1. Introduction

The simple plane pendulum constitutes an important physical system whose analytical solutions are well known. Historically the first systematic study of the pendulum is attributed to Galileo Galilei, around 1602. Thirty years later he discovered that the period of small oscillations is approximately independent of the amplitude of the swing, property termed as isochronism, and in 1673 Huygens published the mathematical formula for this period. However, as soon as 1636, Marin Mersenne and René Descartes had stablished that the period in fact does depend of the amplitude ¹. The mathematical theory to evaluate this period took longer to be established.

The Newton second law for the pendulum leads to a nonlinear differential equation of second order whose solutions are given in terms of either Jacobi elliptic functions or Weierstrass elliptic functions^{1,2,3,4,5,6,7}. There are several textbooks on classical mechanics ^8,9,10, and recent papers ^11,12,13, that give account of these solutions. From the mathematical point of view the subject of interest is the one of elliptic curves such as y ² = (1-x²)(1-k² x²), with k2≠0,1, the corresponding elliptic integrals∫0xdx/y and the elliptic functions which derive from the inversion of them. Generically the domain of the elliptic functions is the complex plane C and they depend also on the value of the modulus k. The theory began to be studied in the mid eighteenth century and involved great mathematicians such as Fagnano, Euler, Gauss and Lagrange. The cornerstone in its development is due to Abel ¹⁴ and Jacobi ^15,16, who replaced the elliptic integrals by the elliptic functions as the object of study. Since then they both are recognized jointly as the mathematicians that developed the elliptic functions theory in their current form and to the theory itself as one of the jewels of nineteen-century mathematics.

Because the solutions to the simple pendulum problem are given in terms of elliptic functions and the founder fathers of the subject taught us all the interesting properties of these functions, it can be concluded that all the characteristics of the different type of motions of the pendulum are known. This is strictly true, however most of the references on elliptic functions (see for instance ^2,3,4,5,6,7 and references therein) focus, as it should be, on its mathematical properties, applying just some of them to the simple pendulum as an example. In this paper we review part of the analysis made by Klein ¹⁷, who studied the properties that the transformations of the modular group Γ and its congruence subgroups of finite index Γ(N) have on the modular parameter τ, being the latter a function of the quarter periods K and K_c which in turn are determined by the value of the square modulus k². Our main interest in this paper is to accentuate the physical meaning that these transformations have in the specific case of the simple pendulum, in our opinion this is a piece of analysis missing in the literature.

For our purposes the relevant mathematical result is that the congruence subgroup of level 2, denoted as Γ(2), is of order six in Γ and therefore a fundamental cell for Γ(2) can be formed from six copies of any fundamental region F of Γ produced by the action of the six elements on the set of modular parameters τ that belong to F. Each of these copies is distinguished from each other, according to the functional form of the modulus k² the six transformations leave invariant, being they: k², 1- k², 1/ k², 1-1/ k², 1/(1- k²) and k²/(k² -1). Interestingly these kind of relations appear in other areas of physics under the concept of duality transformations, nomenclature we will use here. This result can be understood from different mathematical points of view and provides a link between concepts such as lattices, complex structures on the topological torus T2, the modular group Γ and elliptic functions. In the appendices we review briefly the basics of these concepts in order to keep the paper self contained as possible, emphasizing in every moment its role in the solutions of the simple pendulum. From the physical point of view, the pendulum can follow basically two kind of motions (with the addition of some limit situations), the specific type of motion depends entirely on the value of the total mechanical energy kE2, if 0<kE2<1 the motion is oscillatory and if 1<kE2<∞ the motion is circulatory. Therefore in the problem of the simple pendulum, there are two relevant parameters, the square modulus k² of the elliptic functions that parameterize its solutions in terms of the time variable, and the total mechanical energy of the motion kE2. As we will discuss throughout the paper, the relation between these two parameters is not one-to-one due to the duality relations between the different invariant functional forms of k². For instance, for an oscillatory motion whose energy is 0<kE2<1, it is possible to express the solution in terms of an elliptic Jacobi function whose square modulus is k², 1 - k², 1/k², etc., in other words, the duality symmetries between the functional forms of the square modulus k² induce different equivalent ways to write the solution for a specific physical motion of the pendulum. The nature of the time variable also plays an important role in the equivalence of solutions, it turns out that whereas some solutions are functions of a real time, others are functions of a pure imaginary time. In this paper we will discuss all these issues and we will write down explicitly several equivalent solutions to describe a specific pendulum motion. These results constitute an example in classical mechanics of a broader concept in physics termed under the name dualities. It is worth mentioning that some of the results we present here are already scattered throughout the mathematical literature but our exposition collects them together and is driven by a golden rule in physics that demands to explore all the physical consequences from symmetries. Notwithstanding some formulas have been worked out specifically for building up the arguments given in here and to the best knowledge of the author they are not present in the available literature. As an example, we obtain a single solution that describes the motions of the simple pendulum as function of a pure imaginary time parameter, and we show it can be obtained through an S-duality transformation from a single formula that describes the motions of the simple pendulum for all permissible values of the total energy and which is function of a real time variable.

In a general context the duality symmetries we refer to, involve the special linear group SL(2,Z) and appear often in physics either as a symmetry of a theory or as a relationship among two different theories. Typically these discrete symmetries relate strong coupled degrees of freedom to weakly coupled ones and vice versa, and the relationship is useful when one of the two systems so related can be analyzed, permitting conclusions to be drawn for its dual by acting with the duality transformations. There is a plethora of examples in physics that obey duality symmetries, which have led to important developments in field theory, gravity, statistical mechanics, string theory etc. (for an explicit account of examples see for instance ¹⁸ and references therein). As a manner of illustration let us mention just two examples of theories that own duality symmetries: i) in string theory appear three types of dualities, and the one that have the properties described above goes by the name S-duality, being the S group element, one of the two generators of the group SL(2,Z) ¹⁹. In this case the modular parameter τ is given by the coupling constant and therefore the S-duality relates the strong coupling regime of a given string theory to the weak coupling one of either the same string theory or another string theory. It is conjectured for instance that the type I superstring is S-dual to the SO(32) heterotic superstring, and that the type IIB superstring is S -dual to itself. ii) In 2D systems there is a broad class of dual relationships for which the electromagnetic response is governed by particles and vortices whose properties are similar. In particular for systems having fermions as the particles (or those related to fermions by the duality) the vortex-particle duality implies the duality group is the level-two subgroup Γ₀(2) of PSL(2,Z) ²⁰. The so often appearance of these duality symmetries in physics is our main motivation to heighten the fact that in classical mechanics there are systems like the simple pendulum whose motions can be described in different equivalent forms related by duality symmetries.

The structure of the paper is as follows. In Sec. 2 we summarize the real time solutions of the simple pendulum system in terms of elliptical Jacobi functions. The relations between solutions with real time and pure imaginary time in terms of the S group element of the modular group Γ are exemplified in Sec. 3 and the whole web of dualities is discussed in Sec. 4. We make some final remarks in 5. There are two appendices, Appendix A is dedicated to define the modular group, its congruence subgroups and its relation to double lattices whereas in Appendix B we give some properties of the elliptic Jacobi functions that are relevant for the analysis of the solutions of the simple pendulum.

2. Real time solutions

The Lagrangian for a pendulum of point mass m and length l, in a constant downwards gravitational field, of magnitude -g g>0, is given by

where θ is the polar angle measured counterclockwise respect to the vertical line and θ˙ stands for the time derivative of this angular position. Here the zero of the potential energy is set at the lowest vertical position of the pendulum, for which θ=2nπ, with n∈Z. The equation of motion for this system is

This equation can be integrated once giving origin to a first order differential equation, whose physical meaning is the conservation of energy

Physical solutions exist only if E≥0. We can rewrite this equation of conservation, in dimensionless form, in terms of the dimensionless energy parameter: kE2≡(E/2mgl), and the dimensionless real time variable: x≡(g/l)t∈R, obtaining

Analyzing the potential, it is concluded that the pendulum has four different types of solutions depending of the value of the constant kE2. The analytical solutions in two of the four cases are given in terms of Jacobi elliptic functions and can be found for instance in ^{5,7,8,9,10,11,12,13}. The other two cases can be considered just as limit situations of the previous two. The Jacobi elliptic functions are doubly periodic functions in the complex z-plane (see Appendix B for a short summary of the basic properties of these functions), for example, the function sn(z, k) of square modulus 0 < k² < 1, has the real primitive period 4K and the pure imaginary primitive period 2iK_c , where the so called quarter periods K and K_c are defined by the Eqs. (B.2) and (B.5) respectively. The properties of the different solutions are as follows:

where the square modulus k² of the elliptic functions is given directly by the energy parameter: k2≡kE2. Here x₀ is a second constant of integration and appears when Eq. (4) is integrated out. It means physically that we can choose the zero of time arbitrarily. Derivatives of the basic Jacobi elliptic functions are given in (B.11).

Without loss of generality, in our discussion we consider that the lowest vertical point of the oscillation corresponds to the angular value θ=0, and therefore that θ takes values in the closed interval [-θm,θm], where 0<θm<π is the angle for which θ˙m=0. This means that: sinθ2∈[-sinθm2,sinθm2], where according to Eq. (4), sin2(θm/2)=kE2<1. Now according to (5) the solution is obtained by mapping ²⁷: sin(θ/2)→kE sn(x-x0,kE), where x-x0∈[-K,K], or equivalently: snx-x0,kE∈[-1,1]. With this map we describe half of a period of oscillation. To describe the another half, without loss of generality, we can extend the mapping in such a way that for a complete period of oscillation, x-x0∈[-K,3K]. Because the Jacobi function cn x-x0,kE∈[-1,1], the dimensionless angular velocity ω(x) is restricted to values in the interval [-2kE,2kE].

As an example we can choose x₀ = K, so at the time x = 0, the pendulum is at minimum angular position θ(0)=-θm, with angular velocity ω(0) = 0. The pendulum starts moving from left to right, so at x = K it reaches the lowest vertical position θ(K)=0 at highest velocity ω(K) = 2k_E and at x = 2K it is at maximum angular position θ(2K)=θm with velocity ω(2K) = 0. At this very moment the pendulum starts moving from right to left, so at x = 3K it is again at θ(3K)=0 but now with lowest angular velocity ω(3K) = -2k_E and it completes an oscillation at x = 4K when the pendulum reaches again the point θ(4K)=-θm with zero velocity (see Fig. 1). We can repeat this process every time the pendulum swings in the interval [-θm,θm], in such a way that the argument of the elliptic function sn(x, k_E ), becomes defined in the whole real line R. It is clear that the period of the movement is 4K, or restoring the dimension of time, 4Kg/l.

Figure 1. The first set of graphs represents an oscillatory motion of energy kE2=3∕4 with x0=K≈2.1565. θx is given by the magenta graph and it oscillates in the interval θx∈-2π/3,2π∕3. The angular velocity ω(x) is showed in blue and takes values in the interval ωx∈-3, 3. Both graphs have period 4K. The second set of graphs represents a couple of circulating motions both of energy kE2=4∕3 with x0=K/kE≈1.8676. For the motion in the counterclockwise direction, the monotonic increasing function θx is plotted in magenta, for the time interval x∈0, 2x0 it takes values in the interval θx∈[-π, π, whereas for the interval x∈2x0, 4x0 it takes values in the interval θx∈[π, 3π. The angular velocity is showed in blue, it has period 2K/k_E , is always positive and takes values in the interval ωx∈2∕3, 4∕3. The other two plots represent a similar motion in the clockwise direction. 

Of course the value of x₀ can be set arbitrarily and it is also possible to parameterize the solution in such a way that at zero time x = 0, the motion starts in the angle θm instead of -θm. In this case the mapping of a complete period of oscillation can be defined for instance in the interval x-x0∈[K,5K] and the initial condition can be taken as x₀ = -K. In the discussion of the following section we will set x₀ = 0, so at time x = 0, the pendulum is at the lowest vertical position (θ(0)=0) moving from left to right.

● Asymptotical motion (kE2=1 and θ˙≠0): In this case the angle θ takes values in the open interval (-π, π) and therefore, sin(θ/2)∈(-1,1). The particle just reach the highest point of the circle. The analytical solutions are given by

The sign ± corresponds to the movement from (∓π→±π). Notice that tanh(x-x ₀), takes values in the open interval (-1,1) if: x-x0∈(-∞,∞). For instance if θ→π, x - x₀→ ꝏ and tanh(x - x₀) goes asymptotically to 1. It is clear that this movement is not periodic. In the literature it is common to take x₀ = 0.

● Circulating motions (kE2>1): In these cases the momentum of the particle is large enough to carry it over the highest point of the circle, so that it moves round and round the circle, always in the same direction. The solutions that describe these motions are of the form

where the global sign (+) is for the counterclockwise motion and the (-) sign for the motion in the clockwise direction. The symbol sgn (x) stands for the piecewise sign function which we define in the form

and its role is to shorten the period of the function sn(k_E (x-x₀), 1/k_E ) by half, as we argue below. This fact is in agreement with the expression for the angular velocity ω(x) because the period of the elliptic function dn(k_E (x-x₀), 1/k_E ) is 2K/k_E instead of 4K/k _E , which is the period of the elliptic function sn(k_E (x-x₀), 1/k_E ).

The square modulus k ² of the elliptic functions is equal to the inverse of the energy parameter 0<k2=1/kE2<1. Without losing generality we can assume both that kEx-x0∈[-K,K), where K is defined in (B.2) and evaluated for k2=1/kE2 and that arcsin[sn kEx-x0,1kE]∈[-π/2,π/2). Because in this interval the function sgn[cn(k_E (x - x₀), 1/k_E )] = 1, the angular position function θ(x)∈]∓π,±π) for the global sign (±) in (9). As for the interval kEx-x0∈[K,3K), we can consider that the function arcsin[sn(kE(x-x0),1/kE)]∈(-3π/2,-π/2], and because the function sgn[cn(k _E (x - x ₀), 1/k _E )] = -1, it reflects the angular position interval, obtaining finally that θ∈[±π,±3π) for the global (±) sign in (9) (see Fig. 1). We stress that the consequence of flipping the sign of the angular interval through the sgn function is to make the function θ(x) piecewise periodic, whereas the consequence of taking a different angular interval for the image of the arcsin function every time its argument changes from an increasing to a decreasing function and vice versa, is to make the function θ(x) a continuous monotonic increasing (decreasing) function for the global sign + (-). Explicitly the angular position function changes as

It is interesting to notice that if we would not have changed the image of the arcsin function, the angular position function θ(x) would have resulted into a piecewise function both periodic and discontinuous.

The angular velocity is a periodic function whose period is given by Tcirculating=2(K/kE)g/l which means as expected that higher the energy, shorter the period. Because the image of the Jacobi function dn x,k∈[1-1kE2,1], the angular velocity takes values in the interval ω∈[2kE2-1,2kE]. An interesting property of the periods that follows from solutions (6) and (10) is that Toscillatory=kETcirculating where kE2 is the energy of a circulating motion and k2=1/kE2 is the modulus used to compute K in both cases. This is a clear hint that a relation between circulating and oscillatory solutions exists.

These are all the possible motions of the simple pendulum. It is straightforward to check that the solutions satisfy the equation of conservation of energy (4) by using the following relations between the Jacobi functions (in these relations the modulus satisfies 0 < k ² < 1) and its analogous relation for hyperbolic functions (which is obtained in the limit case k = 1)

3. Imaginary time solutions and S-duality

The argument z of the Jacobi elliptic functions is defined in the whole complex plane C and the functions are doubly periodic (see Appendix B), however in the analysis above, time was considered as a real variable, and therefore in the solutions of the simple pendulum only the real quarter period K appeared. In 1878 Paul Appell clarified the physical meaning of the imaginary time and the imaginary period in the oscillatory solutions of the pendulum ^{7,8,9,10,11,12,13,14,15,16,17,18,19,20,21}, by introducing an ingenious trick, he reversed the direction of the gravitational field: g→-g, i.e. now the gravitational field is upwards. In order the Newton equations of motion remain invariant under this change in the force, we must replace the real time variable t by a purely imaginary one: τ≡±it. Implementing these changes in the equation of motion (2) leads to the equation

Writing this equation in dimensionless form requires the introduction of the pure imaginary time variable y≡±τg/l=±ix. Integrating once the resulting dimensionless equation of motion gives origin to the following equation

which looks like Eq. (4) but with an inverted potential (see Fig. 2).

Figure 2. In the figure we show the pendulum potential (blue) for the dynamical motion parameterized with a real time variable. The inverted potential (magenta) corresponds to a dynamics parameterized by a pure imaginary time variable. 

We can solve the equation in two different but equivalent ways: i) the first option consists in writing down the equation in terms of a real time variable and then flipping the sign of the whole equation in order to have positive energies, the resulting equation is of the same form as Eq. (4), ii) the second option consists in solving the equation directly in terms of the imaginary time y. Because both solutions describe to the same physical system, we can conclude that both are just different representations of the same physics. These two-ways of working provide relations between the elliptic functions with different argument and different modulus. As we will discuss the relations among different time variables and modulus can be termed as duality relations and because the mathematical group operation beneath these relations is the S generator of the modular group PSL(2,Z), we can refer to this duality relation as S-duality.

3.3. Equivalent solutions

In the following discussion we will assume without losing generality that 0<k2≤1/2 and therefore that its complementary modulus is defined in the interval 1/2≤kc2<1. The cases where 1/2≤k2<1 and therefore where 0<kc2≤1/2 can be obtained from the case we are considering by interchanging to each other the modulus and the complementary modulus k2↔kc2.

● Oscillatory motion: Let us consider oscillatory solutions for total mechanical energy 0<kE2=k2≤1/2. Solutions for these motions can be expressed in terms of either i) a real time variable and given by Eq. (20), or ii) in terms of a pure imaginary time variable. In the latter case the suitable constant is ỹ0=iK-Kc and according to the Eq. (24) and due to the equivalence of solutions we have

θkx≡2arcsin⁡k snx, k=2 ×arcsin⁡dn ix -iK+Kc, kc≡θkcix. (25)

This result is very interesting, it is telling us that any oscillatory solution can be represented as an elliptic function either of a real time variable or a pure imaginary time variable and although they have the same energy, they differ in the value of its modulus. For solutions with real time the square modulus coincides with the energy kE2 and for solutions with pure imaginary time, the square modulus is equal to 1-kE2. It is clear that the modulus of the two representations of an oscillatory solution satisfies the relation

k2+kc2=1. (26)

As discussed in Appendix B, the elliptic function dn(z, k_c ) has an imaginary period 4iK, therefore the period of the imaginary time oscillatory motion is 4iKg/l, which is in complete agreement with the periods 4Kg/l for the solutions with real time. From Eqs. (25) it is straightforward to compute the angular velocity in terms of an elliptic function whose argument is a pure imaginary time variable (see Table I). A similar result is obtained for an oscillatory motion with energy kE2=kc2.

Table I. The third column shows the solutions to the simple pendulum problem in terms of a real time variable when the total mechanic energy of the motion and the square modulus of the Jacobi elliptic function are the same. The fourth column shows its S-dual solutions in terms of a pure imaginary time variable. 

Two final comments are necessary, first in the general solutions (20) and (24) the ± signs appeared, however in (25) there is not reference to them. This happen because they are explicitly necessary only in the circulating motions. In the case of oscillatory motions the (-) sign can be absorbed in the solution by rescaling the time variable in both cases (real and pure imaginary time). Regarding the elliptic function inside the sign function it does not appear because in the case of (20) we have sgn[dn(x, k)] = 1 and also in (24) sgn[k_c sn(ix-iK_c +K, k)] = 1.

● Circulating motion: For the circulating motion we must also separate the energy intervals in two cases. If we are considering the solutions (20) which have real time variable, the corresponding energy intervals are 1<kE2=1/kc2≤2 and 2≤kE2=1/k2<∞. On the other side, if the solution involves a pure imaginary time variable (Eq. (23)) the relevant energies take values in the intervals -1≤1-kE2=-k2/kc2<0, and -∞<1-kE2=-kc2/k2≤-1. Explicitly we have for the first interval

θ1∕kcx≡2 sgn dn x, 1kcarcsin⁡1kc sn x, 1kc=2 sgn i, kkc sn ix-iK, ikkc×arcsin⁡dn ix -iK, ikkc≡θik/kcix. (27)

Notice that in a similar way to the oscillatory case, we have the following relations between the sum of the square modulus

1kc2-k2kc2=1. (28)

Analogous relations can be found for the solutions with energy kE2=1/k2 and for motions in the clockwise direction.

3.4. S group element as member of PSL(2,Z)

It is possible to reach the same conclusions as in the previous subsection but this time following a slightly different path. In Appendix B we have summarized the action of the different group elements of PSL(2,Z) on the Jacobi elliptic functions, in particular the action of the S group element. Starting for instance with a solution involving a real time variable and applying the action of the S group element, it is possible to obtain the corresponding solution in terms of a pure imaginary time variable. As we will show, the obtained results coincide with the ones we have discussed.

● Oscillatory motion: In this case the starting point is the solution (5) and its time derivative (6) which depends on a real time variable and describe an oscillatory pendulum solution with energy kE2. To fix the discussion we choose x ₀ = 0. Applying the Jacobi’s imaginary transformations Eqs. (B.13) which are the transformations generated by the S generator of the PSL(2,Z) group, we obtain

k sn x, k=-ik sc ix, kc=-k ndix +iK, kc=dn ix-iK+Kc, kc (29)

k cn x, k=k nc ix, kc=-ik kc sdix +iK, kc=-ikc cn ix-iK+Kc, kc (30)

recovering relation (25) with their respective expressions for its time derivative. Notice that although the transformed functions have modulus k_c they satisfy

dn2(ix-iK+Kc,kc)-kc2cn2(ix-iK+Kc,kc)=k2, (31)

which is telling that the solution is indeed of oscillatory energy kE2=k2 as it should be. An analogous result is obtained if we start instead with a solution of modulus kc2 and real time variable.

● Circulating case: In the circulating case we have a similar story, under an S transformation the circulating solutions (9)-(10) lead to the set

sgn dn kx, 1k1ksn kx, 1k=sgn ikck sn ix-iK, ikck×dn ix-iK, ikck, (32)

cn kx, 1k=kc cn ix-iK, ikck (33)

which coincide with the solutions (24) for a choice of the constant ỹ0=iK.

4. Web of dualities

4.2 The lattices domain

At this point it is convenient to discuss the domain of the lattices that play a role in the elliptic Jacobi functions and therefore in the solutions of the simple pendulum. As discussed in the Appendix A, the quarter periods of a Jacobi elliptic function whose square modulus is in the interval 0<k2≤1/2, generate vertical lattices represented by a modular parameter of the form τ = iK_c /K. The point τ = i is associated to the case where the rectangular lattice becomes square and corresponds to the value k² = 1/2. The set of all these lattices (black line in Fig. 3) is represented in the complex plane by the left vertical boundary of the region F1 (Fig. 5) since the quotient KcK∈[1,∞). Acting on these values of the modular parameter with the six group elements of PSL(2,Z/2Z) produce the whole set of values of the modular parameter (Fig. 3) that are consistent with the elliptic Jacobi functions. For example, acting with the S group element of Γ on the vertical line τ = iK_c /K, generates the blue vertical line described mathematically by the set of modular parameters τ = iK/K_c , with K/Kc∈(0,1]. It is clear that the set of six lines is a subset of the F2 fundamental region and constitutes the whole lattice domain of the elliptic Jacobi functions.

Figure 3. Figure shows the whole domain of values that the modular parameter τ can take for the Jacobi elliptic functions. This domain is a subset of the F2 fundamental region (Fig. 2). Black dots represent the values of the square modulus k² = 1/2, 1 - k² = 1/2, 1/k² = 2, 1/(1 - k²) = 2, 1 - 1/k² = -1 and k²/(k² - 1) = -1. 

Figure 4. Tessellation of H. The fundamental region F is represented by the shaded area and the heavy part of the boundary. This region is mapped to the whole upper plane C by the modular group Γ. The region can be viewed as a complete list of the inequivalent complex structures on the topological torus since conformal equivalence of tori is determined by the modular equivalence of their period ratios. In the figure we show some copies of the fundamental region obtained by application of some group elements of PSL(2,Z). 

Figure 5. Fundamental cell F2 for Γ(2). The heavy part of the figure is retained, the rest is not. In particular the cusps -1, 0, 1 and iꝏ are excluded. 

In Table II we give the numerical values (approximated) of the generators of the fundamental cell as well as the modular parameter for some values of the square modulus.

Table II. Approximated numerical values of the periods and the modular parameter for some real values of the modulus of the Jacobi elliptic functions. The values k = 0 and k = 1 correspond to limit situations where one of the two periods is lost. The value k² = 1/2 is known as a fixed point, it belongs both to the boundary of the regions F1 and S of the Fig. 5 and is represented by the black dot whose coordinates are (0, i) in Fig. 3. The value k² = 2 is degenerated in the sense it can be represented by two different types of fundamental cells, in one case the cell belongs to the boundary of the region ST S and in the another case it belongs to the boundary of ST^-1. The fundamental cell for some values of k² in this table are plotted in Fig. 6. 

As a conclusion, for every value of the parameter 0<k2≤1/2 there are six normal lattices related one to each other by transformations of the modular group. Therefore each solution of the simple pendulum with real time variable, showed in Table I, can be written in six different but equivalent ways, where each one of the six forms is in one to one correspondence with one of the six normal lattices. Their S-dual solutions (see Table I) which are functions of a pure imaginary time are just one of the six different ways in which solutions can be written.

4.4. A single normal lattice

It is natural to wonder about the minimum number of normal lattices needed to express all the solutions of the simple pendulum. Due to the duality symmetries between lattices this number is one. As an example, if we now use the S-duality to relate the normal horizontal lattice iL^* to the normal vertical lattice L^* , the horizontal lines that compose the domain in the horizontal lattice becomes vertical lines in the vertical lattices, which means to consider solutions with imaginary time in L ^* . Thus we can end up with only one normal lattice and in order to have the four different types of solutions, it is necessary to consider the whole domain of the lattice, i.e. both vertical lines (imaginary time) and horizontal lines (real time) and on each set of lines to consider two different solutions one oscillatory and one circulating. For completeness in Table III we give the four type of solutions in terms of only one value of the modulus.

Table III. Solutions to the simple pendulum problem written in a unique lattice of square modulus 0 < k²≤ 1/2. 

It is clear that we can express all the solutions also for the other five different functional forms of the square modulus.

5. Final remarks

In this paper we have addressed the meaning of the fact that the complex domain of the solutions of the simple pendulum is not unique and in fact they are related by the PSL(2,Z/2Z) group, finding that the important issue for express the solutions is the relation between the values of the square modulus k² of the Jacobi elliptic functions, and the value of the total mechanical energy kE2 of the motion of the pendulum. Due to the symmetry we conclude that there are six different expressions of the square modulus that are related one to each other trough the six group elements of PSL(2,Z/2Z). These six group actions can be termed as duality-transformations and therefore we have six dual representations of k². As a consequence there are six different but equivalent ways in which we can write a specific pendulum solution, and abusing a little bit of the language we could say there are duality relations between solutions. This analysis teach us the lesson that we can restrict the domain of lattices to the ones whose modular parameter is in the pure imaginary interval τ∈i(1,∞), or equivalently that we can express every solution of the simple pendulum either oscillatory or circulating with Jacobi elliptic functions whose value of the square modulus is in the interval 0<k2≤1/2 (see Table III).

It is well known that there are several physical systems in different areas of physics whose solutions are also given by elliptic functions, for instance in classical mechanics some examples are the spherical pendulum, the Duffing oscillator, etc., in Field Theory the Korteweg de Vries equation, the Ising model, etc., ^12,18. It would be very interesting to investigate on similar grounds to the ones followed here, the physical meaning of the symmetries of the elliptic functions in these systems.