Classical Particle in Presence of Magnetic Field, Hyperbolic Lobachevsky and Spherical Riemann Models

Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases three integrals of motions are constructed and equations of motion are solved exactly in the special cylindrical coordinates on the base of the method of separation of variables. In Lobachevsky space there exist trajectories of two types, finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions, based on the geometric symmetry group, SO(3,1) and SO(4) respectively.


Introduction
In the paper exact solutions for classical problem of a particle in a magnetic field on the background of hyperbolic Lobachevsky H 3 and spherical Riemann S 3 space models will be constructed explicitly.
The grounds to examine these problems are as follows: these both are extensions for a wellknown problem in theoretical physics -a particle in a uniform magnetic field [15]; they can be used to describe behavior of charged particles in macroscopic magnetic field in the context of astrophysics. The form of the magnetic field in the models H 3 and S 3 was introduced earlier in [2,3,4] where the quantum-mechanical variant (for the Shrödinger equation) of the problem had been solved as well and generalized formulas for Landau levels [14,15,16] had been produced. A part of results of the paper was presented in the talk given in [13].
Previously, the main attention was given to Landau problem in 2-dimensional case: many important mathematical and physical results were obtained, see in [1,5,6,8,9,12,17,19]. Comprehensive discussion of the general problem of integrability of classical and quantum systems in Lobachevsky and Riemann 3D and 2A models see in [7,10,11] and references therein. It is known that 2D and 3D systems exhibit properties which are radically different. Our treatment will concern only a 3-dimensional case.
2 Newton second law in Lobachevsky space Motion of a classical particle in external electromagnetic and gravitational fields is described by the known equation [15] mc 2 d 2 x α ds 2 + Γ α βσ dx β ds dx σ ds = eF αρ U ρ , (2.1) where Christoffel symbols are determined by metrical structure of a space-time (the signature + − − − is used). In (2.1) it is useful to perform (3 + 1)-splitting In (2.2)-(2.5) usual SI units are used, so the Christoffel symbols Γ i jk are measured in (meter) −1 . With conventional notation [15] equations (2.2)-(2.5) give d dt Firstly, we will be interested in non-relativistic case 1 , when all equations become simpler d dt 3 Particle in a uniform magnetic f ield, hyperbolic model H 3 Let us start with the known 4-vector potential of a uniform magnetic field in flat space [15] A = 1 2 cB × r, B = (0, 0, B), A a c = B 2 (0; −r sin φ, r cos φ, 0). (3.1) From (3.1), after transformation to cylindric coordinates we obtain The only non-vanishing constituent of the electromagnetic tensor reads which satisfies the Maxwell equations Now we are to extend the concept of a uniform magnetic field to the Lobachevsky model H 3 . Thirty four orthogonal coordinate systems in this space were found by Olevsky [18]. An idea is to select among them some curved analogue for cylindric coordinates and determine with their help an appropriate solution to Maxwell equations in Lobachevsky space. In [18], under the number XI we see the following coordinates dS 2 = c 2 dt 2 − ρ 2 cosh 2 z dr 2 + sinh 2 rdφ 2 + dz 2 , z ∈ (−∞, +∞), r ∈ [0, +∞), φ ∈ [0, 2π], u 1 = cosh z sinh r cos φ, u 2 = cosh z sinh r sin φ, u 3 = sinh z, u 0 = cosh z cosh r, the curvature radius ρ is taken as a unit length. In the limit ρ → ∞ the coordinates (3.2) reduce to ordinary cylindric ones in the flat space. By definition, the uniform magnetic field in the Lobachevsky space is given by 4-potential of the form It behaves properly in the limit ρ −→ ∞, besides it corresponds to an electromagnetic tensor which evidently satisfies the Maxwell equations in H 3 In the absence of an electric field, the non-relativistic equations (2.7)-(2.9) read where the Christoffel symbols are It makes sense to recall a similar problem in flat space, here the Christoffel symbols are simpler In fact, a simplest solution of these equations is well-known: the particle moves on a cylindric surface oriented along the axis z according to the law φ(t) = ωt + φ 0 , correspondingly equations (3.4) take the form and the simplest solution is given by Now, let us turn to the problem in Lobachevsky model -equations (3.3) give It should be stressed that in (3.5)

Particular solutions in Lobachevsky model
Let us construct simple solutions when imposing the following constrain r = r 0 = const, equations (3.5) give The second equation is an identity 0 ≡ 0. With the notation two remaining equations in (4.1) read When B > 0, the angular velocity dφ/dt < 0; and when B < 0, the angular velocity dφ/dt > 0. Second equation in (4.2) means that there exists effective repulsion to both sides from the center z = 0. One can resolve the second equation Below we will see that the constant ǫ can be related to a squared velocity V 2 /c 2 or differently to the integral of motion -the energy of the non-relativistic particle 3 Meaning of the signs ± is evident: it corresponds to a motion along axis z in opposite directions. Further we get For motions of the type I, trajectories run through z = 0; for motions of the type II, the particle is repulsed from the center z = 0 at the points sinh z 0 = ± A/ǫ − 1. Existence of 3 Relationship (4.3) in the limit of flat space will read (dz/dt) 2 = ǫ − A which means that A corresponds to a transversal squared velocity V 2 ⊥ /c 2 . Also, equation (4.3) points out that in Lobachevsky model the transversal motion should vanish (to be frozen) when z → ±∞.
these two different regimes of motion along the axis z correlates with the mentioned effective repulsion along the axis z. Now let us consider the case ǫ = A dz dt We immediately see a trivial solution it corresponds to rotation of the particle along the circle r = r 0 in the absence of any motion along the axis z. Also there are non-trivial solutions to equation (4.4) here we have two different ones depending on sign (+) or (−). Continuous solutions of (4.5) exist only for z > 0 and z < 0 with different and peculiar properties. In the case of sign (+) we have at any positive initial z 0 > 0 the particle goes to +∞; and at any negative initial z 0 < 0 the particle goes to −∞. In the case of sign (−), we get quite different behavior at any positive initial z 0 > 0 the particle moves to z = 0 during infinite time t; at any negative initial z 0 < 0 the particle moves to z = 0 during infinite time t. Now we are to turn to the first equation in (4.2) and find φ(t) One may note that when t → +∞ we obtain a finite value for the rotation angle In the same manner one should find φ(t) in case (4.6) Again a peculiarity in the limit t → +∞ may be noted And the function φ(t) in case (4.7) Before going farther, let us note that from equation (2.6) it follows the conservation of squared velocity (or the energy) in the presence of a magnetic field ǫ = cosh 2 z dr dt 2 + sinh 2 r dφ dt For trajectories with constant r = r 0 , the energy looks simpler (which coincides with (4.3)) The above elementary treatment seems not to be completely satisfactory because we cannot be sure that all possible motions of the particle in a magnetic field in the Lobachevsky space have been found. So we turn to the Lagrange formalism.

Particle in a magnetic f ield and Lagrange formalism in H 3
Now, let us consider the problem using the Lagrange formalism [15] Euler-Lagrange equations read which coincide with equations (3.5). Second equation in (5.1) evidently determines a new conserved quantity We will obtain more from Lagrange formalism, if we use three integrals of motion. Two of them are already known Having remembered equation (4.3), it is easy to guess the third To see that A indeed conserves it suffices to rewrite A as A = cosh 2 zV r 2 + 1 sinh 2 r cosh 2 z sinh 2 rV φ 2 , and takes into account the first and second equations in (5.1) then we arrive at dA/dt = 0. With the help of three integrals of motion we can reduce the problem in its most general form (without any additional and simplifying assumptions) to calculating several integrals. Indeed, from (5.2) it follows Substituting it into (5.3) we get In turn, from (5.3) it follows Dividing (5.5) by (5.6), we obtain sinh rdr which represents trajectory equation in the form dF (r, z) = 0. In turn, dividing (5.5) by (5.4), we get trajectory equation in the form dF (r, φ) = 0 Thus, the solution of the problem -particle in a magnetic field on the background of Lobachevsky space -reduces to the following integrals The last five relations are valid for all possible solutions of the problem under consideration. In particular, let us show that the restriction r = r 0 = const is compatible with equations (5.7)-(5.11). Indeed they give It should be noted that in Section 3 we had other representations for α and A However they are equivalent. Indeed, equating two expressions for α 14) and substituting this into A in (5.12) we get which coincides with (5.13). It should be specially noted that from (5.14) we must conclude that to the case of the most simple motion r 0 = const, there corresponds the inequality From (5.15) we conclude that the motion with r = r 0 is possible if

Possible solutions in H 3 , radial f inite and inf inite motions
Now, with the use of general relationships (5.7)-(5.11), let us examine the general case without restriction r = r 0 . Let us A = ǫ. First, as shown above equation (5.9) results in

Equation (5.8) can be rewritten as
.
Integral in the right-hand side is known -see (4.8) For the integral in the left-hand side we get It should be noted the identity Depending on the values of a, b, c there may be realized solutions of different types.

Finite radial motions
Let it be a < 0, B 2 − A > 0. The roots x 1 and x 2 are and inequality b 2 − 4ac > 0 reduces to the following restriction ( In general, at a < 0 we might expect several possibilities   Physically interesting cases, Figs. 1a-1c, can be characterized additionally by constrains on a, b, c Therefore, only Figs. 1a and 1b correspond to the physically possible solutions (with two turning points in radial variable) Fig. 1a, Now let us turn to the integral (6.2) when (6.3) holds Therefore, equation (6.1) gives (let ǫ > A) As expected, the variable cosh r runs within a finite segment or (see Fig. 1a) which at I = 0 takes a more simple form (see Fig. 1b) It should be noted that when according to (6.4) the motion within the segment [x 1 , x 2 ] reduces to the motion with a fixed value r 0 which coincides with the expression for cosh r 0 given by (5.14).

Infinite radial motions (a > 0)
One special case arises when a = A − B 2 = 0, indeed then we have which corresponds to an infinite motion in radial variable (let ǫ > A) Now let us examine other possibilities related to inequality a > 0. When a = A − B 2 > 0, the roots x 1 , x 2 are defined by and inequality b 2 − 4ac > 0 reduces to In general, at the restrictions a > 0 we might expect the following cases As concerns Fig. 2e, we should examine the following inequality Therefore, only Figs. 2b and 2c correspond to physically possible solutions (motions infinite in radial variable r). Integrating (6.2) reduces to the elementary calculation Therefore, equation (6.1) gives (with ǫ > A) Evidently, equation (6.7) leads to (see Fig. 2c) from whence at I = 0 it follows (see Fig. 2b) cosh r > +1, A > 0. sinh rdr Its right-hand side gives The left-hand side gives (two different possibilities depending on (B 2 , A) relation) Therefore, the trajectories F (r, z) (5.10) have the form (four different cases)

shifts in Lobachevsky space
Now, let us consider the trajectory equation F (r, φ) (5.11) With the help of a new variable, the integral in the left-hand side reads In connection to this integral we must require [(A − B 2 ) + (I + B) 2 ] − u 2 > 0, which can be transformed to the form or (see Section 6) x = cosh r, ax 2 + bx + c > 0; therefore, all analysis given in Section 6 is valid here as well. In particular, we should remember about necessary condition Integral ( One may assume that equation (8.3) when B 2 > A will describe a circle of fixed radius with shifted center in Lobachevsky space. As we see below this assumption is true. Indeed, let us introduce two coordinate systems in Lobachevsky space H 3 u 1 = cosh z sinh r cos φ, u 2 = cosh z sinh r sin φ, u 3 = sinh z, u 0 = cosh z cosh r; In particular, β-shifted circle with fixed value r ′ = r ′ 0 will be described in coordinates (r, φ) by the following equation cosh β cosh r + sinh β sinh r cos φ = cosh r ′ 0 .
Relations (8.6) and (8.7) coincide if the parameter β and the radius of the shifted trajectory r ′ 0 are defined according to Correspondingly, the main invariant reads Depending on the sign of this invariant one can reach the most simple description by means of appropriate shifts 1) B 2 − A > 0 (finite motion) One special case exists, see (6.5), (6.6). By symmetry reasons, Lorentzian shifts of the type (0-2) will manifest themselves in the same manner.
9 Lorentzian shifts and symmetry of a magnetic f ield in H 3 Let us turn again to a pair of coordinate systems in space H 3 u 1 = cosh z sinh r cos φ, u 2 = cosh z sinh r sin φ, related by the shift (0-1) , or in cylindric coordinates (direct and inverse formulas) sinh r ′ cos φ ′ = sinh β cosh r + cosh β sinh r cos φ, cosh r ′ = cosh β cosh r + sinh β sinh r cos φ; With respect to that coordinate change (r, φ) =⇒ (r ′ , φ ′ ), the uniform magnetic field transforms according to so that the magnetic field transforms by means of the Jacobian It is convenient to represent the coordinate transformation in the form φ = arctan sinh r ′ sin φ ′ − sinh β cosh r ′ + cosh β sinh r ′ cos φ ′ = arctan A, r = arccosh cosh β cosh r ′ − sinh β sinh r ′ cos φ ′ = arccosh B; correspondingly the Jacobian looks Taking into account the identity and the formulas for the Jacobian we get It is matter of simple calculation to verify that the denominator and numerator (cosh β sinh r ′ − sinh β cosh r ′ cos φ ′ )(− sinh β cosh r ′ cos φ ′ + cosh β sinh r ′ ) + sinh 2 β sin 2 φ ′ = −2 cosh β sinh β cosh r ′ sinh r ′ cos φ ′ + cosh 2 β sinh 2 r ′ + sinh 2 β cosh 2 r ′ cos 2 φ ′ + sinh 2 β sin 2 φ ′ are equal to each other. Thus, the Jacobian of the shift (0-1) in hyperbolic space is J = sinh r ′ sinh r and therefore this shift leaves invariant the uniform magnetic field under consideration By symmetry reason, we can conclude the same result for the shifts of the type (0-2). However, in that sense shifts of the type (0-3) behave differently. Indeed, (9.1) Equation (9.1) leads to relation between (r, z) and (r ′ , z ′ ) cosh z ′ cosh r ′ = cosh β cosh z cosh r + sinh β sinh z, Electromagnetic field is transformed according to so the uniform magnetic field in the space H 3 is not invariant with respect to the shifts (0-3).
One may describe electromagnetic field in terms of 4-potential, and the rule to transform the field with respect to the shift (0-1) looks In flat space, the shift (0-1) generates a definite gauge transformation By analogy reason, one could expect something similar in the case of Lobachevsky space as well It is indeed so. Let us demonstrate this. Accounting for two formulas we conclude that the gauge function Λ in (9.2) is defined by its partial derivatives in accordance with The integrability condition ∂ 2 Λ/∂φ ′ ∂r ′ = ∂ 2 Λ/∂r ′ ∂φ ′ in an explicit form looks It is the matter of simple direct calculation to verify it. Now we are to find an explicit form of the gauge function Λ. For better understanding, it is helpful first to consider a similar problem in flat space Integrating the first equation we obtain and further from the second equation we derive Therefore, the gauge function is The similar problem in space H 3 should be considered by the same scheme. Let us integrate the first equation in (9.3) Introducing the notation cosh β = c, sinh β = s, c 2 − s 2 = 1, and new variable tanh r ′ 2 = y, dr ′ = 2 cosh 2 r ′ dy, Now we are to calculate After substituting into the second equation in (9.3) the expression for Λ gives Therefore, λ(φ) = −2Bφ + λ 0 and the gauge function Λ(r ′ , φ ′ ) is found as remembering that c = cosh β, s = sinh β.
10 Particle in a magnetic f ield, spherical Riemann model S 3 In [18] under number XI we see the following system of coordinates in spherical space S 3 u 1 = cos z sin r cos φ, u 2 = cos z sin r sin φ, u 3 = sin z, u 0 = cos z cos r, u 0 2 + u 1 2 + u 2 2 + u 3 2 = 1.
In these coordinates, let us introduce a magnetic field which satisfies the Maxwell equations in S 3 1 cos 2 z sin r ∂ ∂r cos 2 z sin r 1 cos 4 z sin 2 r B sin r ≡ 0.
The Christoffel symbols read The non-relativistic equations of motion (2.9) in coordinates (r, φ, z) in a magnetic field look The last equation in (10.1) points that along the axis z the effective attractive force acts to the center z = 0.

Simplest solutions in spherical space
Let us search for solutions with a fixed radius r = r 0 ; equations (10.1) give where the second equation is an identity 0 ≡ 0. With notation α = −B/ cos r 0 , A = B 2 tan 2 r 0 , the problem reduces to The second equation gives the constant ǫ will be related to the squared velocity. First, let A = ǫ, then equation (11.2) gives (in contrast to Lobachevsky model, now only one possibility is realized: Two different signs (±) correspond to the motion of the particle in opposite directions along the axis z. The calculations below are evident (let t 0 = 0) Let us examine a special case when ǫ = A, equation ( it corresponds to the rotation with constant angular velocity around the circle r = r 0 in the absence of any motion along the axis z. Now we are to turn to the first equation in (11.1) and find φ(t) Thus, we construct the following solution Distinctive feature of the motion is its periodicity and closeness of corresponding trajectories. The period T is determined by the energy ǫ 12 Particle in a magnetic f ield and Lagrange formalism in S 3 Let us consider the problem of a particle in a magnetic field in S 3 on the base of the Lagrange function Euler-Lagrange equations in the explicit form are which coincide with equations (10.1). Two integrals of motion in S 3 are known the third one can be constructed as follows With the help of tree integrals of motion one can readily transform the problem under consideration to calculating several integrals dφ dt = 1 cos 2 z I + B(cos r − 1) sin 2 r , (12.1) sin rdr [I + B(cos r − 1)]dr sin r A sin 2 r − [I + B(cos r − 1)] 2 = dφ. (12.5) For the most simple case when r = r 0 = const, these relations give In Section 11 for α and A we had other expressions which were equivalent to the present ones. Indeed, from the identity α = α we get substituting it into (12.6) we arrive at which coincides with (12.7).
The integral in the right-hand side is known, see (11.4), We may expect physical solutions when Also, we require From the first and second inequalities, it follows respectively: Therefore, (13.3) will be satisfied if Therefore, equation (12.2) results in Now let us consider the trajectory equation (12.4) in the form F (r, z) = 0. Its right-hand side after integration gives Therefore, the trajectory equation (12.4) is Now, let us consider equation (12.5)  Let us consider the behavior of this equation with respect to (Euclidean) shifts (0-1) in spherical space. To this end, let us introduce two coordinate systems in S 3 u 1 = cos z sin r cos φ, u 2 = cos z sin r sin φ, u 3 = sin z, u 0 = cos z cos r, Equation (13.7) gives the relation between two cylindric coordinate systems sin r ′ cos φ ′ = − sin α cos r + cos α sin r cos φ, cos r ′ = cos α cos r + sin α sin r cos φ, and inverse ones sin r cos φ = sin α cos r ′ + cos α sin r ′ cos φ ′ , cos r = cos α cos r ′ − sin α sin r ′ cos φ ′ .
with respect to Euclidean shifts (0-1) in S 3 parameters J, C transform according to This parametric shift generated by symmetry of the system leaves invariant the following (Euclidean) combination of two parameters By a special choice of a shift one can translate the above equation (13.9) to more simple forms: for instance, to 14 Space shifts in space S 3 and gauge symmetry in S 3 Let us introduce two cylindric coordinate systems u 1 = cos z sin r cos φ, u 2 = cos z sin r sin φ, u 3 = sin z, u 0 = cos z cos r; related by the shift (0-1) sin r ′ cos φ ′ = − sin α cos r + cos α sin r cos φ, cos r ′ = cos α cos r + sin α sin r cos φ.
Under this change of variables, (r, φ) =⇒ (r ′ , φ ′ ), the uniform magnetic field is transformed according to or in terms of the Jacobian The coordinate transformation can be presented as φ = arctan sin r ′ sin φ ′ sin α cos r ′ + cos α sin r ′ cos φ ′ = arctan A, r = arccos cos α cos r ′ − sin α sin r ′ cos φ ′ = arccos B.
From whence it follows J = sin r ′ sin r , therefore, a magnetic field is invariant under the shift (0-1) in space S 3 By symmetry reason, the same behavior of a magnetic field will take place for shifts of the type (0-2). However, for shifts of the type (0-3) cos z ′ cos r ′ = cos α cos z cos r + sin α sin zz, sin z ′ = − sin α cos z cos r + cos α sin z, cos z ′ sin r ′ = cos z sin r, φ ′ = φ.
electromagnetic field is transformed as therefore a magnetic field is not invariant under these shifts (0-3).
In terms of 4-potentials, the transformation rule reads and we may expect Accounting for two relations for a 4-potential in shifted coordinates we have from whence it follows sin r ′ (cos α sin r ′ + sin α cos r ′ cos φ ′ ) 1 + cos α cos r ′ − sin α sin r ′ cos φ ′ , sin α sin φ ′ 1 + cos α cos r ′ − sin α sin r ′ cos φ ′ .
The gauge function Λ in (14.1) is defined by its partial derivatives Integrability condition in the explicit form reads Now we are to find the gauge function Λ. Let us integrate the first equation in (14.2) Λ = −B sin α sin φ ′ dr ′ 1 + cos α cos r ′ − sin α sin r ′ cos φ ′ + λ(φ ′ ).

Extension to relativistic case
Now let us briefly consider an extension to the relativistic problem -particle in a magnetic field in spaces H 3 and S 3 . It suffices to discuss the hyperbolic case, then the relativistic equations have the form or in the explicit form Therefore, all calculations performed for non-relativistic case are valid for relativistic case as well with the only change B =⇒ λB and additional restriction ǫ < 1.

Discussion
Let us summarize the main results of the paper.
Motion of a classical particle in 3-dimensional Lobachevsky and Riemann spaces is studied in the presence of an external magnetic field which is analogous to a constant uniform magnetic field in Euclidean space. In both cases the equations of motion are solved exactly in special cylindrical coordinates. In Lobachevsky space there exist trajectories of two types: finite and infinite in radial variable, in Riemann space all motions are finite and periodical. The invariance of the uniform magnetic field in tensor description and gauge invariance of corresponding 4-potential description is demonstrated explicitly. The role of the symmetry is clarified in classification of all possible solutions given, based on the geometric symmetry group, SO(3, 1) and SO(4) respectively.
Several additional points should be mentioned. Magnetic fields introduced in the models H 3 , S 3 are not invariant under geometrical shift of the type (0-3), instead these geometrical transformations generate some additional electric fields. So, instead of four symmetry generators in Euclidean space, (P 1 , P 2 , P 3 , J z ), in curved space models we have only 3 generators for symmetries, (J 01 , J 02 , J z ). This means that the magnetic fields under consideration in curved spaces are "less uniform" than in Euclidean space.
The choice of special coordinate systems is a matter of principal importance when exploring any problem, and our special choice of cylindrical coordinates is not accidental but turns to be decisive one. For instance, in Lobachevsky and Riemann models there exist other cylindrical coordinates in which Maxwell equations can be solved as well. In this case, electromagnetic potential is (in H 3 model) dS 2 = dt 2 − dr 2 − dr 2 − sinh 2 dφ 2 − cosh 2 dz 2 , A φ = const [ln(cosh r)] u 0 = cosh r cosh z, u 3 = cosh r sinh z, u 1 = sinh r cos φ, u 0 = sinh r sin φ, such a potential has a good limiting behavior at vanishing curvature limit and admits separating of variables, but it hardly could be brought to explicit analytical solutions when dealing with a particle behavior in this field.
Up to now, the most attention in the literature was given to a Kepler problem (quantum mechanical and classical) in Lobachevsky and Riemann models and to the magnetic monopole problem. Corresponding electromagnetic potentials (in spherical coordinates) are H 3 , A 0 = e tanh r , A φ = g cos θ; S 3 , A 0 = e tan r , A φ = g cos θ.
Magnetic field potentials used by us are They all provide us with solutions of the Maxwell equations. So all three potentials are equally correct. They are equally interesting as problems simple enough for their analytical treatment and as extensions of classical physical problems in flat space. At present time, we think, the third one is most interesting because one may expect new results on this field. For instance, as shown in [4] there exists special additional electric field that allows for solutions of corresponding Schrödinger equations in terms of hypergeometric functions, so one may expect the respective solutions of the classical equations in the presence of these additional electric fields as well.
In this case one might expect to extend symmetry operations governing the structure of all solutions in presence of both uniform magnetic and electric fields.