The curvature κ, of a given curve γ : I → Mn in a Riemannian manifold, can be interpreted as the tension that γ receives at each point as a result of the way it is immersed in the surrounding space. In 1740, Bernoulli proposed a simple geometric model for an elastic curve in , according to which an elastic curve or elastica is a critical point of the elastic energy functional . Elastic curves in were already classified by Euler in 1743 but it was not until 1928 that they were also studied in by Radon, who derived the Euler-Lagrange equations and showed that they can be integrated explicitly. The elastica problem in real space forms has been recently considered using different approaches (see  and ).
Are there other interesting elastic curves? This question has been answered affirmatively by Marinov et al. in  where the Sturmian spirals in the Euclidean plane were described explicitly. There they have studied also the curves which belong to the class of the so called generalized Sturmian spirals which obey to the elastica equation. Eventually, they found analytical formulas for their parameterizations and presented a few illustrative plots.
Here, we are interested in elastic spacelike and timelike curves with tension in the Lorentz-Minkowski plane and derive the explicit vectorial equations of the respective generalized Sturmian spirals which belong to this class. By going to the three-dimensional pseudo-Euclidean space one can also study the third class of elastic Sturmian spirals on the lightlike cone in the spirit of , but this task is beyond the present study. The description of the generalized Sturmian spirals in the Lorentz-Minkowski plane can be found in .
The Lorentz-Minkowski plane is the Euclidean plane equipped with an indefinite flat metric g given by the infinitesimal distance
where (x1, x2) are the rectangular coordinates in . Recall that any given vector can be either spacelike if g(v, v) > 0, timelike if g(v, v) < 0 or null (lightlike) if g(v, v) = 0. The norm of a vector v is provided by the formula Two vectors v and w are said to be orthogonal, if g(v, w) = 0. An arbitrary curve γ(s) in , can locally be spacelike or timelike, if all its velocity vectors γ′(s) are, respectively, spacelike or timelike. Any spacelike or timelike curve γ can be parametrized by the so called arc-length parameter s for which g(γ′(s), γ′(s)) = ±1 (see ). In particular, every spacelike curve α(s) in the Lorentz-Minkowski plane can be represented in the form (cf. [11, 12] and )(1)
and the corresponding Frenet vector fields along it are given by the formulas(2)
It is an easy task to see that they obey to the relations(3)
in which the function(4)
is the curvature of the curve.
The intrinsic equation of the spacelike elastic curves in the Lorentz-Minkowski plane reads(5)
in which the overdots mean derivatives with respect to s and λ is the tension constant.
Similarly, any timelike curve β(s) in the Lorentz-Minkowski plane can be parameterized as follows(6)
This time the Frenet vector fields(7)
and the curvature
satisfy the relations(8)
The respective intrinsic equation of the timelike elastic curves is(9)
with the same notation.
3 Elastic curves in the Lorentz-Minkowski plane
Below we will find the explicit parameterizations (up to quadratures) of the spacelike and timelike elastic curves in the Lorentz-Minkowski plane.
Let be a spacelike elastic curve in the Lorentz-Minkowski plane. Then its position vector is given by the formulas
where λ is the tension and the constants c1, c2 ∈ ℝ are such that .
Let β : be a timelike elastic curve in Lorentz-Minkowski plane. Then the position vector is given by the formulas
in which λ is the tension and the constants c1, c2 ∈ ℝ are such that .
4 Spacelike elastic Sturmian spirals
Let α : be a spacelike elastic Sturmian spiral in Lorentz-Minkowski plane which means that its curvature is given by the function κ = σ/r, where and σ ∈ ℝ+. Before making use of the intrinsic equation (5) let us integrate it with respect to s. This produces the equation(19)
in which E denotes the integration constant which can be interpreted as energy. For the special case of the Sturmian spirals it can be rewritten in the form(20)
As the tension λ is a physical property it can be assumed to be a positive constant. Therefore, there are only two cases to be considered, depending on the sign of E.
Case 1. Let us start by assuming that both E and λ are positive. We will write this fact formally as
with a and c being non-zero positive constants which comply with the condition a > c. Substituting the above expressions of E and λ in (20) we obtain(21)
As a consequence, the solutions of the differential equation (21) are either(22)
where the first slot in the Jacobian sinus elliptic function sn ) is reserved for the argument, the second one for the so called elliptic modulus which is a real number between zero and one.
Now, when (22) holds we have(24)
The equality on the right hand side is satisfied for s0 = 2K(k)/a, where K(k) is the complete elliptic integral of the first kind and κ(s0) = a. Then using (24) in (13) we can conclude that (c1, c2) = ((c2-a2})/4,0) . Taking into account all above and Theorem 3.1 we end up with the parameterization(25)
where (.,.) is the Jacobian amplitude function, cn and dn are the remaining Jacobian elliptic functions and F(.,.), E(.,.) denote the incomplete elliptic inegrals of the first, respectively second kind (for more details see ).
Now, let us switch to the solution presented in (23). Proceeding in the same manner we obtain(26)(27)
Case 2. When E is negative, we can write respectively(28)
Inserting E and λ in (20) we obtain the equation(29)
The solution of the above differential equation is(30)
5 Timelike elastic Sturmian spirals
Let β : be a timelike elastic Sturmian spiral in the Lorentz-Minkowski plane. Integrating the intrinsic equation (9) we have(33)
where E as before is the constant of integration (the energy). Since β is a Sturmian spiral, the curvature function κ is given by the function , where and σ ∈ ℝ+. Rewriting appropriately the equation (33), we find(34)
Again, depending on the sign of E there are two cases.
Case 1. Both E and λ are positive which is ensured by writing
Substituting these values of E and λ in (34), we obtain(35)
The solution of the above differential equation is of the form(36)
Therefore we get(37)(38)
Case 2. Assuming that E is negative, that is(39)
we find that the intrinsic equation of the sought curves is(40)
Regarding the solution of the above differential equation we have(41)
From (41) we have also(42)(43)
Both cases of timelike elastic Sturmian spirals in the Lorentz-Minkowski plane are pictured in Figure 2 for a specific choice of the parameters.
The Lorentz-Minkowski plane is equipped with an indefinite metric and one should expect different types of curves to those in the Euclidean case. Elsewhere Marinov et al  have studied the elastic Sturmian spirals in the Euclidean plane. By drawing inspiration from this work we have considered here the spacelike and timelike elastic Sturmian spirals in the Lorentz-Minkowski plane . Our main results are the derivation of the explicit parametric equations for spacelike and timelike elastic Sturmian spirals in . We have presented also in Figure 1 and Figure 2 some graphical illustrations of these curves that are realized using MathematicaO.
Last but not least, we believe that the results presented in this paper suggest that some other curves like the elastic Serret's curves, elastic Bernoulli's Lemniscate and the generalized elastic curves whose curvature depend on the distance from the origin deserve to be studied in some details in the Lorentz-Minkowski plane as well. We are planning to report on realization of this program elsewhere.
The first author would like to thank TUBITAK for the financial support during his PhD study. The third named author is partially supported by TUBITAK (The Scientific and Technological Research Council of Turkey) within the frame of Programme 2221.
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Published Online: 2016-12-30
Published in Print: 2016-01-01