The Gauss-Landau-Hall problem on Riemannian surfaces

arXiv:math-ph/0404060v1 26 Apr 2004 The Gauss-Landau-Hall problem on Riemannian surfaces Manuel Barros1, Jos´e L. Cabrerizo2, Manuel Fern´andez2 and Alfonso Romero1 1Departamento de Geometr´ ia y Topolog´ ia, Facultad de Ciencias Universidad de Granada, 18071-Granada, Spain. E-mail addresses: [email protected], [email protected] 2Departamento de Geometr´ ia y Topolog´ ia, Facultad de Matematicas Universidad de Sevilla, 41012-Sevilla, Spain. E-mail addresses: [email protected], [email protected] Abstract We introduce the notion of Gauss-Landau-Hall magnetic field on a Riemannian surface. The corresponding Landau-Hall problem is shown to be equivalent to the dynamics of a massive boson. This allows one to view that problem as a globally stated, variational one. In this framework, flowlines appear as critical points of an action with density depending on the proper acceleration. Moreover, we can study global stability of flowlines. In this equivalence, the massless particle model correspond with a limit case obtained when the force of the Gauss-Landau-Hall increases arbitrarily. We also obtain new properties related with the completeness of flowlines for a general magnetic fields. The paper also contains new results relative to the Landau-Hall problem associated with a uniform magnetic field. For example, we characterize those revolution surfaces whose parallels are all normal flowlines of a uniform magnetic field. 1 From a classical picture to a general setting Classically, the Landau-Hall problem consists of the motion study of a charged particle in the presence of a static magnetic field, H. In this setting, free of any electric field, a particle, of charge eand mass m, evolves with velocity vsatisfying the Lorentz force law, [1], dP dt =e cv×H, where cdenotes the light speed, P= (ǫ/c2)vstands for the momentum of the particle, and ǫ=mc2[1 −(kvk2/c2)]−1/2is its energy. Since dP/dt is orthogonal 1 to P, then (d/dt)(kPk2) = 0. This implies the constancy of both kvkand ǫ. Assume His stationary, i.e., His a time-independent vector of the Euclidean space R3. With the choice of a suitable orthonormal reference system, we may assume that H=h(0,0,1), for some h∈R. In this framework, we have d dt v1(t) = ω v2(t),d dt v2(t) = −ω v1(t),d dt v3(t) = 0, where ω= (ehc)/ǫ is constant. Then x1(t) = x0 1+rsin(ωt +α), x2(t) = x0 2+rcos(ωt +α), x3(t) = x0 3+v0 3t, where r=kvk/ω. In particular, if v0 3= 0, then the particle describes a circle in the plane x3=x0 3, with center (x0 1, x0 2, x0 3) and radius r. Now, in this plane we consider the 2-form Fdefined by F(X, Y ) = ε < X ×Y, H >, where ε=±1 is the sign of h/ω. It is clear that Fis covariantly constant, and therefore it is a constant multiple of the area element, indeed F=εh dx1∧dx2. Now, consider the metric gon the plane defined by g:= ε(h/ω)g0, where g0=< , > denotes the Riemannian metric on the plane induced by the usual one of R3. Define the operator Φ, g-equivalent to F, by g(Φ(X), Y ) = F(X, Y ). Then, the Lorentz force law can be expressed in terms of this form by d dt v(t) = Φ(v(t)).(1) This approach to the classical picture can be obviously extended to a more general setting. In fact, it seems natural to define a magnetic field on a n(≥2)- dimensional Riemannian manifold (M, g), as a closed 2-form Fon M. The Lorentz force of a magnetic background (M, g, F) is defined to be the skew-symmetric operator, Φ, given by g(Φ(X), Y ) = F(X, Y ),(2) for any couple of vector fields X, Y on M. Let us remark that Φ is metrically equivalent to F, so no information is lost when Φ is considered instead F. In classical terminology, it is said that Φ is obtained from Fby raising its second index, and Φ and Fare then said to be physically equivalent. On the other hand, there exists another operator Φ′defined from Fvia gin a similar way, namely g(X, Φ′(Y)) = F(X, Y ),but it is easily seen that Φ′=−Φ. So, the choice from among Φ or Φ′to represent F, using g, is not relevant. Along this paper, we will use Φ to denote the Lorentz force induced from (M, g, F). A (smooth) curve γin (M, g) is called a flowline of the dynamical system associated with the magnetic field F(or simply a flowline of F, or a magnetic curve of (M, g, F)), if its velocity vector field, γ′, satisfies the following (Landau-Hall) differential equation, ∇γ′γ′= Φ(γ′),(LH) 2 where ∇is the Levi-Civita connection of g[compare with Eq. (1)]. For the trivial magnetic field, F= 0, the case without the force of a magnetic field, magnetic curves correspond with the geodesics of (M, g). As it is well known, they are nicely characterized as critical points of an energy action and so they represent the trajectories for free fall particles (moving under the influence of only gravity). In the general case, however, magnetic flows are important examples of dynamical systems on Riemannian manifolds whose flowlines, being the trajectories of charged particles in (non trivial) magnetic fields, are not geodesics (Proposition 2.1) but, as we will see later, they are closely related with the Riemannian structure. Nevertheless, the magnetic curves of (M, g, F) can be also viewed, at least locally, as the solutions of a variational principle. In fact, let Ube an open subset of M where F=dω for some potential 1-form ω(this open subset could be the whole M when H2(M) = 0). For any two fixed points p, q ∈U, we consider the space Γpq of smooth curves in Uthat connect these two points. Now, we choose the action LH : Γpq →Rdefined by LH(γ) = 1 2Zγ g(γ′, γ′)dt −Zγ ω(γ′)dt. (3) The tangent space of Γpq in γis made up of the smooth vector fields, V, along γthat vanish at the end points p, q ∈U. An standard computation involving integration by parts allows one to compute the first variation of this action to be δ(LH)(γ)[V] = −Zγ g(∇γ′γ′−Φ(γ′), V )dt. As a consequence, we get δ(LH)(γ)[V] = 0,for any V∈TγΓpq if and only if γis a solution of (LH). This argument shows that the differential equation (LH) is indeed the Euler-Lagrange equation associated with the functional LH. However, it seems natural to realize the old idea of characterizing magnetic curves from a global variational principle. In other words, to obtain the magnetic trajectories of (M, g, F) as solutions of a variational problem that neither it does not involves any local potential nor it does not constraint the topology of M. This is, in general, an interesting open problem. One of the main aim of this paper is just to solve it for certain magnetic fields on surfaces. To be precise, we introduce the notion of a Gauss-Landau-Hall magnetic field (in brief, GMF) on an oriented Riemannian surface (M, g). First, we do it in the natural context that surfaces are immersed in Euclidean space R3using the Gauss map. However, we notice that the notion of GMF is absolutely intrinsic so it can be considered on surfaces even if they are not regarded in R3. Then, we are able to obtain an amazing result which characterizes the normal flowlines of a GMF as the solutions of a variational principle globally stated. Therefore, those flowlines appear as critical points of an action whose Lagrangian density involves the proper acceleration of particles (relativistic particles with rigidity of order one, in the sense 3 of Plyushchay, [2],[3]). A priori, these actions describe a massive relativistic boson. However, massless particles with arbitrary helicity are obtained as a limit case, just when the Lorentz force of the GMG increases arbitrarily. Other details on the paper are the following. We first provide in Section 2 an analysis of the existence, uniqueness, extendibility and completeness of the magnetic curves associated with a given (M, g, F). Section 3 deals with uniform magnetic fields on Riemannian surfaces, while the particular case when (M, g) is a revolution surface is studied in Section 4. In Section 5, a one-parameter family Fmof functionals is considered on an appropriate space of curves Λ in the surface. The Euler-Lagrange equation associated to the variational problem is then obtained. In Section 6 we define a Gauss-Landau-Hall magnetic field on a surface, first in R3, and then in general. In this section, we obtain the main result, Theorem 6.1, which asserts that the normal flowlines of a GMF coincide with the critical points of the appropriate functional Fm. Stability of the field equation solutions is also studied. In Section 7, we show a characterization theorem for those revolution surfaces whose parallels are all normal magnetic curves associated to a GMF. We close the section studying some particular examples. 2 Completeness of magnetic curves and more An early property of the magnetic curves is the following conservation’s law. Particles evolve with constant speed, and so constant energy, along the magnetic trajectories d dt g(γ′, γ′) = 2g(Φ(γ′), γ′) = 0.(4) In particular, a magnetic curve γis said to be normal if it has unit energy, i.e., kγ′k2≡1. The existence and uniqueness of geodesics, remains true when one considers magnetic curves. Thus, for each p∈Mand v∈TpMthere is exactly one inextendible (i.e., maximal) magnetic curve, γ: (−a, a)−→ M, of (M, g, F) with γ(0) = pand γ′(0) = v, (see for instance [4], p. 91). Since the proof of this result does not make use neither the definiteness of gnor the skew-symmetry of Φ, one has a present determines the future type result for an indefinite metric, Lorentzian in particular, and for any smooth operator. Even more, the result also works for solutions of a differential equation that extends that of Landau-Hall in the following terms [5], ∇γ′γ′= Φ(γ′) + X◦γ, where Xis a vector field on a semi-Riemannian manifold. This setting includes the important case in Mechanics where X=−∇V, and Vstanding for smooth function on M, ([6], Proposition 3.7.4). Nevertheless, the well known homogeneity result for geodesics, works quite different in non trivial magnetic fields. Therefore, if γis the inextendible magnetic curve of (M, g, F) determined from the initial data (p, v), the curve β, defined by β(t) = γ(λt), λ ∈R\{0}, is a magnetic trajectory of (M, g, λF) and also, when 4 λ > 0, of (M, (1/λ)g, F), in both cases determined from initial data (p, λv). Furthermore, the whole families of magnetic curves of (M, g, F) and (M, λg, λF) coincides, for any constant λ > 0. Consequently, we have Proposition 2.1 Let Fbe a non trivial magnetic field on a Riemannian manifold, (M, g). Then, there exists no affine connection on Mwhose geodesics are the magnetic curves of (M, g, F). A magnetic field (M, g, F) with Lorentz force Φ, provides, in a similar way as in [7], Prop. 3.28, with a unique vector field QΦon the tangent bundle TM. This is defined to have integral curves being the lifting to T M of the magnetic curves, that is, t7→ (γ(t), γ′(t)), where γis a magnetic curve of (M, g, F) (compare with [8]). Certainly this vector field is nothing but the geodesic flow when F= 0. Once more, neither the definiteness of gnor the skew-symmetry of Φ is needed to define QΦ,[5]. On the other hand, the fact that any integral curve of QΦis the velocity of its projection on M, allows us to think of QΦas a nice example of the classically so-called second order differential equation on M. Because the comment previous to Proposition 2.1, QΦis not an spray, in general. A dynamical system with complete trajectories is often thought in Physics to be persisting eternally. But in many circumstances one has to deal with incompleteness. So, because of its importance, we next give criteria to assert when it holds true. An important tool to study the completeness of the inextendible magnetic curves, i.e., under what assumptions all the inextendible magnetic curves are defined on all R, is the vector field QΦ. By using Lemma 1.56 in [7], it is easily seen the following result. Proposition 2.2 Let (M, g)be a Riemannian manifold, Fa magnetic field on M and γ: [a, b)−→ M,a < b, a magnetic curve of F. The following are equivalent: (a) γis extendible to bas a magnetic curve. (b) There exists a sequence {tn} −→ b,tn∈[a, b)such that the sequence of velocities {γ′(tn)}converges in TM. Accordingly, a magnetic curve γ: (a, b)−→ M, a, b ∈R, a < b, of (M, g, F) can be extended to some open interval I, (a, b)⊂I, if and only if γ(a, b) is contained in a compact subset of M. Therefore, we get (compare with Theorem 2.1.18 in [6]) Proposition 2.3 Let γbe an inextendible magnetic curve of (M, g, F)such that γ(a, b)lies in a compact subset of M, for every finite interval (a, b)in its domain. Then, γmust be complete. In particular, if Mis assumed to be compact, then we get that any inextendible magnetic curve of (M, g, F) must be complete. This fact can be also obtained as a consequence of Corollary 2.4, and it will be stated in Remark 2.5 (a), from a different approach. 5 Now, let γ: [a, b)−→ Mbe a magnetic curve. Its length L(γ) satisfies L(γ)≤ (b−a)√e, where eis the (constant) energy of γ. For each t∈[a, b), the distance between γ(a) and γ(t) satisfies d(γ(a), γ(t)) ≤L(γ|[a,t])≤(b−a)√e, which shows that γ([a, b)) is contained in the closed metric ball Bcentered at γ(a) and with radius (b−a)√e. Therefore, γ′([a, b)) ⊂ {(p, v)∈TM :p∈B, g(v, v) = e} ⊂ TM. Then, we have, Corollary 2.4 Let Fbe any magnetic field on a geodesically complete Riemannian manifold (M, g). Then, all the inextendible magnetic curves of (M, g, F)are complete. Proof. If (M, g) is assumed to be geodesically complete, then the Hopf-Rinow theorem implies that Bmust be compact. Hence {(p, v)∈TM :p∈B, g(v, v) = e} is a compact subset of TM. Take now a sequence {tn} −→ b,tn∈[a, b), then {γ′(tn)}lies in a compact subset of TM. So, by passing to a subsequence of {tn}, we are under the assumption (b) of Proposition 2.2, concluding that γis extendible to bas a magnetic curve. Remark 2.5 (a): If Mis assumed to be compact (therefore (M, g) is geodesically complete for any Riemannian metric gon M), then we can give an alternative proof of Corollary 2.4. In fact, the previous conservation’s law [Eq. (4)] for the length of velocity vectors of magnetic curves, implies that the vector field QΦon TM can be restricted to each spherical tangent bundle UeM={(p, v)∈TM : g(v, v) = e} ⊂ TM, e > 0. But UeMis compact whenever Mis compact, and hence the restriction of QΦto UeMis a complete vector field. This proves that all the inextendible magnetic curves of Fare complete. (b): Proposition 2.1 has shown a remarkable difference between magnetic curves and geodesics. The following non-connecteness fact complement that result. Let us consider the unit 2-sphere S2(1) endowed with its standard round metric g, and let Fbe the magnetic field F=µΩ2,where Ω2is the area 2-form and µ∈R, µ 6= 0.As we will show later (see the comment after Proposition 3.2), the associated magnetic curves with energy eare circles on S2(1) with radius r= [1 + (µ2/e)]−1/2.Then, as r < 1,any two antipodal points can not be connected by a magnetic curve of (S2, g, F).Moreover, for any p∈S2, all the inextendible magnetic curves γof (S2\ {−p}, g, F) such that γ(0) = pare complete. (c): Let (M, g) be a Riemannian manifold where gis an incomplete metric. If Fis a magnetic field on (M, g),then there exists a pointwise conformal metric f2g such that the inextendible magnetic curves of (M, f2g, F) are complete. In fact, there exists f∈C∞(M), f > 0,such that f2gis geodesically complete, [9]. Therefore, Proposition 2.4 gives that the magnetic curves of (M, f2g, F) are complete. 6 (d): It should be observed that the closedness assumption on the 2-form Fin Proposition 2.4 was not used. On the other hand, the skew-symmetry of the tensor field Fhas played a crucial role [recall the conservation’s law (4)]. In fact, consider the tensor field F=−2x dx2on the Euclidean plane (R2, g0=dx2+dy2). If Φ denotes the operator defined from Fusing Eq. (2), then Φ(∂/∂x) = −2x(∂/∂x) and Φ(∂/∂y) = 0. Therefore, γ(t) = (x(t), y(t)) satisfies Eq. (LH) if and only if x′′(t) + 2 x(t)x′(t) = 0 and y′′(t) = 0. So, γ(t) = (1/t, t) is an inextendible incomplete trajectory of (R2, g0, F). (e): Finally, let us point out that Proposition 2.4 cannot be also extended to the indefinite case. In fact, consider R2endowed with the Lorentzian metric gL=dx2−dy2,and define the magnetic field F=−xdx ∧dy. A curve (x(t), y(t)) is a magnetic curve of (R2, gL, F) if and only if it satisfies x′′(t) = x(t)y′(t), y′′(t) = x(t)x′(t).Then, γ(t) = (2/t, −2/t) is an inextendible magnetic curve which is defined on (0,∞). 3 Uniform magnetic fields From now on, Mwill be an oriented Riemannian surface with standard complex structure J, and area element Ω2so that Ω2(X, JX) = 1 for any unit vector field X in M. Given a curve γin Msuch that g(γ′, γ′) = e > 0 is constant, its Frenet apparatus is {T= (1/√e)γ′, N =JT}. If κdenotes the curvature function, we have the following well-known Frenet equations ∇γ′T=κ√e N, ∇γ′N=−κ√e T. Obviously, any magnetic field on a surface, M, is determined from a smooth function, f(the strength), by F=fΩ2. Therefore, the matrix of Φ in any orthonormal frame, {X, JX}is given by 0−f f0. In particular, along a magnetic curve γof (M, g, F), with energy e, and relative to its Frenet frame, the Lorentz force is obtained to be 0−κ√e κ√e0. Therefore, we get, Proposition 3.1 The curvature of the magnetic curves with energy eis given by κ=f/√e. So, the curvature of the normal magnetics curves completely determines the Lorentz force, i.e., f=κalong these flowlines. A parallel magnetic field F, i.e., a magnetic field with constant strength f=µ, is called a uniform magnetic field. This class of magnetic fields has been extensively 7 considered in the literature from different points of view ([8],[10]-[16], etc.). The geometric partner of the Landau-Hall problem, for uniform magnetic fields, is nothing but the computation of curves with constant curvature. To be precise, we have, Proposition 3.2 Let F=µΩ2be a uniform magnetic field, with constant strength µ, on a Riemannian surface (M, g). A curve γin M, with constant energy e, is a magnetic curve of (M, g, F)if and only if it has constant curvature κ=µ/√e. On surfaces of constant Gauss curvature, the feature of the normal flowlines of a non-trivial uniform magnetic field F=µΩ2is well-known for any uniform magnetic field. On the Euclidean plane, R2, they are circles with radius 1/|µ|. On the 2-sphere of radius r,S2(r), flowlines with energy eare circles with radius (r√e)/pe+r2µ2 (< r). In these two backgrounds, the flowlines are always closed. On the other hand, the situation in a hyperbolic plane is quite different. Let H2(−G) be the upper half-plane (in R2) endowed with the Lobatchevski metric of curvature −G,G > 0, that is, the Poincar´e plane. We use Proposition 3.2 joint the basic knowledge of the curves of constant curvature in H2(−G) (see any basic text of Riemannian geometry) to make trivial the following description of the flowlines which is due to A. Comtet, [11], and has been mentioned along a large list of references. The behaviour of normal magnetic curves changes according to the ratio between the strength, µ, and the curvature of H2(−G). Namely, •If |µ|/√G > 1, then the trajectories are geodesic circles, and therefore they are closed curves. •If |µ|/√G≤1, then the trajectories are non-closed curves which intersect the boundary line, ∂H2(−G), of the upper half-plane. In particular, they are tangent to this boundary, and so they are horocycles when |µ|=√G. Remark 3.3 (a): Let γbe a curve with constant geodesic curvature κ6= 0 in any of the three previous constant curvature surfaces. Then, for a given uniform magnetic field F=µΩ2,a suitable fitting of the constant speed (and hence, the energy) of γmakes this curve to be a magnetic curve of F. (b): Let (M, g) be again one of the three above space forms and F=µΩ2a uniform magnetic field on (M, g).Then, any magnetic curve γwith energy eof (M, g, F) can be then considered as a normal magnetic curve of M, (1/e)g, (1/e)F(see the comment previous to Proposition 2.1). 4 The Landau-Hall problem in a surface of revolution Let α(s) = (f(s), h(s)), a < s < b,f(s)>0, be a parametrization by the arclength of a curve, C, contained in the {xz}-plane of R3. We rotate C around the z-axis to obtain a surface of revolution, say Mα, with canonical parametrization in R3 X(s, v) = (f(s) cos v, f(s) sin v, h(s)) ,0≤v≤2π. (5) 8 Of course we consider that Mαis endowed with the induced metric gof the Euclidean one of R3. Each point of C describes a parallel, γs, which can be parametrized by arclength in the following way γs(t) = f(s) cos t f(s), f(s) sin t f(s), h(s), where 0 ≤t≤2πf(s). The curvature, κs, of γsin Mα, is computed to be κs(t) = k∇TsTsk=f′(s) f(s), where Ts=γ′ sand ∇is the Levi-Civita connection of Mα. In particular, κsis constant along γs,and so this curve is a good candidate to be a flowline of a suitable uniform magnetic field on Mα. Let F=µΩ2be a uniform magnetic field on Mαwith constant strength µ. Then γsis a normal magnetic flowline of (Mα, g, F =µΩ2) if and only if κs=µ (Proposition 3.2). Therefore, the set of magnetic parallels of (Mα, g, F =µΩ2) can be identified with the following subset of the interval (a, b) Γµ={s∈(a, b) : f′(s) = µf(s)}. To determine those surfaces of revolution whose parallels are all normal magnetic curves of a given uniform magnetic field (that is, those with Γµ= (a, b)) we need to solve the ordinary differential equation f′(s) = µf(s). Obviously, we have two possibilities. The trivial one, corresponding with the case of a trivial magnetic field (the strength vanishes), the flowlines are then geodesics, and the surface of revolution is a right circular cylinder. Otherwise, since the Gauss curvature of a surface of revolution (in the canonical parametrization) is given by G(s, t) = −f′′(s) f(s),(6) we get that G(s, t) = −µ2, an hence the surface has constant negative curvature. In particular, we have, Proposition 4.1 The parallels of a surface of revolution, Mα, are all normal magnetic flowlines of a uniform magnetic field, F=µΩ2, if and only if either: 1. Mαis a right circular cylinder (when µ= 0), or 2. Mαis a bugle surface with Gaussian curvature −µ2. 9 made up of a three parameter family of surfaces which includes the bugle surface too. Theorem 7.1 The normal flow of a GMF, (G/m) Ω2, in a surface of revolution, Mα, is invariant under rotations if and only if the profile curve of Mαlies in the following three parameter family of arclength parametrized plane curves α(s) = f(s),Zs 0p1−f′(s)2ds, where f(s) = 1 ma+cexp(−m s +b);a, b, c ∈Rwith a > 0. Observe that the general solution of the ordinary differential equation (13) has the form f(s) = a+cexp(−m s +b)/m, so the proof of the last result becomes obvious. Observe also that the above characterized class of surfaces of revolution includes the bugle surface (a= 0) and the right circular cylinder (c= 0) too. It should be noticed the following coupling phenomenon in a surface of revolution, Mα, between the GMF, F1= (G/m) Ω2and the uniform magnetic field F2=−mΩ2, for some values of the coupling constant m. Suppose, for example, that Mα= T(r, R) is a torus of revolution and ρ= (R2−r2)−1/2(notation as in Section 4). Then, F1always has two parallels being normal magnetic curves, no matter the value of m. Now, we use Proposition 4.2 to obtain the following statement, Proposition 7.2 If −m∈(ρ, ρ), then both F1and F2have two normal magnetic parallels coming from points alternatively placed in the profile circle. Moreover they collapse when −mgoes to −ρor ρ. Proof. For any value of min R, (T(r, R), g, F1) has two normal magnetic parallels obtained by rotation of the two antipodal points in C, defined by cot (s/r) = −r m. These two points are just those determining the diameter D−mthat separates the two magnetic parallel of (T(r, R), g, F2) when −m∈(−ρ, ρ). The second part of this statement follows similarly when use points 1 and 2 of Proposition 4.2. We finish the paper showing several examples. Example 7.3 Let β(s) be an arclength parametrized curve contained in a plane, Π (with unit normal vector B0), in R3. We denote by {T(s), N(s)}a Frenet frame along β(s), so that T(s)∧N(s) = B0, and κ(s) will stand for its curvature function. For a suitable r > 0, we define a tube of radius r, say Tβ(r), as the surface given by X(s, v) = β(s) + rcos(v)N(s) + sin(v)B0. We denote by Λβ={γv,:v∈[0,2π]}the family of curves in the tube obtained when we make vconstant. The curvature of these curves in Tβ(r) can be obtained, from a direct computation, to be 16 κv(s) = κ(s) sin(v) 1−r κ(s) cos(v). Notice that it is not constant unless β(s) is chosen to be constant curvature. On the other hand, the Gauss curvature of the tube Tβ(r) is computed to be G(s, v) = −κ(s) cos(v) r1−rκ(s) cos(v). Now, we can apply these formulas together with the Euler-Lagrange equations associated with the GMF, F= (G/m) Ω2(Propositions 5.1, 5.2), to see that there exist exactly two curves (clamped or closed) in Λβthat are normal magnetic trajectories. They are obtained for cot(v) = −r m and this is, formally, the same result that we have obtained for a torus of revolution (Proposition 4.2) which can be regarded as a tube around a circle. Example 7.4 Similarly, for a curve, β(s), in R3with Frenet frame {T(s), N(s), B(s)}, curvature κ(s) and torsion τ(s), one can define the tube Tβ(r) by X(s, v) = β(s) + rcos(v)N(s) + sin(v)B(s). In particular, if β(s) is a helix (κand τare both constant) then the curvature function, κv(s) of the curves in Λβ={γv,:v∈[0,2π]}satisfy κ2 v=κ2sin2(v) 1−r κ cos(v)2+r2τ2. Now, the curves in Λβthat are normal flowlines of (G/m) Ω2on the helicoidal tube Tβ(r) correspond with the zeroes of the function ϑ:S1→Rdefined by ϑ(v) = 1−r κ cos(v)2cos2(v)−r2m2sin2(v)+r2τ2cos2(v). However, we have ϑ(0) = ϑ(π) = (1 −r κ)2+r2τ2>0 and ϑ(π 2) = ϑ(3π 2) = −r2m2<0. Therefore, there exist four curves of Λβin the flow of (G/m) Ω2. Example 7.5 On the cathenoid (Example 4.3), the GMF given by (G/m) Ω2has a unique normal magnetic parallel for all m. If m > 0, it is obtained for a t0<0. If m < 0, then it is obtained for a t′ 0=−t0>0. Example 7.6 On the hyperboloid of revolution obtained from Eq. (5) by putting f(t) = cosh tand h(t) = sinh t, the GMF given by (G/m) Ω2has also a unique normal magnetic parallel for all m, analogously to the previous case. 17 Example 7.7 On the cicloidal surface (Example 4.4), the GMF given by (G/m) Ω2 has a unique normal magnetic parallel for m∈(−∞,−1/(4a)) S(1/(4a),∞). If m > 0, t0= arccos ((1/(4am)), whereas if m < 0, then t′ 0= 2π−t0. Conclusions Oriented surfaces, M, in R3admit two natural 2-forms. First, the area element, Ω2, associated with the induced metric, g. Second, the area element, N∗(dσ2), of its spherical image under the Gauss map, N:M→S2. It is well known that these 2-forms are nicely related by N∗(dσ2) = GΩ2, where Genotes the Gaussian curvature of g. In particular, both 2-forms are intrinsic and then they are defined once we know a Riemannian metric, g, on M. Associated with these 2-forms appear two classes of magnetic fields on (M, g), 1. The class made up of the constant multiples of the former one, C1={µΩ2: µ∈R}, provides that of uniform magnetic fields, with strength µ, on (M, g). The corresponding Landau-Hall problem has been widely studied along the literature. Even in this paper, we have obtained some new information relative to uniform magnetic field essentially in a surface of revolution. For example, we have characterized right circular cylinders and bugle surfaces as the only surfaces of revolution whose parallels are all normal magnetic flowlines of a uniform magnetic fields. 2. The class of the constant multiples of the later one, C2={µ N∗(dσ2) : µ∈R}, constitutes a class of magnetic fields that in this paper are introduced under the terminology of Gauss-Landau-Hall magnetic fields (GMF). In this case the strength is given by µ G and obviously both classes coincide when (M, g) has constant curvature. In this paper, we wish to state the importance and nice interest of GMF on surfaces. In fact, the chief result of the paper appears when we study the Landau-Hall problem associated with a GMF (which we call the Gauss-Landau-Hall problem). Then, we are be able to show that this problem is equivalent to the dynamics of a massive relativistic boson. This provides an amazing relationship between two, a priori, quite different physical phenomena. Therefore, we can use two points of view to study each of the two involved problems. On one hand, one can study completeness, homogeneity and so on, in the dynamical study of bosonic worldlines. By the way, we have introduced a section with new results on these topics. But on the other hand, the Gauss-Landau-Hall problem can be regarded as a variational problem globally stated. In this setting, flowlines are critical points of an action which has been used to model relativistic particles with order one rigidity. In particular, we can talk about, and so we study, global stability of normal flowlines of a GMF. Say finally that under this equivalence, 18 the model to describe a massless particle with arbitrary helicity correspond with a limit case obtained when the force of the GMF increases arbitrarily. We believe that this new point of view in the study of GMF is physically remarkable and it could be extended to other classes of magnetic fields. Acknowledgments Research partially supported by MCYT FEDER Grant BFM 2001-2871-C04 and by Acci´on Coordinada Grupos de Investigaci´on FQM-324 and FQM-327, Junta de Andaluc´ ia. References [1] L.D. Landau and E. M. Lifschitz, Course of Theoretical Physics, Vol. 1. Mechanics. 3rd edition. (Pergamon Press, Oxford-New York-Toronto, 1976). [2] M. S. Plyushchay, Phys. Lett. B, 243, 383 (1990). [3] M.S. Plyushchay, Int. J. 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