Magnetic geodesics on surfaces with singularities

We prove that, generically, magnetic geodesics on surfaces will turn away from points with lightlike tangent planes, and we motivate our result with numerical solutions for closed magnetic geodesics.


Introduction
A magnetic geodesic describes the trajectory of a charged particle in a Riemannian manifold M under the influence of an external magnetic field. Numerical experimentation suggests that almost all magnetic geodesics tend to avoid any lightlike singularities (points where the tangent spaces are lightlike) that M may have, regardless of choice of bounded smooth external magnetic field. Our primary result is a mathematically rigorous confirmation of this behavior. Initially, we take M to be a complete, orientable Riemannian manifold without boundary of dimension n and Riemannian metric ·, · . For a given two form Ω defined on M we associate a smooth section Z ∈ Hom(T M, T M ) defined via η, Z(ξ) = Ω(η, ξ) for all η, ξ ∈ T M . We will investigate the existence of closed curves γ = γ(t) satisfying the following equation (1.1) Note that, in contrast to geodesics, which correspond to Z = 0, the equation for magnetic geodesics is not invariant under rescaling of t. In the case that M is a surface, that is n = 2, we know that every two-form Ω is a multiple of the volume form Ω 0 associated with ·, · . Thus, every two-form can be written as Ω = κΩ 0 for some function κ : M → R. We can exploit this fact to rewrite the right hand side of (1.1) as Z(γ ′ ) = κJ 90 γ (γ ′ ), (1.2) where J 90 γ represents rotation in the tangent space T γ M by angle π/2, see [10]. Due to this fact one often refers to (1.2) as the prescribed geodesic curvature equation, and κ is proportional to the geodesic curvature function. We will always assume that κ is a smooth and bounded function.
Note that a solution of (1.1) has constant speed, which follows from ∂ ∂t due to the skew-symmetry of the two-form Ω. For magnetic geodesics on surfaces, several existence results are available, employing techniques from symplectic geometry [5], [6] and from the calculus of variations [14]. In the papers of Schneider [11], [12], and the paper by Schneider and Rosenberg [13], existence results for closed magnetic geodesics on Riemann surfaces are given by studying the zeros of a certain vector field.
Here rather, we give an approach more aimed at usefulness for numerics, and then proceed to produce examples of closed magnetic geodesics numerically. We then study the behavior of magnetic geodesics near singular points of a surface by proving our main result Theorem 3.1, and our proof employs the fact that magnetic geodesics have constant speed parameterizations. This article is organized as follows: In section 2 we derive several numerical examples of magnetic geodesics. Moreover, we provide several analytic statements that support our numerical calculations. In section 3 we focus on magnetic geodesics on almost-everywhere-spacelike surfaces with lightlike singularities and show that they will tend to turn away from the singularities unless they enter the singular sets at specific angles, which is the content of Theorem 3.1.

Closed magnetic geodesics on surfaces in Euclidean and Minkowski 3-spaces
Before we turn to the numerical integration of (1.1) let us make the following observations. By the Theorem of Picard-Lindeloef we always get a local solution to (1.1). However, similar to the classical Hopf-Rinow theorem in Riemannian geometry we can show be a curve in M with geodesic curvature κ(γ(t)) at γ(t), in other words, γ is a nontrivial solution to Then the domain (a, b) can be extended to all of R.
Proof. To show that the maximal interval of existence of (2.1) is indeed all of R we assume that there is a maximal interval of existence and then show that we can extend the solution beyond that interval. Thus, assume that γ : (a, b) → M is a magnetic geodesic with maximal domain of definition. Since |γ ′ | 2 is constant we know that the curve γ has constant length L[γ]. Then we have for a sequence γ(t i ) i∈N where d denotes the Riemannian distance function. Hence, γ(t i ) i∈N is a Cauchy sequence with respect to d. It is easy to see that the limit is independent of the chosen sequence. As a next step, we show that we may extend γ ′ to (a, b]. To this end we use the local expression for (2.1), that is Now, consider the expression Using that |γ ′ | is constant it follows that γ ′ (t i ) forms a Cauchy sequence and converges to some γ ′ ∞ . Again, the limit is independent of the chosen sequence. By differentiating the equation for magnetic geodesics and using the same method as for estimating |γ ′ (t i ) − γ ′ (t j )| L ∞ we can show that also γ ′′ (t i ) forms a Cauchy sequence. Now, assume thatγ : (β − a, β + a) → M is a magnetic geodesic withγ(β) =γ(β) andγ ′ (β) = γ ′ (β). Since magnetic geodesics are uniquely determined by their initial values,γ andγ coincide on their common domain of definition. This yields a continuation of γ as a magnetic geodesic on (a, b + β), which contradicts the maximality of b.
Remark 2.2. We will be looking for closed solutions of (1. For our numerical studies of (1.2) we need the following where n denotes a normal to the surface compatible with J 90 and × denotes the cross product in R 3 .
We now consider a surface S(u, v) parametrized by coordinates (u, v) in a subdomain of R 2 , and a curve γ(t) = S(u(t), v(t)) on the surface. We can rewrite (2.2) and (2.3): Expanding to obtain and taking n = S u × S v , and using However, if the surface is conformally parametrized, that is the system (2.4) and (2.5) simplifies to Using the formulations ( Note that κ = 0 will give great circles of course, and clearly κ a nonzero constant will give a circle in the sphere that is not a great circle. κ = sin u can give a curve as in Figure 2.1.
Examples are found in Figure 2

2.4.
Example: minimal Enneper surface. The Enneper minimal surface in R 3 can be conformally parametrized as with f = (1 + u 2 + v 2 ) 2 . This yields the system An example is found in Figure 2.5.

2.5.
Minkowski 3-space. Let R 2,1 denote the Minkowski 3-space {(x, y, s) | x, y, s ∈ R} with Lorentzian metric of signature (+, +, −). Spacelike surfaces with mean curvature identically zero are called maximal surfaces, and the next example is such a surface. Our primary result (Theorem 3.1) is about spacelike surfaces in R 2,1 , with singularities at which the tangent planes become lightlike. Proposition 2.4 is true for spacelike surfaces in R 2,1 as well, once R 3 is replaced by R 2,1 , the cross product for R 3 is replaced by the cross product for R 2,1 , and the induced connection ∇ for surfaces in R 3 is replaced by the induced connection ∇ for surfaces in R 2,1 . The statement is as follows: where n denotes a normal to the surface compatible with J 90 and × denotes the cross product in R 2,1 .
2.6. Example: maximal Enneper surface. In this case we can choose in R 2,1 . This parametrization can be obtained from the Weierstrass-type representation for maximal surfaces (see, for example, [9]), which states that where g is a meromorphic function and η is a holomorphic 1-form on a Riemann surface. This surface is conformally parametrized wherever it is nonsingular, and has spacelike tangent planes at nonsingular points. The singularities occur whenever |g| = 1, and the metric for the surface is (1 − |g| 2 ) 2 |η| 2 .
Since, for any magnetic geodesic γ(t) = S(u(t), v(t)), we have the term u ′2 + v ′2 would have to diverge whenever γ approaches a singular point. It follows that magnetic geodesics cannot be extended, as solutions of the magnetic geodesic equation, into singular points. The effect of this fact is that magnetic geodesics tend to avoid singular points, as we will see in Theorem 3.1. Examples of magnetic geodesics in the maximal Enneper surface are shown in Figures 2.6, 2.7 and 2.8.
Typically, even at their singularities, maximal surfaces can be described as smooth graphs of functions over domains in the horizontal spacelike coordinate plane of R 2,1 (see [4], [7], [8] for example), and thus Theorem 3.1 will apply to maximal surfaces.

2.7.
Example: rotated cycloids. In the case of surfaces in R 3 , magnetic geodesics will generally not avoid singular sets on those surfaces, and the final example here illustrates this. We consider rotated cycloids in R 3 , which have cuspidal edge singularities. We choose the following parametrization S(u, v) = ((2 + cos u) cos v, (2 + cos u) sin v, u − sin u).
The system becomes An example of a magnetic geodesic that meets the singular set is shown in Figure 2.9.

Avoidance of lightlike singularities by magnetic geodesics on surfaces
In our numerical investigations of magnetic geodesics on the maximal Enneper surface we have seen that magnetic geodesics avoid the singular set of the surface. In this section we will generalize this conclusion not only to arbitrary maximal surfaces, but we will also mathematically confirm this behavior on general spacelike surfaces in R 2,1 at points where the tangent planes degenerate to become lightlike. More precisely, we will consider the case that the tangent plane T p M becomes lightlike and the surface is a graph of a function over a domain U with immersable boundary ∂U in the horizontal spacelike coordinate plane of R 2,1 whose second derivatives are finite and not all zero at the projection of p into U . This is the content of the following theorem: Theorem 3.1. Suppose that (M, g) is an almost-everywhere-spacelike smooth surface in R 2,1 that becomes singular at a non-flat point p ∈ M . Then there are only at most six directions within T p M to which any magnetic geodesic meeting p with C 1 regularity and bounded geodesic curvature must be tangent. Two of these at most six directions are the lightlike directions.
Proof. We may parametrize the surface as a graph, that is S(u, v) = (u, v, f (u, v)) for some function f (u, v), and we can consider a curve γ(t) = S(u(t), v(t)). The surface is spacelike, with the exception of a measure zero set in the surface at which the tangent planes are lightlike. Without loss of generality, we assume (1) the tangent plane at u = v = 0 is lightlike, (2) the surface is placed in R 2,1 in such a way that the curve γ(t) on the surface satisfies for some value of θ ∈ R \ πZ, (4) the tangent planes to f at the points γ(t) for t > 0 are spacelike. We assume that γ is a magnetic geodesic, thus γ ′ , γ ′ is a positive constant for t > 0. We set . First, we examine the limiting behavior of u ′′ (t) and v ′′ (t) as t approaches 0. Because γ ′ , γ ′ is constant for t > 0, by property (3) above we have γ ′ , γ ′ = sin 2 θ for all t ≥ 0. We can assume |n| = 1 for t > 0. We then have γ ′′ , γ ′ = 0 (3.1) and, by Proposition 2.5, γ ′′ , γ ′ × n = κ sin 2 θ .
, v(t))) and using , we can take the limit as t → 0 in Equation 3.1 to obtain the finite limit Noting that A| t=0 = 0 and B| t=0 = sin θ = 0, we see that only these two cases can occur: (1) u ′′ is bounded at t = 0 and lim t→0 v ′′ = cot θ · R| t=0 , or (2) there exists a sequence t j > 0 converging to zero so that |u ′′ (t j )| diverges to infinity and |v ′′ (t j )/u ′′ (t j )| converges to zero as j → ∞. In the second case, we can obtain the conclusion by examining Examining the behavior as t → 0 of Equation 3.2, we see that is bounded near t = 0. In the first case (1) above with bounded u ′′ , T converges asymptotically to √ hR sin θ, and this can be bounded only if R| t=0 = 0. In the second case (2) above with unbounded u ′′ , we can write T at t j as Since u ′ (t j ) and v ′ (t j ) are bounded, and v ′′ (t j )/u ′′ (t j ) and h −1 (t j ) converge to zero, and since (v ′ f u − u ′ f v )| t=t j converges to sin θ, as j → ∞, this term is asymptotically equal to ( √ hR sin θ)| t=t j were R| t=0 = 0, and again we conclude T is bounded only if R| t=0 = 0. Thus, in either case, we must have (f uu cos 2 θ + 2f uv cos θ sin θ + f vv sin 2 θ) u=v=0 = 0 . (3. 3) If f vv = 0, resp. f uu = 0, the angle θ must satisfy If f uu = f vv = 0, then θ = π/2 + kπ for some integer k. Thus there are at most four possible values for the angle θ ∈ [0, 2π) in addition to θ = 0, π for which the magnetic geodesic can approach the singular point p.
Remark 3.2. Theorem 3.1 can be generalized to almost-everywhere-spacelike submanifolds of general dimensional Minkowski spaces, with the corresponding conclusion being that generically the possible directions in which a magnetic geodesic can approach a point with a lightlike tangent space form a subset in the space of all directions that has codimension at least 1.