Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

We construct integrable Hamiltonian systems with magnetic fields of the ellipsoidal, paraboloidal and conical type, i.e. systems that generalize natural Hamiltonians separating in the respective coordinate systems to include nonvanishing magnetic field. In the ellipsoidal and paraboloidal case each this classification results in three one–parameter families of systems, each involving one arbitrary function of a single variable and a parameter specifying the strength of the magnetic field of the given fully determined form. In the conical case the results are more involved, there are two one–parameter families like in the other cases and one class which is less restrictive and so far resists full classification.

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Introduction
In the last half of a century, mathematical physics had seen a growing activity in the study of integrable and superintegrable Hamiltonian systems, starting from the seminal papers by Smorodinsky and Winternitz et al [1,2].In most cases, these studies were aimed at the socalled natural Hamiltonians, i.e. on physical systems with conservative, position-dependent forces.For review of these results we refer the readers to the review paper [3] and a recent monograph [4].A few exceptions were papers constructing integrals of motion for particular configurations of the magnetic field, e.g.several monopole systems [5][6][7][8][9].
The first systematic attempt to study the systems with vector potentials seems to be [10], followed by several papers investigating integrable and superintegrable systems with magnetic fields in two spatial dimensions, see [11][12][13].In three spatial dimensions two of the present authors (AM and L Š) together with Winternitz started to address the problem in [14] and then followed this line of research with various collaborators in [15][16][17][18][19][20][21].In several of these papers we have seen that the one-to-one relation between integrability with integrals at most quadratic in the momenta and separability of the Hamilton-Jacobi equation in the configuration space (which was established for natural systems in [2] by direct comparison of the obtained systems with the results of Eisenhart on separability [22,23]) no longer holds in the presence of magnetic fields.This can be seen from a direct comparison with results of [24] or from the results of [25] where it has been shown that in the presence of nonvanishing magnetic field the separability requires existence of at least one first order integral in the momenta.
In addition in [15] we have seen that even the leading order terms in the integrals may be more general in the presence of magnetic fields than those of [2].Thus, in [26] Marchesiello and Šnobl have classified possibilities for the leading order terms allowed algebraically, organizing them into 11 classes; some of those generalize the structure of leading order terms of [2] by inclusion of additional terms, some are completely new and unrelated to any orthogonal coordinate system (the so called nonstandard cases).Afterwards, we have searched for new systems in the respective classes.In [27] we have classified systems with integrals generalizing the cylindrical and spherical case, which together with the analysis of the corresponding standard cases in [16,18] completely exhaust these two physically very important cases.In [28] we constructed all integrable systems of one nonstandard case of [26] (case (i) therein), leading to one family of solutions depending on 5 parameters.In [29] we considered further four classes with generalized structure of the integrals, showing that for the prolate / oblate spheroidal, circular parabolic and nonstandard class (g) of [26] the inclusion of additional terms does not lead to any new system in addition to the standard structures of [17], and fully classified systems with generalized elliptic cylindrical integrals.Curiously, the standard elliptic cylindrical case (as well as parabolic cylindrical one) remain unclassified.This is closely related to the fact that also in two dimensions elliptic and parabolic type integrable systems resist complete classification, despite the attempt in [13].Integrable systems with the Cartesian type integrals were also classified [30], however of their generalized form, class (k) of [26], only particular examples are so far known, see [15].
In this paper we address the systems of ellipsoidal, paraboloidal and conical types.As it was shown in [26], the corresponding leading order terms in the integrals are of the same form as in the absence of the magnetic field, see [2], no extra terms in the leading order structure of the integrals are possible.In addition, from the results of [25] it follows that the arising integrable systems cannot be separable in these coordinates, as none of the integrals can reduce to a first order one.
The structure of the paper is the following: first, in section 2 we introduce the setting and necessary notation.Next, in sections 3-5 we construct the integrable systems of the ellipsoidal, paraboloidal and conical type, respectively.We conclude our paper with a summary and conclusions in section 6.

The Hamiltonian systems and integrals of motion
Let us consider a charged particle moving in an external static electromagnetic field in three dimensional Euclidean space as a classical nonrelativistic Hamiltonian system where is the position-dependent vector potential and W(⃗ x) is the electrostatic potential.The physical dynamics of such a system is invariant under time-independent gauge transformations thus the vector potential is not a physically observable quantity and the physically relevant magnetic field ⃗ B(⃗ x) can be derived from the curl of the vector potential ⃗ A(⃗ x) through the relation where ε jkl is the completely antisymmetric tensor with ϵ 123 = 1.The scalar potential of the Hamiltonian system, i.e. the momentum-free term in (2.1), reads and like the vector potential is affected by the gauge transformation (2.2).We focus on the problem of integrability of the system (2.1) with integrals of motion X 1 and X 2 , which both are at most quadratic polynomials in the momenta.The leading order terms of any quadratic integral are polynomials of second order in the enveloping algebra of the Euclidean algebra e(3) with the basis p 1 , p 2 , p 3 , l 1 , l 2 , l 3 , where l k = ∑ m,n ε kmn x m p n are the components of the angular momentum, k = 1, 2, 3. Thus the considered quadratic integrals can be expressed in terms of covariantized momenta where and must satisfy where { , } P.B. denotes the Poisson bracket.The functions s xj a (⃗ x), m a (⃗ x) and the numerical coefficients α a mn ∈ R have to be determined.(Notice that x j in the upper index of s xj a (⃗ x) is the spatial coordinate canonically conjugated to the momentum p j .Similarly below we shall have function s u a (s, t, u) multiplying the canonical momentum p u in the ellipsoidal coordinates etc.) Two of the authors of the present paper have classified in [26] the leading order structure of hypothetical quadratic integrals, up to Euclidean transformations, into 11 classes.Some of these classes are more general compared to the case without magnetic field, see [2]; however, in this paper we shall focus on the classes (b) and (e) of [26] which correspond directly to the ellipsoidal, paraboloidal and conical separable cases in the absence of magnetic field, as constructed in [2].In order to facilitate comparison, we shall use the structure of the leading order terms of the commuting integrals as in [2]; for their relation to the forms presented in the classification theorem of [26] we refer the readers to [26].Thus we will consider the following three possibilities for the commuting integrals: ) (2.13)

The ellipsoidal case
In order to find all integrable systems with integrals of motion of the form (2.8) and (2.9) we start by expressing the integrals of motion in the ellipsoidal coordinates.The standard relation between the Cartesian and ellipsoidal coordinates is where s > b 2 > t > a 2 > u > 0. In the new coordinates (s, t, u) we express the canonical 1−form λ = p 1 dx + p 2 dy + p 3 dz as p s ds + p t dt + p u du and find the relation ) , ) , In the same way transform also the components of the vector potential 1-form as well as the covariantized momenta p A k .The transformation of the vector potential also implies how the components of the magnetic field in these coordinate systems are related through its exterior derivative B = dA, namely where B s , B t and B u are functions of the coordinates s, t, u.To write down the Hamiltonian system in the ellipsoidal coordinates, we also need the metric computed via the pullback of the Cartesian metric.It reads and the corresponding inverse metric is Thus the Hamiltonian system (2.1) expressed in the ellipsoidal coordinates reads Using the transformations introduced above, the integrals X 1 and X 2 of (2.8) and (2.9) can be written as (3.9)

The determining equations and the resulting integrable systems in the ellipsoidal case
In order to classify all new integrable systems allowing integrals of motion of the form (3.8) and (3.9), we have to solve the involutivity conditions (2.7) of the Hamiltonian and the integrals of motion, that is, We first write the leading order, i.e. quadratic, determining equations coming from {H, X 1 } P.B. = 0: ) , (where for the sake of brevity we suppress the dependence of the functions s α 1,2 (s, t, u) on the coordinates s, t, u), and from {H, X 2 } = 0: ) , ) , ) .(3.12) We solve for the components of the magnetic field from the equations (3.12), finding Plugging (3.13) into the remaining equations (3.12), we express three of the derivatives of s j 2 (s, t, u), j = s, t, u: ) , ) .(3.14) Inserting (3.11), (3.13) and (3.14) into the commutativity conditions {X 1 , X 2 } P.B. = 0, the leading order terms imply That is, we have found three algebraic relations and three differential equations.It turns out that it is more convenient to solve the differential equations in (3.15) first: introducing arbitrary functions s t 1 (t, u), s u 22 (t, u), s u 21 (s, u), s u 22 (t, u) and s u 1 (u) of the indicated variables.The solution of the three algebraic constraints in (3.15) now reads involving the remaining so far arbitrary functions s u 1 (u), s u 22 (t, u) and s u 21 (s, u).Substituting (3.17) into (3.16)we see that the equations (3.15) are in full generality solved by The complete solution of (3.20) via compatibility conditions on the PDEs reads ) where s u 220 is an integration constant and s u 21 (u) is an arbitrary function of u.Using all information about the functions s s,t,u 1,2 (s, t, u) obtained so far, the equations (3.14) imply one ODE, namely which is easily solved by together with a compatible pair of PDEs for ∂s t 2 ∂s (s, t, u) and involving an arbitrary function s t 2 (u).Plugging (3.24) into the last remaining leading order equation (3.19) from {H, X 1 } P.B. = 0, we find that s t 2 (u) must be a constant times u, that is, Thus the general solution of all the leading order equations reads depending on three integration constants s t 20 , s u 220 , s u 210 .We now proceed with lower order terms in the involutivity conditions (3.10).First from the commutativity {X 1 , X 2 } P.B. = 0 at the first order level, we have conditions which depend only on the functions s α 1 (s, t, u) and s α 2 (s, t, u), α = s, t, u.Plugging (3.25) into (3.26)we arrive at complicated expressions that must vanish for all values of s, t, u and do not involve any arbitrary functions.Isolating functionally independent terms in these and solving for vanishing coefficients, we find a set of algebraic equations whose solutions imply that two of the three integration constants in (3.25) must vanish and lead to three very similar, however due to different ranges on the coordinates s, t, u not equivalent, solutions: We proceed here in detail with one of them as the other two are very similar and just summarized at the end.Plugging (i) into the solution set (3.25), we find The zeroth order term in {X 1 , X 2 } P.B. = 0 using (3.27) reads which implies the potential of the form The other two zeroth order equations, i.e. coming from {H, X 1 } P.B. = 0 and {H, X 2 } P.B. = 0 are equivalent to (3.28).
It remains to solve the first order conditions coming from {H, X 1 } P.B. = 0 and {H, X 2 } P.B. = 0.These can be solved with respect to the first order derivatives of m 1 (s, t, u) and m 2 (s, t, u) and in turn determine compatibility conditions for them, i.e. interchangeability of the mixed derivatives, The two sets of compatibility conditions coming from the conditions on m 1 (s, t, u) and m 2 (s, t, u) are actually equivalent and read Finally we compute m 1 (s, t, u) and m 2 (s, t, u) from the first order determining equations coming from {H, X 1 } P.B. = 0 and {H, X 2 } P.B. = 0, and thus we find the complete solution for the first subcase.The resulting integrable system has the magnetic field and the potential (3.32).Its integrals are determined by the equations (3.27) and , )) . (3.34) In a completely analogous fashion the other two subcases lead to two new integrable systems characterized by (ii) the magnetic field and the potential (3.36) its integrals are determined by equations (3.25) with s t 20 = s u 210 = 0 and ) , (3.37) and (iii) the magnetic field and the potential (3.39) in this case the functions in the integrals are expressed as in equation (3.25) with s t 20 = s u 220 = 0 and (3.40)

The paraboloidal case
Let us start by expressing the integrals (2.10) and (2.11) in the paraboloidal coordinates.The standard relation between the Cartesian and paraboloidal coordinates (µ, ν, λ) is where µ > b > λ > a > ν > 0. The components of linear momentum transform as components of 1−form, namely 2) The components of the vector potential and covariantized momenta transform in the same way.The Cartesian and paraboloidal components of the magnetic field are related through where B µ , B λ and B ν are functions of the coordinates µ, λ, ν.The metric in the paraboloidal coordinates is given by (4.4) Thus the Hamiltonian system (2.1) is expressed as where p A α = pα + Aα(µ, ν, λ), α = µ, ν, λ.The integrals (2.10) and (2.11) in the paraboloidal coordinates become ) (4.7)

The determining equations and the resulting integrable systems in paraboloidal case
In order to classify integrable systems with the integrals of the form (4.6) and (4.7), we have to solve the involutivity conditions (2.7).Since they must be satisfied for all values of the momenta, they similarly as above imply PDEs for the functions s α 1,2 (µ, ν, λ) and m 1,2 (µ, ν, λ) as well as for the magnetic field and the electrostatic potential as the coefficients of second, first and zeroth order monomials in the momenta in (2.7).
Let us first consider the second order (i.e.leading order) conditions coming from {H, X 1 } P.B. = 0: and from {H, X 2 } = 0: ) , ) . (4.9) We first solve for the components of the magnetic field from the equations (4.8), finding  (4.11) i.e. again as in the ellipsoidal case three algebraic equations and three differential equations.We first solve the differential equations, then we plug them into the algebraic equations and thus we find a general solution of (4.11) in the form involving arbitrary functions s ν 2 (µ, ν) and s ν 1 (ν, λ) of their respective variables.Now plugging (4.12) into (4.8) and (4.9) and solving them, we arrive at the complete solution of all the leading order equations, involving three integration constants s ν 11 , s ν 21 , s λ 21 .We now proceed with lower order terms in the involutivity conditions (3.10) in the paraboloidal case.First we solve the commutativity of X 1 and X 2 , i.e. {X 1 , X 2 } P.B. = 0 at the first order level assuming (4.10) and (4.13).We find again that they imply three possibilities for the integration constants in (4.13): and as above we proceed with one of them as the others are very similar and just summarized at the end.Plugging (i) into the solution set (4.13), we find and the magnetic field (4.10) becomes The zeroth order term in {X 1 , X 2 } P.B. = 0 by using (4.14) reads which implies the potential of the form The other two zeroth order equations, i.e. coming from {H, X 1 } P.B. = 0 and {H, X 2 } P.B. = 0, are again equivalent to (4.17).We look for m 1 (µ, ν, λ) and m 2 (µ, ν, λ) from the first order determining equations coming from {H, X 1 } P.B. = 0, namely and from {H, X 2 } P.B. = 0: We first compute the compatibility conditions for m 1 and m 2 in (4.18) and (4.19), and plugging the potential of the form into them and solving, we find where w 0 (ν) is an arbitrary function of ν.Next, by using (4.18) and (4.19) we find the corresponding functions m 1 and m 2 .Thus, solving all the determining equations we arrive in the case (i) to the integrable system with the magnetic field (4.15) and the potential (4.20).Its integrals (4.6)-(4.6)are determined by (4.14) and Similarly as in the ellipsoidal case, the other two choices (ii) and (iii) of the integration constants lead to other two similar, however due to inequalities between the coordinates non-equivalent, solutions: (ii) the system characterized by the magnetic field and the potential where s ν 21 is a nonvanishing constant and w 0 (λ) is an arbitrary function of λ.Its commuting integrals (4.6) and (4.7) are determined by equation (4.13) with s ν 11 = s λ 21 = 0 and (iii) the system characterized by the magnetic field and the potential where s ν 11 is a nonvanishing constant and w 0 (µ) is an arbitrary function of µ.Its commuting integrals (4.6) and (4.7) are determined by equation (4.13) with s ν 21 = s λ 21 = 0 and ) . (4.27)

The conical case
Let us now focus on the last class of integrable systems that we shall study in this paper, namely the conical case, corresponding to the separation in the conical coordinates in the absence of the magnetic field.As above, we start by expressing the integrals of motion (2.12) and (2.13) in the conical coordinates.We use the same relation between the Cartesian and conical coordinates (r, θ, λ) as in [2], namely where Similarly as in the previous cases, the transformation of the coordinates (5.1) implies the transformation of the components of the canonical momenta and from {H, X 2 } = 0: ) . (5.9) Initially, the computation closely resembles the ones in the ellipsoidal and paraboloidal cases.We solve for the components of the magnetic field from the equations (5.9), finding (5.10) and out of (5.9) we have two equations left determining the derivatives together with the independence of s r 2 (r, θ, λ) on r, i.e.
Inserting (5.8), (5.10) and (5.11) into the commutativity conditions {X 1 , X 2 } P.B. = 0, the leading order terms imply five conditions, namely two algebraic relations (instead of three as in the previous cases) and three differential equations (5.12) We also have the unresolved conditions (5.15) from the leading order terms.The complete solution of this system of PDEs is not known.Using the transformation (5.29) and integrals (5.6) and (5.7) with where w 0 (r) is an arbitrary function of r and ω is an arbitrary real parameter.
However, we remind the readers that these systems in case (iii) are only particular solutions of the determining equations and other integrable systems based on other solutions of the equation (5.25) may and probably do exist, e.g. containing some special functions.

Conclusions
We have presented the results of our analysis of integrable systems in magnetic fields which possess quadratic integrals of motion of the ellipsoidal, paraboloidal and conical types.Let us briefly summarize our results and compare them to the respective systems without magnetic field, as classified in [2].

Ellipsoidal case
There exist three classes of integrable systems with the integrals of the form (2.8) and (2.9), namely 1. the system (3.32) and (3.33), i.e.
Comparing the potentials of these systems to the potential of the ellipsoidal type system without a magnetic field, namely (see [2]) we see that each of our classes contains precisely one arbitrary function of the nonmagnetic potential, and one arbitrary real parameter that determines the strength of the magnetic field.In each case, the variable on which the free function in the potential W(s, t, u) depends fixes the structure of the magnetic field up to a single parameter β which determines its strength.The systems look very similar to each other and can be formally obtained one from the other by permutation of the ellipsoidal coordinates, which is however not allowed by the ranges imposed on them by definition (3.1).

Paraboloidal case
The results are structurally similar to the ellipsoidal case.Again, there exist three classes of integrable systems with the integrals of the form (2.10) and (2.11), namely 1. the system (4.15) and (4.20), i.e.
and one subcase that resists full classification but we were able to obtain some particular solutions.This situation is reminiscent of the elliptic and parabolic type integrable systems in two dimensions and the corresponding three dimensional elliptic and parabolic cylindrical cases for which full classification is also not known, see [13,29].
Thus, the results in the conical case have somewhat different structure.Comparing the resulting potentials with the non-magnetic potential W(r, θ, λ) = f(r) + g(θ) + h(λ) r 2 (θ 2 − λ 2 ) , (6.12) see [2], we see that also without the magnetic field the radial coordinate r plays a role different from the remaining two.The two explicit solutions have a structure similar to the ellipsoidal and paraboloidal cases, i.e. involve one arbitrary function of a single variable (θ or λ, respectively) in the potential and one arbitrary real parameter determining the strength of the magnetic field.The other subclass was shown to involve systems with an arbitrary function of r and in case (5.29) also an arbitrary real parameter in the potential W(r, θ, λ), one parameter determining the strength of the magnetic field, whose form however is different in the two examples found, and in the case (5.27) also an additional parameter present in the integrals only.This parameter indicates the existence of an additional independent linear integral (the coefficient of the parameter κ 2 ) of the form is (at least) minimally superintegrable and must lie at the intersection of the spherical and conical type systems, see [16].
Let us observe that in none of the considered three cases the leading order structure of the commuting integrals allows to reduce any combination of them to a first order integral.Therefore, by results of [25] the arising systems are not separable in the respective (e.g.ellipsoidal) orthogonal coordinate system.This matches with the fact that in the list of quantum systems with magnetic fields separating in orthogonal coordinates on the three-dimensional Euclidean space, see [24], there is no system separating in the ellipsoidal, paraboloidal or conical coordinates.As a further consistency check, we searched for a hypothetical additional first order integral for each of the constructed integrable systems in the ellipsoidal and paraboloidal cases and for the conical type systems (5.17) and (5.19) for generic values of the parameters a, b (or b, c, respectively) and did not find any.In this, we had to rely on the computer algebra system Maple to solve complicated overdetermined systems of homogeneous linear equations for the parameters multiplying the leading order terms, with coefficients depending on the nonvanishing strength of the magnetic field and the parameters a, b (b, c, respectively).As long as we can trust this computer result, none of these systems is for generic values of the parameters a, b separable in the sense of additive separation of variables in the Hamilton-Jacobi equation in any orthogonal coordinate system.
One of the obvious, however open, questions is how the integrable systems presented here look like in the usual, Cartesian coordinates.The difficulty in answering this question stems from the fact that inverse transformation, expressing the ellipsoidal, paraboloidal and conical coordinates (3.1), (4.1) and (5.1) in terms of the Cartesian ones, is not known in any closed form.Thus, the arbitrary functions in the potential cannot be expressed in the Cartesian coordinates.We tried to find at least the Cartesian components of the magnetic fields of the obtained systems, see (3.4), (4.3) and (5.3), however, we did not succeed in expressing them as functions of the Cartesian coordinates despite the fact that the magnetic fields are explicitly given algebraic functions of the ellipsoidal, paraboloidal and conical coordinates, respectively.
The above mentioned difficulty in expressing the resulting systems in the Cartesian coordinates also implies that search for hypothetical superintegrability of the constructed integrable systems is a very complicated task, despite the fact that in the conical case we have found a superintegrable example (5.27).Expressing the hypothetical additional integral X 3 of the general form (2.5) in the respective curvilinear coordinates and computing the Poisson bracket {H, X 3 }, one arrives at very cumbersome conditions which seem to be rather impossible to analyze.At the same time, it is impossible to express these conditions in the Cartesian coordinates since the magnetic field and the potential are not known in those.However, a definite verdict on this question is still pending and we are looking for some novel way of addressing it.