives V1(x1) = 0. From the first-order equations (4.4) we find c0 = 0. Nothing new here. If c1 is not zero, the compatibility condition (A.8), (A.9) together with the zero-order equa- tion (4.5) imply that either V1 and V2 are both second-order polynomials or V1 = 0 and the magnetic field is constant and aligned along the x2-axis. Both these cases are of no interest in this section since they are considered elsewhere, in Section 8 and Appendix C. A.3.2 α33 = 0, β13 6= 0 By translation in x1 we can assume γ12 = 0. The compatibility condition (A.5) reduces to −β23 ( x2u (3) 1 (x2) + 4u′′1(x2) ) = 0, β13u ′′ 1(x2) = 0. (A.12) Since β13 is not zero, (A.12) gives u1(x2) = a1x2, a1 ∈ R. The compatibility conditions (4.8) for m3 imply that V2 is a second-order polynomial. This case is of no interest here and it is studied in Section 8. A.3.3 α33 = 0, β13 = 0, β23 6= 0 Still, by translation in x1, we can set γ12 = 0. Equations (4.3) imply u1(x2) = a2 6x22 . In order to have nonvanishing magnetic field and nonvanishing Poisson bracket in (6.1), we find from equations (4.4) and (4.5) that we must have S = 0, a2 = 0 and c0 = c1s11 γ13 . Looking for a solution of (4.5) for V2 not second or lower-order polynomial we find ~B(~x) = ( 0,− c1 γ13 , 0 ) , V1(x1) = c21x 2 1 2γ213 , V2(x2) = c21x 2 2 8γ213 − v21 2x22 , which is the already known system (3.8). Classical Superintegrable Systems in a Magnetic Field 23 A.3.4 α33 = β13 = β23 = 0, γ12 6= 0 The compatibilities (A.5)–(A.6) reduce to the sole equation γ12u (3) 1 (x2) = 0, therefore u1(x2) = a1x2 + a2x 2 2, aj ∈ R. The compatibility conditions (4.8), together with the zero-order equation, imply that either the magnetic field is constant and V2 is a second-order polynomial, or V1(x1) = u2(x1) = 0 and X2 reduces to a first-order integral. Both these cases are of no interest here. A.3.5 α33 6= 0 By translation in x1 and x2 we can set β13 = β23 = 0. Thus, (A.6) gives u1(x2) = a 2x22 , a ∈ R. By plugging this solution into (4.3) we obtain that it must be c1 = 0 and therefore by (A.5) γ12 = 0. By looking for a solution of the equations (4.8) and (4.4) we find that also c0 = 0, i.e., {X, p3} = 0, and this case is of no interest here. A.4 β33 = γ23 = γ13 = c1 = 0 In this case equation (A.1) is trivially satisfied for both uj . The remaining second-order equations give for Sj : S1(x1, x2) = Sx2 + s11, S2(x1, x2) = −Sx1 + s21, S3(x1, x2) = −x1 2 (2α33x1x2 − β13x1 + 2β23x2 − 2γ12)u ′ 1(x2) − 2x2(α33x2 − β13)u2(x1) + S31(x2), S, sij ∈ R, where S31(x2) must solve( α33x 2 1x2 − β13 2 x21 + β23x1x2 − γ12x1 ) u′′1(x2) + 3x1(α33x1 + β23)u ′ 1(x2) − ((β13 − 2α33x2)x1 − β23x2 + γ12)u ′ 2(x1) + 2(2α33x2 − β13)u2(x1)− S′31(x2) = 0 and uj satisfy the compatibilities (4.7), that in this case reduce to (2α33x1x2 − β13x1 + β23x2 − γ12)u′′′1 (x2)− 4(2α33x1 + β23)u ′′ 1(x2) + (2α33x1 + β23)u ′′ 2(x1) + 6α33u ′ 2(x1) = 0, (A.13) 6α33u ′ 1(x2) + (2α33x2 − β13)(u′1(x2) + 8u′2(x1)) − (2α33x1x2 − β13x1 + β23x2 − γ12)u′′′2 (x1) = 0. (A.14) As in the above Appendix A.3, to solve (A.13)–(A.14) we have to distinguish several subcases. 24 A. Marchesiello and L. Šnobl A.4.1 α33 6= 0 By translation in x1 and x2 we can eliminate β13 and β23. By taking the third-order deriva- tives ∂2x1∂x2 , ∂2x2∂x1 of (A.13) and (A.14), respectively we get u2(x1) = a22x 2 1 − a23 x21 + a21x1, u1(x2) = a12x 2 2 − a13 x22 + a11x2, aij ∈ R. By plugging the above solution into the third condition in (4.4) we see that, as a polynomial in x1 and x2, it can be vanishing only if c0 = 0. Since also c1 = 0, there is nothing new here. A.4.2 α33 = 0, β13 6= 0 By translation in x1 we can assume γ12 = 0. By considering the first-order derivative of (A.13) with respect to x2 and of (A.14) with respect to x1, we find u1(x2) = a11x2 + a12x 2 2 + a13x 3 2, u2(x1) = a21x1 + a22x 2 1 + a23 x21 , aij ∈ R. (A.15) We plug (A.15) into (A.13)–(A.14) and we obtain the conditions a23 = a13 = 0 and a22β23 = 0. If β23 6= 0, then a22 = 0. The equation for mx3 in (4.4), together with (4.8), implies that Vj are second-order polynomials and a12 = 0, i.e., the magnetic field is constant. This case is of no interest here, cf. Section 8. If β23 = 0 we again consider the third equation in (4.4), together with (4.8) and their first order and second order mixed derivatives with respect to x1 and x2. If a22 is not vanishing, we can find a solution for both Vj which is not a second-order polynomial (or lower) but only assuming that c0 = 0. Therefore nothing new arises in this case. If a22 = 0, we can find a solution for Vj only if the magnetic field is vanishing, or c0 = 0, or the solution is equivalent to the system (3.8). Anyway, not of interest here (we recall that in Appendix A.4 we have c1 = 0). A.4.3 α33 = β13 = 0, β23 6= 0 We can eliminate γ12 by translation in x2, thus setting γ12 = 0. The compatibility conditions (A.13), (A.14) imply that u1(x2) = a12x 2 2 + a11 x22 , u2(x1) = −4a12x 2 1 + a21x1, aij ∈ R. By imposing that also the remaining equations (4.3), (4.4) and (4.5) are satisfied we find that either c0 = 0 which implies {X, p3} = 0 in contradiction with our assumption, or we find a11 = a12 = 0 and V1(x1) = 1 2 a221x 2 1, V2(x2) = a221 8 x22 − v21 2x22 , i.e., a system equivalent to (3.8). Classical Superintegrable Systems in a Magnetic Field 25 A.4.4 α33 = β13 = β23 = 0, γ12 6= 0 The compatibility conditions (A.13), (A.14) imply that u1(x2) = a12x 2 2 + a11x2, u2(x1) = −a12x21 + a21x1, aij ∈ R. By imposing that also the remaining equations are satisfied we find that necessarily a12 = 0 and V1(x1) = 1 2 ( a211 + a221 ) x21 + v12x1, V2(x2) = 1 2 ( a211 + a221 ) x22 + a21v12 a11 x2. Thus this case was already studied in Section 8. B Solution of the determining equations in Case I for β12, β21 not both vanishing Here we distinguish between the cases (a) β12 and β21 both not vanishing; (b) β12 6= 0 and β21 = 0. Notice that, by a canonical permutation of the variables, this case is equivalent to β21 6= 0 and β12 = 0. Case (a) is treated in the following Appendix B.1. Case (b) follows in Appendix B.2. B.1 β12 6= 0, β21 6= 0 Since β12 and β21 are both nonvanishing, by translation in x1 and x2 we can set γ13 = γ23 = 0. Let us start from the compatibility conditions (4.7), that read (β33x1 + β21x2)u ′′′ 1 (x2) + 4β21u ′′ 1(x2) = 0, β33(u ′′ 1(x2) + u′′2(x1)) = 0, (2α33x1x2 − β13x1 + β23x2 − γ12)u′′′1 (x2) + 4(2α33x1 + β23)u ′′ 1(x2) + (2α33x1 + β23)u ′′ 2(x1) + 6α33u ′ 2(x1) = 0, (β12x1 + β33x2)u ′′′ 2 (x1) + 4β12u ′′ 2(x1) = 0, β21u ′′ 2(x1) + β12u ′′ 1(x2) = 0, (2α33x1x2 − β13x1 + β23x2 − γ12)u′′′2 (x1)− 4(β13 − 2α33x2)u ′′ 2(x1) − (β13 − 2α33x2)u ′′ 1(x2) + 6α33u ′ 1(x2) = 0. (B.1) The above equations could be trivially satisfied for uj or not, depending on the constants αij , βij , γij . This determines a splitting in the computation. B.1.1 β33 6= 0 The second equation in (B.1) implies u1(x2) = a12 2 x22 + a11x2, u2(x1) = −a12 2 x21 + a21x1, aij ∈ R. By imposing that the above solution satisfies the remaining compatibilities (B.1) and that the magnetic field does not vanish, we find that a12 = α33 = 0, therefore the magnetic field is constant. The third second-order equation simplifies to (β33a11 + 3β12a21)x1 + (3β21a11 + β33a21)x2 = c1, 26 A. Marchesiello and L. Šnobl which must hold for all values of x1 and x2. Without loss of generality, we can assume that one component of the magnetic fields is not vanishing, e.g., a11. Thus the above equation implies β33 = −3 β12a21 a11 , β21 = a221 a211 β12, c1 = 0. (B.2) With this assumption the first and second-order equations (4.3) can be solved for S1 and S2. We plug the so found solution into the third equation in (4.4) and take its third-order deriva- tives ∂2x1∂x2 and ∂2x2∂x1 . In this way we obtain the condition β12a21V ′′′ j (xj) = 0, j = 1, 2. Since the magnetic field is already constant, we do not consider here solutions for Vj in the form of at most quadratic polynomials. Thus necessarily a21 = 0. However, from (B.2) we have β21 = 0, which violates our assumption for this subcase. B.1.2 β33 = 0 Equations (B.1) can be solved for nonvanishing magnetic field only if α33 = 0. In this case we find the solution u1(x2) = a1x2, u2(x1) = a2x1, aj ∈ R, corresponding to constant magnetic field. The third second-order equation then reduces to a1β21x2 + a2β12x1 = c1, which must hold for all x1, x2. Since β12 and β21 are assumed to be not vanishing in this section, we conclude that there is no solution for nonvanishing magnetic field. B.2 β12 6= 0, β21 = 0 Again, we start by the compatibilities (B.1), now simplified by the condition β21 = 0. We can proceed as above, considering first the case β33 6= 0. Then, by translation in x1 and x2 we can assume γ23 = γ13 = 0. We proceed as in Appendix B.1.1 with the simplification β21 = 0. Equation (B.2) implies that the magnetic field has to vanish. Therefore, we continue in the following by assuming β33 = 0. Since β12 6= 0, by translation in x1 we can still set γ13 = 0. The computation then splits into two major subcases, according to whether γ23 = 0 or not. B.2.1 γ23 6= 0 The third second-order equation simplifies to 2β12u2(x1) + γ23u ′ 1(x2) + β12x1u ′ 2(x1) = c1. (B.3) By solving for uj we find u1(x2) = a1x2, u2(x1) = a2 2x21 + c1 − a1γ23 2β12 , aj ∈ R. The remaining second-order equations can be solved for Sj only under the condition a1β23 = a2β23 = a2γ12 = a1α33 = 0. We find S1(x1, x2) = −s21x2 − a2β12 x1 + s12, Classical Superintegrable Systems in a Magnetic Field 27 S2(x1, x2) = a1β12 2 x21 − a1γ23x2 + s21x1 + a2γ23 2x21 + s22, S3(x1, x2) = β13a1 2 x21 + γ12a1x1 − α33a2 x22 x21 + β13a2 x2 x21 . Equations (4.4) imply the following constraint on our integration constants: a2s21 = 0. The compatibility conditions (4.8) can be solved for Vj and together with (4.5) imply a2 = 0. In order to have nonvanishing magnetic field we thus must have α33 = β23 = 0. Equation (4.5) further implies s21 = s12 = 0 together with V1(x1) = 1 8 a21x 2 1 − v1 2x21 , V2(x2) = 1 2 a21x 2 2 + ( a21γ 2 23 − a1c1γ23 − 2c0β12 ) x2 2γ23β12 + c0x2 γ23 , i.e., we arrived at a system equivalent to (3.8) (after translation in x2). B.2.2 γ23 = 0 Equation (B.3) does not contain u1 anymore, while for u2 implies u2(x1) = a2 x21 + c1 2β12 , a2 ∈ R. From the last but one equation in (B.1) we see that u1(x2) = a1x2, a1 ∈ R, while the remaining equations (B.1) read a2β23 = 0, α33a1x 5 1 − 4a2(β23x2 − γ12) = 0, (B.4) i.e., a2β23 = α33a1 = a2γ12 = 0. The second-order equations (4.3) can be solved for S1 and S2. We find S1(x1, x2) = Sx2 − 2a2β12 x1 + s11, S2(x1, x2) = −Sx1 + 1 2 a1β12x 2 1 + s21, sij ∈ R. The remaining second-order equations for S3 imply that β23 = 0, otherwise they would imply that the magnetic field must vanish. To proceed, we need to solve (B.4). We have to distinguish between the cases a2 = 0 and a2 6= 0. Let us start by assuming a2 = 0. The compatibility conditions (4.8) together with the zero- order equation (4.5) can be solved only if α33 = γ12 = c1 = 0 and S = s11 = s21 = 0. However, the resulting system is equivalent to the already known maximally superintegrable system (3.8), up to a permutation of the canonical variables. If a2 6= 0, then from (B.4) we have γ12 = 0 and α33a1 = 0. Under these two conditions, also the remaining second-order equations (4.3) can be solved for S3. We find S3(x1, x2) = 1 2 a1β13x 2 1 + 2a2 ( β13 x2 x21 − α33 x22 x21 ) . The compatibility conditions for the first-order equations (4.8) can be solved for V1. They admit a solution for V2 only if α33 = β13 = 0 (and in this way also the remaining (B.4) is satisfied). After solving for V1 and imposing the previous condition, we see that they are satisfied for 28 A. Marchesiello and L. Šnobl any V2. To find V2 we look at the zero-order equation, in which we impose all the conditions we obtained till now and the solutions found for V1 and Sj . We obtain in this way a polynomial in x1 (whose some coefficients contains equations for V2) that must vanish. By collecting the different powers of x1 and impose that they are all equal to zero, we arrive at the condition a2β12 = 0, which cannot be satisfied in the case we are considering here. Thus, no new system can be found. C Solution of the determining equations for Case II Let us start by some preliminary considerations. By taking second-order derivatives with respect to x2 of (4.4) and (4.5) we obtain the conditions s′′1(x2)U ′′ 2 (x1) = 0 and s′′1(x2)W ′(x1) = 0. If s′′1 6= 0, we therefore have U ′′2 = W ′ = 0. As a consequence, new solutions can arise here only for U ′′′3 6= 0 (i.e., for nonconstant magnetic field). If so, (7.5) imply β33 = β31 = 0. However then the compatibility conditions (4.8) cannot be solved for s′′1 6= 0. Thus, necessarily s′′1 = 0, which gives s1(x2) = s11 + s12x2. (C.1) Let us proceed by looking at (7.5) in case β33 6= 0. Then U ′′′2 = 0 and the third equation in (7.5) can be solved for U3. Then the remaining equations imply that the magnetic field and the potential W are constant, which is not of interest here. Thus, necessarily β33 = 0 and (C.1) holds. C.1 α33 6= 0 By translation in both x1 and x2 we can set β13 = β23 = 0. The conditions (7.5) and (4.3) can be solved. We find U2(x1) = a1 2x1 , U3(x1) = a2 2x1 , s3(x1) = 0, s2(x1) = −s12x1, a1, a2, s12 ∈ R, together with c2 = β21 = β31 = γ13 = γ12 = 0. This give the solution for Sj , j = 1, 2, 3. The compatibility conditions (4.8) can then be solved for W and by imposing also (4.5) give W (x1) = − a 8x41 + w x21 , a, w ∈ R, (C.2) together with s11 = s12 = β12 = a2 = 0, where a ≡ a1. The magnetic field is ~B(~x) = ( 0, ax−31 , 0 ) , a ∈ R. (C.3) The first-order equations (4.8) give M(x1, x2) = α33 ( 2wx22 x21 − a2x22 2x41 ) with the condition c0 = c1 = 0. This system is a special case of the systems in Table 1, namely it can be expressed in the forms both E1 and E2. The integral constructed here is included in the ones coming from E1 and E2, i.e., the system is only minimally quadratically superintegrable. From now on we continue our search by assuming α33 = 0. Classical Superintegrable Systems in a Magnetic Field 29 C.2 α33 = 0, β23 6= 0 By translation in x2 we can set γ12 = 0. Then the second and fourth equation in (7.5) imply U2(x1) = a11x 2 1, U3(x1) = a21x 2 1 + a22x 3 1, aij ∈ R. From the compatibility conditions (4.8) and the zero-order equation (4.5) we easily see that the only possibility is polynomial potential at most quadratic and a22 = 0 (and a21a11 = 0), i.e., constant magnetic field. Thus, this is not of interest here. C.3 α33 = β23 = 0, β31 6= 0 Equation (7.4) reads (β12x1 − γ13)U ′′2 (x1) + 2β12U ′ 2(x1)− β31U ′3(x1)− c2 = 0, which, integrated by x1 and neglecting integration constants (which do not affect the magnetic field), gives U3(x1) = 1 β31 ( (β12x1 − γ13)U ′2(x1) + β12U2(x1) + c2x1 ) . (C.4) Conditions (4.8) together with (4.4) and (4.5) seen as a polynomial in x2 imply that unless s12 = 0 the magnetic field vanishes. Since this is of no interest here, we continue in the following by assuming s12 = 0. From the second-order equations (4.3), considered as polynomials in x2, we see that necessarily s3(x1) = s2(x1) = 0. We shall distinguish several subcases. C.3.1 β12 = γ13 = 0 Let us start by the easiest case in which β12 = γ13 = 0. By (C.4) we have here U ′′3 = 0. Thus U ′′2 cannot vanish, otherwise we have no magnetic field. From equations (4.3) we see that necessarily β21 = 0. At this point, all (4.3) are solved, except for the condition (β13x1 + γ12)U ′′′ 2 (x1) + (3β13 + β31)U ′′ 2 (x1) = 0. (C.5) The zero-order equation reads( s11 + c2γ12 β31 ) W ′(x1) = 0. For s11 6= − c2γ12 β31 the above condition implies vanishing potential W (x1). From (4.4) we see that then also U ′′2 is constant. Thus, this solution is of no interest here. Thus, let us set s11 = − c2γ12 β31 . In this case the third equation in (4.4) is solved for c0 = − c1c2 β31 . The remaining first-order equations can be solved for M and give M(x1, x2) = (β13x1 + γ12)x2W ′(x1) for W (x1) satisfying( W ′′(x1) + U ′′2 (x1) 2 ) (β13x1 + γ12) + (c1 − β31U ′2(x1))U ′′2 (x1) + 3β13W ′(x1) = 0. (C.6) Recall that U2 needs to satisfy (C.5), whose solution depends on the constants involved there. We can have 30 A. Marchesiello and L. Šnobl (a) β13 = 0. If γ12 6= 0, (C.5) is solved by U2(x1) = a b2 ebx1 , a, b ∈ R, b = −β31 γ12 ∈ R, giving a new superintegrable system ~B(~x) = ( 0, aebx1 , 0 ) , W (~x) = wx1 + cebx1 − 1 2 a2 b2 e2bx1 , a, w ∈ R, (C.7) where c = −ac1γ12 β2 31 ∈ R. For γ12 = 0, equation (C.5) reduces to β31U ′′ 2 (x1) = 0, implying U ′′2 (x1) = 0 and therefore no magnetic field. (b) β31 6= −β13 6= 1 2 β31, β13 6= 0. By translation in x1 we can set γ12 = 0. After the substitution β31 = −bβ13, the solution of (C.5) in this case reads U2(x1) = axb1 b− 1 , a ∈ R, b 6= 0, 1, 2. (C.8) By solving (C.6) for W , we arrive at a new system determined by ~B(~x) = ( 0, a(b− 2)xb−31 , 0 ) , W (~x) = −a 2x 2(b−2) 1 2 + a(b− 2)cxb−21 + w x21 , (C.9) where a, b, w, c = c1 b(2−b)β13 ∈ R, b 6= 0, 1, 2. Notice that in the limit b→ 0 we reduce (C.9) to the system determined by (C.2) and (C.3), that therefore can be seen as a special case of (C.9) (which is already known to have two second- order integrals, with highest-order terms ( lA3 )2 and pA1 l A 3 , that are however dependent once also the Cartesian integrals (1.9) and the Hamiltonian are taken into account). (c) β13 = −1 2 β31. Since in this case β13 = −1 2β31 6= 0, by translation in x1 we can still set γ12 = 0. Equation (C.5) has solution U2(x1) = ax1 ln |x1|, a ∈ R, that gives the new system ~B(~x) = ( 0, ax−11 , 0 ) , W (~x) = −1 2 a2(ln |x1|)2 + b ln |x1|+ w x21 , a, w ∈ R, (C.10) where b = ac1 β31 − a2. (d) β13 = −β31. As above, we can still set γ12 = 0 by translation. Equation (C.5) admits the solution U2(x1) = −a ln |x1|, a ∈ R. (C.11) By denoting c = − c1a β13 , from (C.6) and (C.11) we obtain ~B(~x) = ( 0, ax−21 , 0 ) , W (~x) = c x1 + w x21 , a, c, w ∈ R. (C.12) Thus, though (C.8) is singular for b = 1 (and we obtain a different solution (C.11) for (C.5) if b = 1), indeed such a singularity does not appear in the solution for ~B and W (it is lost by differentiating) and the system (C.9) becomes (C.12) for b = 1. Classical Superintegrable Systems in a Magnetic Field 31 C.3.2 γ13 6= 0, β12 = 0 In this case it is convenient to start by looking at the zeroth-order equation (4.5), that simplifies to ( γ13 ( (β13x1 + γ12)U ′′ 2 (x1)− (β13 − β31) ) U ′2(x1) + s11β31 + c2γ12 ) W ′(x1) = 0. (C.13) For W ′ 6= 0, we solve the above equation for U2. Once we plug the so found solution into the remaining equations we see that there can be several subcases according to different values of the constants involved. However, in most cases, the solution for U2 is polynomial and at most of first order. By (C.4) also U3 results into a polynomial of order at most one. Thus in these cases we have vanishing magnetic field. The only exception is the solution U2 (x1) = 2γ12a 2 ( b22 + 1 2 ) eb1x1 − ( (2b1b2s11 + c1)a+ w2b 2 1γ12 ) b1x1 b21γ12a ( 2b22 + 1 ) , ~B(~x) = (0, 1, b2) · aeb1x1 , W (~x) = − ( b22 + 1 ) 2b21 a2e2b1x1 + w2e b1x1 + x1w1, M(x1, x2) = −γ12 b1 ( a2 ( b22 + 1 ) e2b1x1 − w2b 2 1e b1x1 − w1b1 ) x2, (C.14) where s12 = 0, β13 = 0, β21 = b1b2γ12, β31 = −b1γ12, γ13 = b2γ12, c0 = γ12b2w1 + b31γ12w 2 2b 3 2 a2(2b22+1)2 − b1b2(2b1b2s11+c1)w2 a(2b22+1)2 + (b1s11−b2c1)(b1b2s11+b22c1+c1) b1γ12(2b22+1)2 and c2 = ab1s11−w2b21b2γ12−ab2c1 a(2b22+1) . By rotation of the coordinate frame around the x1-axis, which preserves the existence of the Cartesian integrals p2 and p3, the system (C.14) can be brought to the form (C.7). For W ′ = 0 the equation (C.13) does not give any condition on U2, that is therefore con- strained only by the remaining second and first-order equations (4.3), (4.4). However, also in this case we always arrive to a solution for U2 at most linear in x1, except for U2 = 1 b2 (a1 sin(bx1) + a2 cos(bx1)) that leads to the superintegrable system already found in [16], whose magnetic field has components B1 = 0, B2 = a1 sin(bx1) + a2 cos(bx1), B3 = a2 sin(bx1)− a1 cos(bx1), where a1, a2 ∈ R and b = β31 γ13 (which must for this case be also equal to b = β21 γ12 ). C.3.3 β12 6= 0 By translation in x1, let us set γ13 = 0. We can proceed as above and, after a long and tedious computation, arrive at two solutions • if W is not identically zero, we have U2(x1) = 1 b1 − 1 ( axb1−11 + b1s11 b2β13 ) , ~B(~x) = (0, 1, b2) · a(b1 − 2)xb1−31 , W (~x) = −a 2 2 ( b22 + 1 ) x 2(b1−2) 1 + w2x b1−2 1 + w1 x21 , M(x1, x2) = β13x2 ( (b1 − 2) ( w2x b1 1 − a 2 ( b22 + 1 ) x 2(b1−2) 1 ) − 2 w1 x21 ) , (C.15) 32 A. Marchesiello and L. Šnobl where c0 = − c2 a w2− (b1−1)b2c22 b21β13 , c1 = −2(b1−1) b2b1 c2− b1 a β13w2, s12 = 0, β21 = b1b2β13, β12 = −b2β13, β31 = −b1β13 and γ12 = 0. By rotation of the coordinate frame around the x1-axis, which preserves the existence of the Cartesian integrals p2 and p3, the system (C.15) can be brought to the form (C.9). • For W (~x) = 0 we find U2(x1) = −a ln |x1|, ~B(~x) = (0, 1, b) · a x21 , W (~x) = 0, M(x1, x2) = 0, where c0 = 0, c1 = 0, s11 = −abβ31, β13 = −β31, β21 = −bβ31 and γ12 = 0. However, this solution can be obtained as a limiting case of (C.15) with b1 = 1, b2 = b, w2 = 0 and w1 = a2(b22+1) 2 . C.4 α33 = β23 = β31 = 0 Equation (7.4) now reads (β12x1 − γ13)U ′′2 (x1) + 2β12U ′ 2(x1)− c2 = 0. (C.16) Thus, it does not imply any relationship between U2 and U3. It only gives a condition on U2 that could be trivially satisfied or not, according to the values of the constants involved. By the same argument used in Appendix C.3, we see that also in this case s12 = 0 and s3(x1) = s2(x1) = 0. C.4.1 β21 6= 0 Let us look at the situation in which β21 6= 0 and recall that we are looking for an integral X of the form (7.1). Its Poisson bracket with X1 = p2, i.e., {X, p2}, is again an integral, by assumption expressible in terms of the known integrals, similarly to equation (7.1). However, these two expressions can be interchanged under the permutation of x2 and x3; thus we can reduce the considered problem into an already discussed one. We have that (p1, p2, p3, l1, l2, l3)→ (p1, p3, p2,−l1,−l3,−l2), while the system (1.7) transforms into W (x1)→W (x1), (u2, u3)→ (−u3, u2). (C.17) Therefore, all the conditions imposed in this section (including β21 6= 0) on the coefficients of the second-order terms in the integral change into α22 = β32 = β21 = β22 = 0, β31 6= 0 and (7.2) turns into a0 = −β13, a1 = 0, a2 = β31, a3 = β21 and α11 = α12 = α13 = α23 = α33 = β11 = β33 = β23 = 0. Thus we recover all the conditions imposed on our parameters in Appendix C.3. By (C.17) any system belonging into this section would transform into some system already found in Appendix C.3. Therefore in the following we can assume that β21 = 0 without lost of generality. We distinguish several subcases depending on whether (C.16) is trivially satisfied or not. C.4.2 β21 = 0, β12 6= 0 By translation in x1 we can set γ13 = 0. Equation (C.16) gives U2(x1) = a1 2x1 + c2 2β12 x1, a1 ∈ R. Classical Superintegrable Systems in a Magnetic Field 33 We look at the remaining second-order equations (4.3), that read (β13x1 + γ12)U ′′′ 3 (x1) + 3β13U ′′ 3 (x1) = 0, β12x 5 1U ′′′ 3 (x1) + 3β12x 4 1U ′′ 3 (x1)− 3γ12a1 = 0. If β13 = 0 and γ12 6= 0 they have a solution only if a1 = 0 and U3 = 0, i.e., the magnetic field vanishes. If β13 = γ12 = 0, the above equations admit solution for U3(x1) = a2 x1 . The zeroth-order equation (4.5) has solution for W (x1) = 0. If β12 6= 0 the remaining equa- tions (4.4) lead to the solution U2(x1) = 0, U3(x1) = a 2x1 , ~B(~x) = ( 0, 0, a x31 ) , W (~x) = −ab ln |x1| x21 − a2 8x41 + w x21 , (C.18) where a1 = 0, c0 = 0, c1 = 2β12b, c2 = 0, s11 = 0, β13 = 0, γ12 = 0 and M(x1, x2) = 0. This system is new when b 6= 0, otherwise it can be turned into a subcase of (C.9) by permutation of x2 and x3. If β13 6= 0 we can set γ12 = 0. We have a solution for U3 given by U3(x1) = 4a2 β13x1 , b ∈ R. Equations (4.4) and (4.5) can be solved for W and M under the conditions s11 = 0 and a2 = −a1β12 8 . We find M(x1, x2) = β13 ( b2 + 1 ) a2 x2 2x41 − β13c 2 ln |x1| − 1 x21 − 2β13w x2 x21 and W (x1) = − ( b2 + 1 ) a2 8x41 + c ln |x1|+ w x21 , w ∈ R, (C.19) where b = −β12 β13 6= 0, a = a1, c = ac1 2β13 and by using the physically irrelevant shift of the potential by an additive constant we set c0 = c2 = 0. The magnetic field reads ~B(~x) = (0, 1, b) · ax−31 , a, b ∈ R, b 6= 0. (C.20) However, the reference frame can be rotated around x1 without affecting the fact that p2 and p3 are integrals. Such a rotation brings the magnetic field (C.20) and the potential (C.19) and the integral into the form (C.18). C.4.3 β21 = β12 = 0, γ13 6= 0 From equation (C.16) we have U2(x1) = −c2x 2 1 2γ13 . From the second and zeroth-order equations we get U3(x1) = ax21, a ∈ R. The first-order equations can be solved for W and give a polynomial solution at most quadratic. Since the magnetic field is constant, nothing of interest can be found here. 34 A. Marchesiello and L. Šnobl C.4.4 β21 = β12 = γ13 = 0 Equation (C.16) implies c2 = 0 and it is trivially satisfied for U2. The remaining second-order equations read β13 ( 3U ′′j (x1) + x1U ′′′ j (x1) ) + γ12U ′′′ j (x1) = 0, j = 2, 3. If β13 = 0 we easily conclude that the only possibility is constant magnetic field and vanishing potential. For β13 6= 0 we can set γ12 = 0 by translation in x1. The above equations imply Uj(x1) = aj 2x1 , j = 2, 3. From the first-order equations, seen as polynomials in x2, we get the conditions a22 x51 + a23 x51 − 2W ′(x1) = 0, a2(2a3β13 + 3s11x1) + c1a3x 2 1 = 0, together with two differential equations for M . From the second equation above we see that we have two possibilities for nonvanishing magnetic field: a2 = c1 = 0 or a3 = s11 = 0. In the first case, when imposing also the remaining zero-order equation (4.5) we get that necessarily a3 = 0, i.e., the magnetic field vanishes. In the second case we have the solution W (x1) = c ln |x1|+ w x21 − a2 8x41 , a, c, w ∈ R, with c = ac1 2β13 , a = a2. The magnetic field reads ~B(~x) = ( 0, ax−31 , 0 ) , a ∈ R. By solving the remaining equations for M , we find M(x1, x2) = a2β13 x2 2x41 − 2β13c ln |x1| x2 x21 + β13(c− 2w) x2 x21 . However, this system is just a special case of (C.19), (C.20) for b = 0. Acknowledgments This paper was supported by the Czech Science Foundation (Grant Agency of the Czech Repub- lic), project 17-11805S. This paper is dedicated to our son Flavio born just after the submission of the original manuscript. References [1] Benenti S., Chanu C., Rastelli G., Variable separation for natural Hamiltonians with scalar and vector potentials on Riemannian manifolds, J. Math. Phys. 42 (2001), 2065–2091. 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Theor. 46 (2013), 295204, 15 pages, arXiv:1303.2345. https://doi.org/10.1088/1751-8121/aa9203 https://arxiv.org/abs/1706.08655 https://doi.org/10.1088/1751-8121/aadc23 https://arxiv.org/abs/1806.06849 https://doi.org/10.1063/1.4792478 https://doi.org/10.1063/1.4792478 https://arxiv.org/abs/1208.2995 https://doi.org/10.1103/PhysRevA.41.5666 https://doi.org/10.1063/1.2988133 https://arxiv.org/abs/0808.2146 https://doi.org/10.1088/1751-8121/aa7fa3 https://arxiv.org/abs/1701.09168 https://doi.org/10.1016/C2013-0-05576-4 http://dx.doi.org/10.1007/BF02755212 https://doi.org/10.1088/1751-8121/aa6f68 https://doi.org/10.3842/SIGMA.2018.092 https://arxiv.org/abs/1804.03039 https://doi.org/10.1088/1751-8113/48/39/395206 https://arxiv.org/abs/1507.04632 https://doi.org/10.1088/1751-8121/aaae9b https://doi.org/10.1063/1.3013804 https://arxiv.org/abs/0807.2858 https://doi.org/10.1063/1.3096708 https://arxiv.org/abs/0811.1568 https://doi.org/10.1088/1751-8121/aa7a67 https://arxiv.org/abs/1703.09751 https://doi.org/10.1088/1751-8121/ab01a2 https://arxiv.org/abs/1810.05793 https://doi.org/10.1088/1751-8113/46/42/423001 https://doi.org/10.1088/1751-8113/46/42/423001 https://arxiv.org/abs/1309.2694 https://doi.org/10.1088/1751-8113/48/40/405201 https://doi.org/10.1088/1751-8113/48/40/405201 https://arxiv.org/abs/1501.00471 https://doi.org/10.1007/s10569-004-1586-y https://doi.org/10.1007/s10569-004-1586-y https://doi.org/10.1063/1.1818721 https://arxiv.org/abs/nlin.SI/0405065 https://doi.org/10.1103/PhysRevE.78.046608 https://arxiv.org/abs/0807.1047 https://doi.org/10.1007/BF00910289 https://doi.org/10.1088/0305-4470/27/3/040 https://doi.org/10.1088/0305-4470/27/3/040 https://doi.org/10.1088/0305-4470/32/29/311 https://doi.org/10.1088/1751-8113/46/29/295204 https://arxiv.org/abs/1303.2345 1 Introduction 2 Minimal superintegrability for Case I when all the integrals commute with one linear momentum 2.1 Example: extension of 2D second-order superintegrable systems 2.2 Example: a family of higher-order superintegrable systems from the 2D caged oscillator 3 Maximal superintegrable class canonically conjugated to natural 2D systems 4 Second-order integrals 5 A necessary condition for second-order superintegrability 6 Quadratic superintegrability in Case I 7 Quadratic superintegrability in Case II 8 Constant magnetic field and second-order polynomial potentials 8.1 v12 and v22 both not vanishing 8.2 v22=0 and v12 not vanishing 8.3 v12=v22=0 9 Conclusions 9.1 Superintegrable systems with second-order integrals 9.2 Superintegrable systems with higher-order integrals A Solution of the determining equations in Case I for 12=21=0 A.1 33=0 A.2 33=0, 13=0, 23=0 A.3 33=23=0, 13=0 A.3.1 33=13=23=12=0 A.3.2 33=0, 13=0 A.3.3 33=0, 13=0, 23=0 A.3.4 33=13=23=0, 12=0 A.3.5 33=0 A.4 33=23=13=c1=0 A.4.1 33=0 A.4.2 33=0, 13=0 A.4.3 33=13=0, 23=0 A.4.4 33=13=23=0, 12=0 B Solution of the determining equations in Case I for 12, 21 not both vanishing B.1 12=0, 21=0 B.1.1 33=0 B.1.2 33=0 B.2 12=0, 21=0 B.2.1 23=0 B.2.2 23=0 C Solution of the determining equations for Case II C.1 33=0 C.2 33=0, 23=0 C.3 33=23=0, 31=0 C.3.1 12=13=0 C.3.2 13=0, 12=0 C.3.3 12=0 C.4 33=23=31=0 C.4.1 21=0 C.4.2 21=0, 12=0 C.4.3 21=12=0, 13=0 C.4.4 21=12=13=0 References
id nasplib_isofts_kiev_ua-123456789-210595
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-12-17T12:04:18Z
publishDate 2020
publisher Інститут математики НАН України
record_format dspace
spelling Marchesiello, Antonella
Šnobl, Libor
2025-12-12T10:35:47Z
2020
Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates. Antonella Marchesiello and Libor Šnobl. SIGMA 16 (2020), 015, 35 pages
1815-0659
2020 Mathematics Subject Classification: 37J35; 78A25
arXiv:1911.01180
https://nasplib.isofts.kiev.ua/handle/123456789/210595
https://doi.org/10.3842/SIGMA.2020.015
We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems that are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher-order integrals.
This paper was supported by the Czech Science Foundation (Grant Agency of the Czech Republic), project 17-11805S. This paper is dedicated to our son Flavio, born just after the submission of the original manuscript.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
Article
published earlier
spellingShingle Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
Marchesiello, Antonella
Šnobl, Libor
title Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
title_full Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
title_fullStr Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
title_full_unstemmed Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
title_short Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
title_sort classical superintegrable systems in a magnetic field that separate in cartesian coordinates
url https://nasplib.isofts.kiev.ua/handle/123456789/210595
work_keys_str_mv AT marchesielloantonella classicalsuperintegrablesystemsinamagneticfieldthatseparateincartesiancoordinates
AT snobllibor classicalsuperintegrablesystemsinamagneticfieldthatseparateincartesiancoordinates

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