Duality symmetries behind solutions of the classical simple pendulum

Transcription

1 EDUCATION Revista Mexicana e Física E 64 08) 05 JULY DECEMBER 08 Duality symmetries behin solutions of the classical simple penulum R. Linares Romero Departamento e Física, Universia Autónoma Metropolitana Iztapalapa, San Rafael Atlixco 86, 09340, Ciua e México, México. lirr@xanum.uam.mx Receive September 07; accepte 6 January 08 Describing the motion of the classical simple penulum is one of the aims in every unergrauate classical mechanics course. Its analytical solutions are given in terms of elliptic functions, which are oubly perioic functions in the complex plane. The inepenent variable of the solutions is time an it can be consiere either as a real variable or as a purely imaginary one, which introuces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions this symmetry is coifie in the functional form of its moulus, an is escribe mathematically by the six imensional coset group Γ/Γ) where Γ is the moular group an Γ) is its congruence subgroup of secon level. A iscussion of the physical consequences that this symmetry has on the motions of the simple penulum is presente in this contribution an it is argue they have similar properties to the ones terme as uality symmetries in other areas of physics, such as fiel theory an string theory. Thus by stuying eeper a very familiar mechanical system, it is possible to get an insight to more abstract physical an mathematical concepts. In particular a single solution of pure imaginary time for all allowe values of the total mechanical energy is given an obtaine as the S-ual of a single solution of real time, where S stans for the S generator of the moular group. Keywors: Penulum solutions; ualities; moular group. PACS: 45.0.D-; 45.0.Jj; a. Introuction The simple plane penulum constitutes an important physical system whose analytical solutions are well nown. Historically the first systematic stuy of the penulum is attribute to Galileo Galilei, aroun 60. Thirty years later he iscovere that the perio of small oscillations is approximately inepenent of the amplitue of the swing, property terme as isochronism, an in 673 Huygens publishe the mathematical formula for this perio. However, as soon as 636, Marin Mersenne an René Descartes ha stablishe that the perio in fact oes epen of the amplitue []. The mathematical theory to evaluate this perio too longer to be establishe. The Newton secon law for the penulum leas to a nonlinear ifferential equation of secon orer whose solutions are given in terms of either Jacobi elliptic functions or Weierstrass elliptic functions [-7]. There are several textboos on classical mechanics [8-0], an recent papers [-3], 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 = x ) x ), with 0,, the corresponing elliptic integrals x x/y an the elliptic functions 0 which erive from the inversion of them. Generically the omain of the elliptic functions is the complex plane C an they epen also on the value of the moulus. The theory began to be stuie in the mi eighteenth century an involve great mathematicians such as Fagnano, Euler, Gauss an Lagrange. The cornerstone in its evelopment is ue to Abel [4] an Jacobi [5,6], who replace the elliptic integrals by the elliptic functions as the object of stuy. Since then they both are recognize jointly as the mathematicians that evelope the elliptic functions theory in their current form an to the theory itself as one of the jewels of nineteen-century mathematics. Because the solutions to the simple penulum problem are given in terms of elliptic functions an the founer fathers of the subject taught us all the interesting properties of these functions, it can be conclue that all the characteristics of the ifferent type of motions of the penulum are nown. This is strictly true, however most of the references on elliptic functions see for instance [-7] an references therein) focus, as it shoul be, on its mathematical properties, applying just some of them to the simple penulum as an example. In this paper we review part of the analysis mae by Klein [7], who stuie the properties that the transformations of the moular group Γ an its congruence subgroups of finite inex ΓN) have on the moular parameter τ, being the latter a function of the quarter perios K an K c which in turn are etermine by the value of the square moulus. Our main interest in this paper is to accentuate the physical meaning that these transformations have in the specific case of the simple penulum, 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, enote as Γ), is of orer six in Γ an therefore a funamental cell for Γ) can be forme from six copies of any funamental region F of Γ prouce by the action of the six elements on the set of moular parameters τ that belong to F. Each of these copies is istinguishe from each other, accoring to the functional form of the moulus the six transformations leave invariant, being they:,, /, /, / ) an / ). Interestingly these in of relations appear in

2 06 R. LINARES ROMERO other areas of physics uner the concept of uality transformations, nomenclature we will use here. This result can be unerstoo from ifferent mathematical points of view an provies a lin between concepts such as lattices, complex structures on the topological torus T, the moular group Γ an elliptic functions. In the appenices we review briefly the basics of these concepts in orer to eep the paper self containe as possible, emphasizing in every moment its role in the solutions of the simple penulum. From the physical point of view, the penulum can follow basically two in of motions with the aition of some limit situations), the specific type of motion epens entirely on the value of the total mechanical energy E, if 0 < E < the motion is oscillatory an if < E < the motion is circulatory. Therefore in the problem of the simple penulum, there are two relevant parameters, the square moulus of the elliptic functions that parameterize its solutions in terms of the time variable, an the total mechanical energy of the motion E. As we will iscuss throughout the paper, the relation between these two parameters is not one-to-one ue to the uality relations between the ifferent invariant functional forms of. For instance, for an oscillatory motion whose energy is 0 < E <, it is possible to express the solution in terms of an elliptic Jacobi function whose square moulus is,, /, etc., in other wors, the uality symmetries between the functional forms of the square moulus inuce ifferent equivalent ways to write the solution for a specific physical motion of the penulum. 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 iscuss all these issues an we will write own explicitly several equivalent solutions to escribe a specific penulum motion. These results constitute an example in classical mechanics of a broaer concept in physics terme uner the name ualities. It is worth mentioning that some of the results we present here are alreay scattere throughout the mathematical literature but our exposition collects them together an is riven by a golen rule in physics that emans to explore all the physical consequences from symmetries. Notwithstaning some formulas have been wore out specifically for builing up the arguments given in here an to the best nowlege of the author they are not present in the available literature. As an example, we obtain a single solution that escribes the motions of the simple penulum as function of a pure imaginary time parameter, an we show it can be obtaine through an S-uality transformation from a single formula that escribes the motions of the simple penulum for all permissible values of the total energy an which is function of a real time variable.. In a general context the uality symmetries we refer to, involve the special linear group SL, Z) an appear often in physics either as a symmetry of a theory or as a relationship among two ifferent theories. Typically these iscrete symmetries relate strong couple egrees of freeom to wealy couple ones an vice versa, an the relationship is useful when one of the two systems so relate can be analyze, permitting conclusions to be rawn for its ual by acting with the uality transformations. There is a plethora of examples in physics that obey uality symmetries, which have le to important evelopments in fiel theory, gravity, statistical mechanics, string theory etc. for an explicit account of examples see for instance [8] an references therein). As a manner of illustration let us mention just two examples of theories that own uality symmetries: i) in string theory appear three types of ualities, an the one that have the properties escribe above goes by the name S-uality, being the S group element, one of the two generators of the group SL, Z) [9]. In this case the moular parameter τ is given by the coupling constant an therefore the S-uality relates the strong coupling regime of a given string theory to the wea coupling one of either the same string theory or another string theory. It is conjecture for instance that the type I superstring is S-ual to the SO3) heterotic superstring, an that the type IIB superstring is S-ual to itself. ii) In D systems there is a broa class of ual relationships for which the electromagnetic response is governe by particles an vortices whose properties are similar. In particular for systems having fermions as the particles or those relate to fermions by the uality) the vortex-particle uality implies the uality group is the level-two subgroup Γ 0 ) of P SL, Z) [0]. The so often appearance of these uality symmetries in physics is our main motivation to heighten the fact that in classical mechanics there are systems lie the simple penulum whose motions can be escribe in ifferent equivalent forms relate by uality symmetries. The structure of the paper is as follows. In Sec. we summarize the real time solutions of the simple penulum system in terms of elliptical Jacobi functions. The relations between solutions with real time an pure imaginary time in terms of the S group element of the moular group Γ are exemplifie in Sec. 3 an the whole web of ualities is iscusse in Sec. 4. We mae some final remars in 5. There are two appenices, Appenix A is eicate to efine the moular group, its congruence subgroups an its relation to ouble lattices whereas in Appenix B we give some properties of the elliptic Jacobi functions that are relevant for the analysis of the solutions of the simple penulum.. Real time solutions The Lagrangian for a penulum of point mass m an length l, in a constant ownwars gravitational fiel, of magnitue g g > 0), is given by Lθ, θ) = ml θ mgl cos θ), ) where θ is the polar angle measure counterclocwise respect to the vertical line an θ stans for the time erivative of this angular position. Here the zero of the potential energy is set at the lowest vertical position of the penulum, for which θ = nπ, with n Z. The equation of motion for this system Rev. Mex. Fis. E 64 08) 05

3 DUALITY SYMMETRIES BEHIND SOLUTIONS OF THE CLASSICAL SIMPLE PENDULUM 07 is θ + g sin θ = 0. ) l This equation can be integrate once giving origin to a first orer ifferential equation, whose physical meaning is the conservation of energy E = θ ml θ + mgl sin = constant. 3) Physical solutions exist only if E 0. We can rewrite this equation of conservation, in imensionless form, in terms of the imensionless energy parameter: E E/mgl), an the imensionless real time variable: x g/l)t R, obtaining θ θ + sin = 4 x E. 4) Analyzing the potential, it is conclue that the penulum has four ifferent types of solutions epening of the value of the constant E. The analytical solutions in two of the four cases are given in terms of Jacobi elliptic functions an can be foun for instance in [5,7-3]. The other two cases can be consiere just as limit situations of the previous two. The Jacobi elliptic functions are oubly perioic functions in the complex z-plane see Appenix B for a short summary of the basic properties of these functions), for example, the function snz, ) of square moulus 0 < <, has the real primitive perio 4K an the pure imaginary primitive perio ik c, where the so calle quarter perios K an K c are efine by the Eqs. B.) an B.5) respectively. The properties of the ifferent solutions are as follows: Static equilibrium θ = 0): The trivial behavior occurs when either E = 0 or E =. In the first case, necessarily θ = 0. For the case E = we consier also the situation where θ = 0. In both cases, the penulum oes not move, it is in static equilibrium. When θ = nπ the equilibrium is stable an when θ = n + )π the equilibrium is unstable. Oscillatory motions 0 < E < ): In these cases the penulum swings to an fro, respect to a point of stable equilibrium. The analytical solutions are given by θx) = arcsin[ E snx x 0, E )], 5) θ x ωx) = E cnx x 0, E ), 6) where the square moulus of the elliptic functions is given irectly by the energy parameter: E. Here x 0 is a secon constant of integration an appears when Eq. 4) is integrate out. It means physically that we can choose the zero of time arbitrarily. Derivatives of the basic Jacobi elliptic functions are given in B.). Without loss of generality, in our iscussion we consier that the lowest vertical point of the oscillation correspons to the angular value θ = 0, an therefore that θ taes values in the close interval [ θ m, θ m ], where 0 < θ m < π is the angle for which θ m = 0. This means that: sinθ/) [ sinθ m /), sinθ m /)], where accoring to Eq. 4), sin θ m /) = E <. Now accoring to 5) the solution is obtaine by mapping [7]: sinθ/) E snx x 0, E ), where x x 0 [ K, K], or equivalently: snx x 0, E ) [, ]. With this map we escribe half of a perio of oscillation. To escribe the another half, without loss of generality, we can exten the mapping in such a way that for a complete perio of oscillation, x x 0 [ K, 3K]. Because the Jacobi function cnx x 0, E ) [, ], the imensionless angular velocity ωx) is restricte to values in the interval [ E, E ]. As an example we can choose x 0 = K, so at the time x = 0, the penulum is at minimum angular position θ0) = θ m, with angular velocity ω0) = 0. The penulum starts moving from left to right, so at x = K it reaches the lowest vertical position θk) = 0 at highest velocity ωk) = E an at x = K it is at maximum angular posi- FIGURE. The first set of graphs represents an oscillatory motion of energy E = 3/4 with x 0 = K.565. θx) is given by the magenta graph an it oscillates in the interval θx) [ π/3, π/3]. The angular velocity ωx) is showe in blue an taes values in the interval ωx) [ 3, 3]. Both graphs have perio 4K. The secon set of graphs represents a couple of circulating motions both of energy E = 4/3 with x 0 = K/ E For the motion in the counterclocwise irection, the monotonic increasing function θx) is plotte in magenta, for the time interval x [0, x 0) it taes values in the interval θx) [ π, π), whereas for the interval x [x 0, 4x 0) it taes values in the interval θx) [π, 3π). The angular velocity is showe in blue, it has perio K/ E, is always positive an taes values in the interval ωx) [/ 3, 4/ 3]. The other two plots represent a similar motion in the clocwise irection. Rev. Mex. Fis. E 64 08) 05

4 08 R. LINARES ROMERO tion θk) = θ m with velocity ωk) = 0. At this very moment the penulum starts moving from right to left, so at x = 3K it is again at θ3k) = 0 but now with lowest angular velocity ω3k) = E an it completes an oscillation at x = 4K when the penulum reaches again the point θ4k) = θ m with zero velocity see Fig. ). We can repeat this process every time the penulum swings in the interval [ θ m, θ m ], in such a way that the argument of the elliptic function snx, E ), becomes efine in the whole real line R. It is clear that the perio of the movement is 4K, or restoring the imension of time, 4K g/l. Of course the value of x 0 can be set arbitrarily an it is also possible to parameterize the solution in such a way that at zero time x = 0, the motion starts in the angle θ m instea of θ m. In this case the mapping of a complete perio of oscillation can be efine for instance in the interval x x 0 [K, 5K] an the initial conition can be taen as x 0 = K. In the iscussion of the following section we will set x 0 = 0, so at time x = 0, the penulum is at the lowest vertical position θ0) = 0) moving from left to right. Asymptotical motion E = an θ 0): In this case the angle θ taes values in the open interval π, π) an therefore, sinθ/), ). The particle just reach the highest point of the circle. The analytical solutions are given by θx) = ± arcsin[tanhx x 0 )], 7) ωx) = ± sechx x 0 ). 8) The sign ± correspons to the movement from π ±π). Notice that tanhx x 0 ), taes values in the open interval, ) if: x x 0, ). For instance if θ π, x x 0 an tanhx x 0 ) goes asymptotically to. It is clear that this movement is not perioic. In the literature it is common to tae x 0 = 0. Circulating motions E > ): 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 roun an roun the circle, always in the same irection. The solutions that escribe these motions are of the form [ θx) = ± sgn cn E x x 0 ), [ arcsin sn E x x 0 ), ωx) = ± E n E x x 0 ), E E E )] )], 9) ), 0) where the global sign +) is for the counterclocwise motion an the ) sign for the motion in the clocwise irection. The symbol sgnx) stans for the piecewise sign function which we efine in the form [ )] sgn cn E x x 0 ), E { + if 4n )K E x x = 0 )<4n + )K, if 4n + )K E x x 0 )<4n + 3)K, ) an its role is to shorten the perio of the function sn E x x 0 ), / E ) by half, as we argue below. This fact is in agreement with the expression for the angular velocity ωx) because the perio of the elliptic function n E x x 0 ), / E ) is K/ E instea of 4K/ E, which is the perio of the elliptic function sn E x x 0 ), / E ). The square moulus of the elliptic functions is equal to the inverse of the energy parameter 0< =/E <. Without losing generality we can assume both that E x x 0 ) [ K, K), where K is efine in B.) an evaluate for = /E an that arcsin[sn Ex x 0 ), / E )] [ π/, π/). Because in this interval the function sgn[cn E x x 0 ), / E )] =, the angular position function θx) [ π, ±π) for the global sign ±) in 9). As for the interval E x x 0 ) [K, 3K), we can consier that the function arcsin[sn E x x 0 ), / E )] 3π/, π/], an because the function sgn[cn E x x 0 ), / E )] =, it reflects the angular position interval, obtaining finally that θ [±π, ±3π) for the global ±) sign in 9) see Fig. ). We stress that the consequence of flipping the sign of the angular interval through the sgn function is to mae the function θx) piecewise perioic, whereas the consequence of taing a ifferent angular interval for the image of the arcsin function every time its argument changes from an increasing to a ecreasing function an vice versa, is to mae the function θx) a continuous monotonic increasing ecreasing) function for the global sign + -). Explicitly the angular position function changes as θ x + n K ) = θx) ± πn. ) E It is interesting to notice that if we woul not have change the image of the arcsin function, the angular position function θx) woul have resulte into a piecewise function both perioic an iscontinuous. The angular velocity is a perioic function whose perio is given by T circulating = K/ E ) g/l which means as expecte that higher the energy, shorter the perio. Because the image of the Jacobi function nx, ) [ /E, ], the angular velocity taes values in the interval ω [ E, E ]. An interesting property of the perios that follows from solutions 6) an 0) is that T oscillatory = E T circulating where E is the energy of a circulating motion an = /E is the moulus use to compute K in both cases. This is a clear hint that a relation between circulating an oscillatory solutions exists. These are all the possible motions of the simple penulum. It is straightforwar to chec that the solutions satisfy Rev. Mex. Fis. E 64 08) 05

5 DUALITY SYMMETRIES BEHIND SOLUTIONS OF THE CLASSICAL SIMPLE PENDULUM 09 the equation of conservation of energy 4) by using the following relations between the Jacobi functions in these relations the moulus satisfies 0 < < ) an its analogous relation for hyperbolic functions which is obtaine in the limit case = ) sn x, ) + cn x, ) =, 3) tanh x) + sech x) =, 4) sn x, ) + n x, ) =. 5) 3. Imaginary time solutions an S-uality The argument z of the Jacobi elliptic functions is efine in the whole complex plane C an the functions are oubly perioic see Appenix B), however in the analysis above, time was consiere as a real variable, an therefore in the solutions of the simple penulum only the real quarter perio K appeare. In 878 Paul Appell clarifie the physical meaning of the imaginary time an the imaginary perio in the oscillatory solutions of the penulum [7-], by introucing an ingenious tric, he reverse the irection of the gravitational fiel: g g, i.e. now the gravitational fiel is upwars. In orer the Newton equations of motion remain invariant uner 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 ) leas to the equation θ τ g sin θ = 0. 6) l Writing this equation in imensionless form requires the introuction of the pure imaginary time variable y ±τ g/l = ±ix. Integrating once the resulting imensionless equation of motion gives origin to the following equation 4 θ y ) sin θ ) = E, 7) which loos lie Eq. 4) but with an inverte potential see Fig. ). We can solve the equation in two ifferent but equivalent ways: i) the first option consists in writing own the equation in terms of a real time variable an then flipping the sign of the whole equation in orer to have positive energies, the resulting equation is of the same form as Eq. 4), ii) the secon option consists in solving the equation irectly in terms of the imaginary time y. Because both solutions escribe to the same physical system, we can conclue that both are just ifferent representations of the same physics. These two-ways of woring provie relations between the elliptic functions with ifferent argument an ifferent moulus. As we will iscuss the relations among ifferent time variables an moulus can be terme as uality relations an because the mathematical group operation beneath these relations is the S generator of the moular group P SL, Z), we can refer to this uality relation as S-uality. 3.. Real time variable If we write own 7) explicitly in terms of the real time parameter x, we obtain a conservation equation of the form 4 θ θ sin = E. 8) x The first feature of this equation is that the constant E is negative E < 0). This happens because as a consequence of the imaginary nature of time, the momentum also becomes an imaginary quantity an when it is written in terms of a real time it prouces a negative inetic energy. On the other sie the inversion of the force prouces a potential moifie by a global sign. Flipping the sign of the whole equation an enoting E = E, leas to the Eq. 4) 4 θ x ) θ + sin = E. 9) We have alreay iscusse the solutions to this equation see Sec..). However because we want to unerstan the symmetry between solutions, it is convenient to write own the ones of the circulating motions 9)-0) relating the moulus of the Jacobi elliptic functions not to the inverse of the energy but to the energy itself, which can be accomplishe by consiering that the Jacobi elliptic functions can be efine for moulus greater than one. So we can write own both the oscillatory an the circulating motions in a single expression [3] θx) = ± E sgn[nx x 0, E )] arcsin[snx x 0, E )]. 0) FIGURE. In the figure we show the penulum potential blue) for the ynamical motion parameterize with a real time variable. The inverte potential magenta) correspons to a ynamics parameterize by a pure imaginary time variable. Here the square moulus = E taes values in the intervals 0 < E < for the oscillatory motions an < E < for the circulating ones. The reason of writing own the circulating solutions in this way is because introucing another group element of P SL, Z), we can relate them to the stanar form of the solution 9) with moulus smaller than one. We shall o this explicitly in the next section. Rev. Mex. Fis. E 64 08) 05

6 0 R. LINARES ROMERO 3.. Imaginary time variable In orer to solve Eq. 7) irectly in terms of a pure imaginary time variable, it is convenient to rewrite the equation in a form that loos similar to Eq. 9), which we have alreay solve, an with this solution at han go bac to the original equation an obtain its solution. We start by shifting the value of the potential energy one unit such that its minimum value be zero. Aing a unit of energy to both sies of the equation leas to 4 θ y ) θ + cos = E. ) The secon step is to rewrite the potential energy in such a form it coincies with the potential energy of 9) an in this way allowing us to compare solutions. We can accomplish this by a simple translation of the graph, for instance by translating it an angle of π/ to the right see Fig. ). Defining θ = θ π, we obtain θ θ + sin = 4 y E. ) Solutions to this equation are given formally as θ sin = ± E sn y ỹ 0, E ). 3) Now it is straightforwar to obtain the solution to the original Eq. 7), by going bac to the original θ angle, obtaining )] θx) = ± sgn [ E sn y ỹ 0, E [ )] arcsin n y ỹ 0, E. 4) In this last expression we are assuming that Eq. 5) is vali for every allowe value of the energy E 0, ), ), or equivalently E, 0) 0, ) see Eqs. B.9) an B.44)). It is important to stress that while E has the interpretation of being an energy, E can not be interprete as such, as we will iscuss below. Notice we have enote to the integration constant in the variable y as ỹ 0 to emphasize that ỹ 0 C an is not necessarily a pure imaginary number. This happen because in contrast to the case of a real time variable where the integral along the real line x R can be performe irectly, when the variable is complex it is necessary to chose a vali integration contour in orer to eal with the poles of the Jacobi elliptic functions []. For instance, the function ny, ) has poles in y = n + )ik c mo K) for n Z, but nix + n + )K, ) is oscillatory for every x R an 0 < <. The sign function in the solutions is introuce again in orer to halve the perio of the circulating motions respect to the oscillatory ones Equivalent solutions In the following iscussion we will assume without losing generality that 0 < / an therefore that its complementary moulus is efine in the interval / c <. The cases where / < an therefore where 0 < c / can be obtaine from the case we are consiering by interchanging to each other the moulus an the complementary moulus c. Oscillatory motion: Let us consier oscillatory solutions for total mechanical energy 0 < E = /. Solutions for these motions can be expresse in terms of either i) a real time variable an given by Eq. 0), or ii) in terms of a pure imaginary time variable. In the latter case the suitable constant is ỹ 0 = ik K c an accoring to the Eq. 4) an ue to the equivalence of solutions we have θ x) arcsin[ snx, )] = arcsin[nix ik + K c, c )] θ c ix). 5) This result is very interesting, it is telling us that any oscillatory solution can be represente as an elliptic function either of a real time variable or a pure imaginary time variable an although they have the same energy, they iffer in the value of its moulus. For solutions with real time the square moulus coincies with the energy E an for solutions with pure imaginary time, the square moulus is equal to E. It is clear that the moulus of the two representations of an oscillatory solution satisfies the relation + c =. 6) As iscusse in Appenix B, the elliptic function nz, c ) has an imaginary perio 4iK, therefore the perio of the imaginary time oscillatory motion is 4iK g/l, which is in complete agreement with the perios 4K g/l for the solutions with real time. From Eqs. 5) it is straightforwar 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 obtaine for an oscillatory motion with energy E = c. Two final comments are necessary, first in the general solutions 0) an 4) the ± signs appeare, however in 5) 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 absorbe in the solution by rescaling the time variable in both cases real an pure imaginary time). Regaring the elliptic function insie the sign function it oes not appear because in the case of 0) we have sgn[nx, )] = an also in 4) sgn[ c snix ik c + K, )] =. Circulating motion: For the circulating motion we must also separate the energy intervals in two cases. If we are consiering the solutions 0) which have real time variable, the corresponing energy intervals are < E = / c an E = / <. On the other sie, if the solution involves a pure imaginary time variable Eq. 3)) the relevant energies tae values in the intervals E = Rev. Mex. Fis. E 64 08) 05

7 DUALITY SYMMETRIES BEHIND SOLUTIONS OF THE CLASSICAL SIMPLE PENDULUM /c < 0, an < E = c/. Explicitly we have for the first interval [ )] [ )] θ /c x) sgn n x, c arcsin sn x, c c ) )] = sgn [i c sn ix ik, i c [ )] arcsin n ix ik, i c θ i/c ix). 7) Notice that in a similar way to the oscillatory case, we have the following relations between the sum of the square moulus c c =. 8) Analogous relations can be foun for the solutions with energy E = / an for motions in the clocwise irection S group element as member of P SL, Z) It is possible to reach the same conclusions as in the previous subsection but this time following a slightly ifferent path. In Appenix B we have summarize the action of the ifferent group elements of P SL, 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 an applying the action of the S group element, it is possible to obtain the corresponing solution in terms of a pure imaginary time variable. As we will show, the obtaine results coincie with the ones we have iscusse. Oscillatory motion: In this case the starting point is the solution 5) an its time erivative 6) which epens on a real time variable an escribe an oscillatory penulum solution with energy E. To fix the iscussion we choose x 0 = 0. Applying the Jacobi s imaginary transformations Eqs. B.3) which are the transformations generate by the S generator of the P SL, Z) group, we obtain snx, ) = i scix, c ) = nix + ik, c ) = nix ik + K c, c ), 9) cnx, ) = ncix, c ) = i c six + ik, c ) = i c cnix ik + K c, c ), 30) recovering relation 5) with their respective expressions for its time erivative. Notice that although the transforme functions have moulus c they satisfy n ix ik+k c, c ) ccn ix ik+k c, c ) =, 3) which is telling that the solution is inee of oscillatory energy E = as it shoul be. An analogous result is obtaine if we start instea with a solution of moulus c an real time variable. Circulating case: In the circulating case we have a similar story, uner an S transformation the circulating solutions 9)-0) lea to the set sgn [ n x, )] [ = sgn i c n x, ) sn ix ik, i c sn ix ik, i c cn x, ) = c cn ix ik, i c )] ), 3) ), 33) which coincie with the solutions 4) for a choice of the constant ỹ 0 = ik. 4. Web of ualities 4.. The set of S-ual solutions We have argue that a symmetry of the equation of motion for the simple penulum leas to the possibility that its solutions can be obtaine in two ways: i) consiering a real time variable an ii) consiering a pure imaginary time variable. The solutions for energies in the interval E 0, ), ) are given by Jacobi elliptic functions, the ones for energies E 0, ) escribe oscillatory motions an the ones for energies E, ) escribe circulating ones. On the other han we also now that the Jacobi elliptic functions are oubly perioic functions in the complex plane C see Appenix B), an aitionally to the complex argument z, they also epen on the value of the moulus whose square taes values in the real line R with exception of the points 0 an. In the previous section we have iscusse that given a type of motion, for instance an oscillatory motion with energy 0 < E /, there are at least two equivalent angular functions escribing it, one with moulus = E an real time enote as θ x) in 5) an a secon one with moulus c = E an pure imaginary time enote as θ c ix). We can refer to this ual escription of the same solution as S-uality. In Table I we give the solutions for all the simple penulum motions oscillatory an circulating) in terms of real time an its S-ual solution given in terms of a pure imaginary time. The fact that the solutions involve either real time or pure imaginary time only, but not a general complex time leas to the conclusion that although the omain of the elliptic Jacobi functions are all the points in a funamental cell, or ue to its oubly perioicity, in the full complex plane C, the penulum solutions tae values only in a subset of this omain. Let us exemplify this fact for a vertical funamental cell, i.e., for values of the square moulus in the interval 0 < < /, which correspon to a normal lattice L see Appenix B). In this case the generators are given by 4K an 4iK c with K c > K. If the time variable x is real, the solutions are Rev. Mex. Fis. E 64 08) 05

8 R. LINARES ROMERO TABLE I. The thir column shows the solutions to the simple penulum problem in terms of a real time variable when the total mechanic energy of the motion an the square moulus of the Jacobi elliptic function are the same. The fourth column shows its S-ual solutions in terms of a pure imaginary time variable. Energy E Variable Real time solution Imaginary time solution 0, /] θ/ arcsin[ snx, )] arcsin[nix ik + K c, c )] ω/ cnx, ) i c cnix ik + K c, c ) [/, ) θ/ arcsin[ c snx, c)] arcsin[nix ik c + K, )] ω/ c cnx, c)] i cnix ik c + K, ), ] θ/ ± sgn[nx, / c )] arcsin[snx, / c )/ c ] ± sgn[i/ c ) snix ik c, i/ c )] arcsin[n ix ik c, i/ c )] ω/ ±/ c ) cnx, / c ) ±/ c ) cn ix ik c, i/ c ) [, ) θ/ ± sgn[nx, /)] arcsin[snx, /)/] ± sgn[i c/) snix ik, i c/)] nix ik, i c/) ω/ ±/) cnx, /) ± c/) cn ix ik, i c/) given by the function snx, ) which owns a pure imaginary perio ik c. The oscillatory solutions on the funamental cell are given generically either by arcsin[ snx x 0, )] or arcsin[ snx x 0 + ik c, )], or in general on the complex plane C the omain of these solutions is given by all the horizontal lines whose imaginary part is constant an given by nik c with n Z. Accoring to Table I, the oscillatory solutions of pure imaginary time on the same funamental cell, have energies in the interval / E = c < an are given generically by arcsin[nix ix 0 + K, ) or arcsin[nix ix 0 + 3K, ). In general the omain of these solutions in the complex plane C are all the vertical lines whose real part is constant an given by n + )K with n Z, which is in agreement with the fact that the function nz, ) owns a real perio K. Any other point in the omain of the elliptic Jacobi functions, ifferent to the ones mentione o not satisfy the initial conitions of the penulum motions. This iscussion can be extene to the horizontal funamental cells normal lattices il ) whose moulus is given by c an the ones that involve an ST S transformation an therefore a Dehn twist see Appenix B). We conclue that if we consier only solutions of real time variable such that the square moulus an the energy coincie the four types of Table I), then the corresponing omains are horizontal lines on the normal lattices L, il, L an i c L. If instea we consier the four solutions of pure imaginary time parameter, the corresponing omains are vertical lines on the normal lattices il, L, il an c L. However ue to the fact that the moular group relates the normal lattices one to each other, we can consier less normal lattices an instea consier other Jacobi functions on the smaller set of normal lattices to obtain the same four group of solutions. We shall aress this issue below. 4.. The lattices omain At this point it is convenient to iscuss the omain of the lattices that play a role in the elliptic Jacobi functions an therefore in the solutions of the simple penulum. As iscusse in the Appenix A, the quarter perios of a Jacobi elliptic fun- FIGURE 3. Figure shows the whole omain of values that the moular parameter τ can tae for the Jacobi elliptic functions. This omain is a subset of the F funamental region Fig. ). Blac ots represent the values of the square moulus = /, = /, / =, / ) =, / = an / ) =. ction whose square moulus is in the interval 0 < /, generate vertical lattices represente by a moular parameter of the form τ = ik c /K. The point τ = i is associate to the case where the rectangular lattice becomes square an correspons to the value = /. The set of all these lattices blac line in Fig. 3) is represente in the complex plane by the left vertical bounary of the region F Fig. 5) since the quotient K c /K [, ). Acting on these values of the moular parameter with the six group elements of P SL, Z/Z) Rev. Mex. Fis. E 64 08) 05

9 DUALITY SYMMETRIES BEHIND SOLUTIONS OF THE CLASSICAL SIMPLE PENDULUM 3 TABLE II. Approximate numerical values of the perios an the moular parameter for some real values of the moulus of the Jacobi elliptic functions. The values = 0 an = correspon to limit situations where one of the two perios is lost. The value = / is nown as a fixe point, it belongs both to the bounary of the regions F an S of the Fig. 5 an is represente by the blac ot whose coorinates are 0, i) in Fig. 3. The value = is egenerate in the sense it can be represente by two ifferent types of funamental cells, in one case the cell belongs to the bounary of the region ST S an in the another case it belongs to the bounary of ST. The funamental cell for some values of in this table are plotte in Fig. 6. Moulus ω /4 ω /4 τ 0 π/ i i / ) i.796 i / ) i i 3/ ) i i i π/ 0 4/ i ) i i ± i) ) i ±0.5 + i i ) i i prouce the whole set of values of the moular 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 escribe mathematically by the set of moular parameters τ = ik/k c, with K/K c 0, ]. It is clear that the set of six lines is a subset of the F funamental region an constitutes the whole lattice omain of the elliptic Jacobi functions. In Table II we give the numerical values approximate) of the generators of the funamental cell as well as the moular parameter for some values of the square moulus. As a conclusion, for every value of the parameter 0 < / there are six normal lattices relate one to each other by transformations of the moular group. Therefore each solution of the simple penulum with real time variable, showe in Table I, can be written in six ifferent but equivalent ways, where each one of the six forms is in one to one corresponence with one of the six normal lattices. Their S-ual solutions see Table I) which are functions of a pure imaginary time are just one of the six ifferent ways in which solutions can be written ST S-uality The form of the solutions for the simple penulum expresse in Table I oes not coincie with the expressions given in Sec., which by the way, are the stanar form in which the solutions are commonly written in the literature. In orer to reprouce the stanar form it is necessary to introuce the ST S transformation see Appenix B). This transformation taes for instance a Jacobi function with moulus 0 < < into a Jacobi function with moulus greater than one < / <. Taing the inverse transformation it is possible to tae a Jacobi function with moulus < / into one with moulus <. Using the relations of the Appenix B it is straightforwar to obtain Eqs. B.35) which written in terms of E instea of remember than in this case < / = E = / E < ), lea to E sn x, E ) = sn E x, / E ), E cn x, E ) = n E x, / E ), n x, E ) = cn E x, / E ). 34) Inserting this relations in the circulating solutions of Table I reprouce solutions 9) an 0). What we have one is to use the ST S-uality between lattices an transform two of them L an i c L into L an il. Restricte to solutions with real time, two of the four type of solutions for which = E >, are transforme to solutions for which = /E <. As we have iscusse the omain of the solutions with real time variable are horizontal lines in the normal lattices L an il, thus in orer to eep the four ifferent types of solutions it is necessary to evaluate two ifferent set of Jacobi functions 5) an 9) on the omain of each one of the two normal lattices L an il. It is clear that this is not the only way we can procee, in fact we can transform the oscillatory solutions with < into oscillatory solutions with moulus grater than. A similar analysis follows if we consier only solutions with imaginary time A single normal lattice It is natural to woner about the minimum number of normal lattices neee to express all the solutions of the simple penulum. Due to the uality symmetries between lattices this number is one. As an example, if we now use the S-uality to relate the normal horizontal lattice il to the normal vertical lattice L, the horizontal lines that compose the omain in the horizontal lattice becomes vertical lines in the vertical lattices, which means to consier solutions with imaginary time in L. Thus we can en up with only one normal lattice an in orer to have the four ifferent types of solutions, it is necessary to consier the whole omain of the lattice, i.e. Rev. Mex. Fis. E 64 08) 05

10 4 R. LINARES ROMERO TABLE III. Solutions to the simple penulum problem written in a unique lattice of square moulus 0 < /. Energy interval E 0, /] E [/, ) E, ] E [, ) Solution θ arcsin[ snx, )] arcsin[nix ik c + K, )] ± sgn[ i/ c ) cnix/ c ik c / c, )] arcsin[/ c ) nix/ c ik c / c, )] ± sgn[cnx/, )] arcsin[snx/, )] both vertical lines imaginary time) an horizontal lines real time) an on each set of lines to consier two ifferent solutions one oscillatory an one circulating. For completeness in Table III we give the four type of solutions in terms of only one value of the moulus It is clear that we can express all the solutions also for the other five ifferent functional forms of the square moulus. 5. Final remars In this paper we have aresse the meaning of the fact that the complex omain of the solutions of the simple penulum is not unique an in fact they are relate by the P SL, Z/Z) group, fining that the important issue for express the solutions is the relation between the values of the square moulus of the Jacobi elliptic functions, an the value of the total mechanical energy E of the motion of the penulum. Due to the symmetry we conclue that there are six ifferent expressions of the square moulus that are relate one to each other trough the six group elements of P SL, Z/Z). These six group actions can be terme as uality-transformations an therefore we have six ual representations of. As a consequence there are six ifferent but equivalent ways in which we can write a specific penulum solution, an abusing a little bit of the language we coul say there are uality relations between solutions. This analysis teach us the lesson that we can restrict the omain of lattices to the ones whose moular parameter is in the pure imaginary interval τ i, ), or equivalently that we can express every solution of the simple penulum either oscillatory or circulating with Jacobi elliptic functions whose value of the square moulus is in the interval 0 < / see Table III). It is well nown that there are several physical systems in ifferent areas of physics whose solutions are also given by elliptic functions, for instance in classical mechanics some examples are the spherical penulum, the Duffing oscillator, etc., in Fiel Theory the Korteweg e Vries equation, the Ising moel, etc., [,8]. It woul be very interesting to investigate on similar grouns to the ones followe here, the physical meaning of the symmetries of the elliptic functions in these systems. Appenix A. The moular group an its congruence subgroups A. The moular group The moular group Γ is the group efine by the linear fractional transformations on the moular parameter τ C see for instance [3-7,,3] an references therein) τ Γτ) = aτ + b cτ +, A.) where a, b, c, Z satisfying a bc =, an the group operation is function composition. These maps all transform the real axis of the τ plane incluing the point at infinity) into itself, an rational values into rational values. The group has two generators efine by the transformations Sτ) /τ, an T τ) + τ. A.) The moular group is isomorphic to the projective special linear group P SL, Z), which is the quotient of the -imensional special linear group SL, Z) by its center {I, I}. In other wors, P SL, Z) = SL, Z)/Z consists of all matrices of the form a b A =, A.3) c with unit eterminant, an pair of matrices A, A, are consiere to be ientical. The group operation is multiplication of matrices an the generators accoringly with A.) are S = 0 0 ), T = 0 ). A.4) These group elements ) satisfy S = ST ) 3 = I I an n T n =. 0 One important property of the moular group is that the upper half plane of C, usually enote as H an efine as H {z C : Imz) > 0}, can be generate by the elements of P SL, Z) from a funamental omain or region F. Mathematically this region is the quotient space F = H/P SL, Z) an satisfies two properties: i) F is a connecte open subset of H such that no two points in F are relate by a Γ transformation A.) an ii) for every point in H there is a group element g Γ such that gτ F. There are many ways of constructing F, an the most common one Rev. Mex. Fis. E 64 08) 05

11 DUALITY SYMMETRIES BEHIND SOLUTIONS OF THE CLASSICAL SIMPLE PENDULUM 5 In this nomenclature the moular group Γ is calle the moular group of level an enote as Γ) [6,3]. A relevant mathematical structure is the coset of the moular group with the congruence subgroups which are isomorphic to P SL, Z/N Z) [3] SL, Z) ΓN) P SL, Z/NZ). A.6) FIGURE 4. Tessellation of H. The funamental region F is represente by the shae area an the heavy part of the bounary. This region is mappe to the whole upper plane C by the moular group Γ. The region can be viewe as a complete list of the inequivalent complex structures on the topological torus since conformal equivalence of tori is etermine by the moular equivalence of their perio ratios. In the figure we show some copies of the funamental region obtaine by application of some group elements of P SL, Z). foun in the literature is to tae the set of all points z in the open region {z : / < Rez) < / z > }, union half of its bounary, for instance, the one that inclues the points: z = / + iy with y sinπ/3), an z = with / Rez) 0 see Fig. 4). It is assume that the imaginary infinite is also inclue. Geometrically, T represents a shift of F to the right by, while S represents the inversion of F about the unit circle followe by reflection about the imaginary axis. As an example, the Fig. 4 represents the transformations of the funamental region F by the elements of the group: {I, T, T, S, T S, T S, ST, ST, ST S, T ST, ST S, T ST } []. Notice that these elements are all the inepenent ones that we can construct as iterative proucts of S, T an T without powers of any of them involve S is simply S S an therefore is not a ifferent moular transformation). The other two transformations we can construct are not inepenent T ST = ST S an T ST = ST S. Further proucts of the generators with these transformations give us the whole tessellation of the upper complex plane. In particular the orbit of the points Imz) are the rational numbers Q an are calle cusps. For the solutions of the simple penulum the relevant congruence subgroup is the one of level : Γ). It turns out that all the groups P SL, Z/NZ) are of finite orer an in particular P SL, Z/Z) is of orer six. In Table IV we give explicitly the six elements of the coset an their corresponing form as group elements of P SL, Z). Analogously to the case of the moular group, a funamental cell for a subgroup ΓN) is a region F N in the upper half plane that meets each orbit of ΓN) in a single point. Because Γ) is of orer six in Γ, a funamental cell for Γ) can be forme from the six copies of any funamental cell F of Γ prouce by the action of the six elements. In Fig. 5 we show the funamental region F of Γ). This cell can be obtaine from the region enote as F which is a ifferent funamental region for Γ as compare to the usual region F of the Fig. 4. F is obtaine if F is replace by its right half, plus inversion of its left half by the S transformation. Thus F consists of the open region {z : 0 < Rez) < / z z/z + z) > } an part of its bounary must be inclue. Geometrically z z/z + z) = represents a unitary circle with center at z =. A possible choice of the bounary inclues the set A. Congruence subgroups Relevant for our iscussion are the congruence subgroups of level N enote as ΓN) or Γ N ). They are efine as subgroups of the moular group Γ, which are obtaine by imposing that the set of all moular transformations be congruent to the ientity mo N { a b ΓN) = SL, Z) : c a b = c } 0 mo N). A.5) 0 FIGURE 5. Funamental cell F for Γ). The heavy part of the figure is retaine, the rest is not. In particular the cusps, 0, an i are exclue. Rev. Mex. Fis. E 64 08) 05