Is the tautochrone curve unique? Pedro Terra,a) Reinaldo de Melo e Souza,b) and C. Farinac) Instituto de Fısica, Universidade Federal do Rio de Janeiro, Rio de Janeiro RJ 21945-970, Brazil (Received 22 February 2016; accepted 9 September 2016) We show that there are an infinite number of tautochrone curves in addition to the cycloid solution first obtained by Christiaan Huygens in 1658. We begin by reviewing the inverse problem of finding the possible potential energy functions that lead to periodic motions of a particle whose period is a given function of its mechanical energy. There are infinitely many such solutions, called “sheared” potentials. As an interesting example, we show that a Poschl-Teller€ potential and the one-dimensional Morse potentials are sheared relative to one another for negative energies, clarifying why they share the same oscillation periods for their bounded solutions. We then consider periodic motions of a particle sliding without friction over a track around its minimum under the influence of a constant gravitational field. After a brief historical survey of the tautochrone problem we show that, given the oscillation period, there is an infinity of tracks that lead to the same period. As a bonus, we show that there are infinitely many tautochrones. VC 2016 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4963770]

3 I. INTRODUCTION instance, Pippard presents a nice graphical demonstration of sheared potentials, whereas Osypowski and Olsson4 provide In classical mechanics, we find essentially two kinds of a different demonstration with the aid of Laplace transforms. problems: (i) fundamental (or direct) problems, in which the Recent developments on this topic, and analogous quantum forces on a system are given and we must obtain the possible cases, can be found in Asorey et al.5 motions, and (ii) inverse problems, in which we know the In this article, our main goal is to analyze inverse prob- possible motions and must determine the forces that caused lems for periodic motions of a particle sliding on a friction- them. Informally, we might say that in inverse problems we less track contained in a vertical plane in a uniform attempt to find the causes by analyzing the effects. A humor- gravitational field. Similar to the sheared potentials, knowl- ous way of stating the difference between these two kinds of 1 edge of the track shape uniquely determines the period of problems is given in Bohren and Huffmann’s book, which motion as a function of the maximum height achieved by the states that in a direct problem, we are given a dragon and particle. However, as we will demonstrate, an infinity of dif- want to determine its tracks, while in an inverse problem, we ferent curves lead to the same period for each maximum start from the dragon’s tracks and attempt to infer what the height. We will show that these curves exhibit a geometrical dragon is like. property akin to shearing that, to our knowledge, has not Some historically relevant examples of inverse problems been previously noted in the literature. are Newton’s determination of the gravitational force from In particular, we are interested in the case where the Kepler’s laws, and Rutherford’s discovery of the atomic period of oscillation does not depend on the maximum height nucleus from the scattering of a particles from a sheet of gold foil. Indeed, inverse problems are extremely frequent (and therefore on the mechanical energy) of the system—the even in contemporary physics. In high-energy Physics, for tautochrone. Because this term is often associated with a instance, we try to understand the fundamental interactions downward only motion, we employ the expression round- between elementary particles by observing the products of trip tautochrone to specify our curves of interest. It has been their scattering. Oil prospecting methods in deep waters are shown by Huygens in 1658 that the cycloid is a tautochrone also inverse problems as they rely on analyzing the proper- curve. A direct consequence of our analysis is that this is not ties of waves (caused by small explosions at the surface of the sole solution of the tautochrone problem. In fact, there the sea) reflected by the presence of interfaces separating are infinitely many round-trip tautochrone tracks, unfolding two different mediums within the surface of Earth. new results for this three-century-old problem. Even in the case of a particle moving in one dimension The remainder of the article is organized as follows. In Sec. II, we review the basic concepts of sheared potentials under the influence of a conservative force, analyzing inverse 2 problems may yield very surprising results. For instance, following Landau and Lifshitz. We also provide two non- consider the periodic motion of a point mass in a potential trivial examples of sheared potentials: (i) power-law poten- well U(x). Although the knowledge of U(x) uniquely deter- tials and (ii) a Poschl-Teller€ potential and the one- mines the period s of oscillation as a function of the mechan- dimensional Morse potentials. In Sec. III, inspired by the ical energy E, the opposite is not true: knowledge of s E properties exhibited by sheared potentials, we consider the does not uniquely determine the potential energy that leadsð Þ periodic motion of a particle sliding without friction on a to the known period. In fact, as explained in Sec. II, it can be track contained in a vertical plane under a constant gravita- shown that a given function s E allows for infinitely many tional force, and ask what curves yield the same period of potential wells that are “sheared”ð Þ from one another. oscillation. We show that there are an infinite number of dif- This beautiful result was obtained in a quite ingenious ferent curves associated with each period, and provide an way by Landau and Lifshitz.2 Since then, it has been revis- interpretation for the condition between these curves as a ited by some authors using different approaches. For new kind of shearing. In Sec. IV, we use the previous result

917 Am. J. Phys. 84 (12), December 2016 http://aapt.org/ajp VC 2016 American Association of Physics Teachers 917

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 146.164.138.88 On: Tue, 22 Nov 2016 17:05:18 to find an infinite family of tautochrone curves. Finally, we 1 U s E dE make some final remarks in Sec. V. xR U xL U ðÞ : (4) ðÞÀ ðÞ¼ pp2m 0 pU E ð À II. BRIEF REVIEW OF SHEARED POTENTIAL The result tells us that,ffiffiffiffiffiffi providedffiffiffiffiffiffiffiffiffiffiffiffiffi two potential wells U(x) ~ WELLS and U x share the same width xR U xL U x~R U ð Þ ð ÞÀ ð Þ¼ ð ÞÀ x~L U for every value of U, the periods of oscillation of a par- Let us consider a particle of mass m moving along the x- ticleð underÞ the influence of these potentials will be described axis under the influence of the resultant force by the same function s E . When this property is satisfied, we x dU x =dx, where U(x) is a generic potential well ð Þ ~ Fð Þ¼À ð Þ say that the potential wells U(x)andU x are “sheared” rela- with a single minimum. Without loss of generality, let us tive to one another. However, if we furtherð Þ require the poten- choose the point x 0 as the minimum of the potential well ¼ tial to be symmetric, so that xR U xL U , then the such that U 0 0. Suppose the particle moves with ð Þ¼À ð Þ s ð Þ¼ potential is uniquely determined by the period function . mechanical energy E; the associated turning points are then The presentation in this section was reproduced from given by the roots of the algebraic equation E U x , which 2 ¼ ð Þ Landau and Lifshitz. Let us illustrate the previous results in we denote by x1 and x2. It is worth emphasizing that x1 and some non-trivial examples that are not usually dealt with in x2 depend on E. Assuming x2 > x1, it is straightforward to the literature. We will start by illustrating the shearing pro- use conservation of (mechanical) energy to obtain the period cess for the power-law case. Then, we will show that the of oscillations as Morse and the Posch-Teller€ potentials are related by shearing (for bounded motions). x2 dx s E p2m : (1) ðÞ¼ x1 E Ux A. Power law potentials ffiffiffiffiffiffi ð À ðÞ It is evident from thispffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi equation that the potential energy Sheared potentials derived from the parabolic potential 6 U(x) and a given mechanical energy E uniquely determine well have been considered by Anton and Brun. These the period of the oscillations. In other words, the function authors obtained anharmonic motions whose periods of s : E s E is uniquely determined by knowledge of oscillations retain the property of being independent of the function! Uð: Þx U x . energy (isochronous oscillations). In order to solve! theð Þ inverse problem—given s E to find In this subsection, we shall consider the more general case the corresponding U(x)—it is tempting (and convenient)ð Þ to of the family of symmetric power law potentials given by  regard the coordinate x as a function of the potential U. U x a x , with a > 0 and  1, whose turning points are ð Þ¼ j j 1=  However, since the function U(x) is not injective, obtaining x6 6 E=a . The period of oscillation may be computed its inverse requires defining two functions, namely, using¼ Eq.ð (1)Þ x : U x U , for the left branch of U(x)(x < 0) and L L x : 7! ð Þ þ dx xR U xR U , for the right branch of U(x)(x 0). s p 7! ð Þ  E 2m  Performing a change of variables, we recast Eq. (1) into the ðÞ¼ x E a x ð À À j j form ffiffiffiffiffiffi 1=2 (5) x  À =2 8mpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiþ x À E 0 x 1 dx; dxR dU dxL dU ¼ þ a 0 À x s E p2m rffiffiffiffiffiffi ð "#þ ðÞ¼ dU pE U þ dU pE U "#ð0 ðE (2) E À À where we have used the fact that the potential well is sym- ffiffiffiffiffiffi dxR dxL dU p2m ffiffiffiffiffiffiffiffiffiffiffiffiffi :ffiffiffiffiffiffiffiffiffiffiffiffiffi metric and that E ax . Making a change of variables u ¼ 0 dU À dU pE U  ¼ þ ¼ ð À x=x and recalling the definition for x , we rewrite the ffiffiffiffiffiffi periodð þÞ as þ To get rid of the integral on the right-handffiffiffiffiffiffiffiffiffiffiffiffiffi side, we multiply both sides of Eq. (2) by dE=pa E, where a is a constant 1= 1=2 p8m I  parameter, and then integrate fromÀ 0 to a. On the right-hand s E EðÞÀ ðÞ; (6) ðÞ a1=  side, we are then left with an integralffiffiffiffiffiffiffiffiffiffiffiffi in U followed by an ¼ ffiffiffiffiffiffi integral in E. Changing the order of integration, which 1 1= 1 1=2 where I  0 u À 1 u À du is simply a numerical requires a subtle change in the integration limits in order to factor. Althoughð Þ¼ the computationð À Þ may be continued to give preserve the integration region in the EU-plane, we obtain an exact result,Ð2,7 it suffices for the purpose of this article to regard the period’s dependence on the energy. Moreover, a s E dE a dx dx a dE p R L Eq. (6) shows that for  2 the exponent on E vanishes and, ðÞ 2m dU : ¼ 0 pa E ¼ 0 dU À dU U pa EpE U as a consequence, the period is independent of the energy, ð À ð ð À À ffiffiffiffiffiffi (3) which is the case of the harmonic oscillator. ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi Let us denote by D the distance between the turning points Fortunately, the integral over E may be exactly calculated for a given energy E; that is, D E x E x E .We and does not depend on U. In fact, it is straightforward to may try to construct a sheared potentialð Þ¼ þð forÞÀU xÀð Þa x , ~ show that this integral is equal to p (it can be computed, for denoted by U x , by defining different expressionsð Þ¼ for posi-j j ð Þ instance, by completing the square). As a consequence, the tive and negative x integration on U is immediate. From these results, and rewriting a U, we conclude that a given period function s  b x : x < 0 does not determine¼ a single potential U, but rather an infinite U~ x j j (7) ð Þ¼ c x  : x 0 family of potentials satisfying the relation  j j 

918 Am. J. Phys., Vol. 84, No. 12, December 2016 Terra, de Melo e Souza, and Farina 918

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which has roots u6 16 1 E=U0,or D~ E x~ E x~ E E=b 1= E=c 1= : (8) ¼ þ ð Þ¼ þð ÞÀ Àð Þ¼ð Þ þð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 6 Imposing the shearing condition, D E D~ E for any E, x6 a ln 1 1 E=U0 : (14) ð Þ¼ ð Þ ¼ þ we obtain the following condition between coefficients a, b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and c: Thus, the distance between these two turning points, DM E x E x E , for an arbitrary negative energy is 1 1 1 1 ð Þ¼ þð ÞÀ Àð Þ : (9) given by a1= ¼ 2 b1= þ c1=  1 1 1 E=U0 D E ln þ þ Indeed, if we explicitly compute the period of oscillations of MðÞ a ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 E=U0! the particle under the influence of the potential well U~ x by À þ ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 suitably adapting Eq. (6) we obtain 1 1 E=U0 1 þ þ (15) a ln0 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE=U0 1 1 1 1= 1=2 I  À ~s E EðÞÀ p8m ðÞ: (10) @ A ðÞ¼ 2 b1= þ c1=  2 U U  ln 0 0 1 : ffiffiffiffiffiffi ¼ a À E þ À E À ! Combining Eqs. (9) and (10), it becomes evident that rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~s E s E , as expected since we required the two poten- Let us now do the same thing for the Poschl-Teller€ poten- ð Þ¼ ð Þ ~ tials U(x) and U x to be sheared relative to each other. tial. An analogous procedure to determine the roots of the ð Þ equation E UPT x yields ¼ ð Þ B. Morse and Poschl-Teller€ potentials 1 x6 6 arcosh U0=E : (16) Let us now consider two less trivial potentials: the one- ¼ a À dimensional Morse potential UM x and the Poschl-Teller€  ð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi potential UPT x , given by Therefore, the distance between these two turning points ð Þ is given by 2ax ax UM x U0 eÀ 2eÀ (11) ð Þ¼ ð À Þ DPT E x x 2=a arcosh U0=E : (17) U0 ð Þ¼ þ À À ¼ð Þ ð À Þ UPT x ; (12) ðÞ¼Àcosh2 ax At first glance, Eqs. (15) and (17) do notp appearffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi to coincide. ðÞ However, making use of the mathematical identity 2 where U0 and a are positive constants (see Fig. 1). These two arcosh x ln x px 1 , it becomes evident that ð Þ¼ ð þ À Þ potentials have importance beyond the context of classical DM E DPT E . This means, as we had anticipated, that mechanics. For instance, the Morse potential plays an impor- theseð Þ¼ two potentialsð Þ areffiffiffiffiffiffiffiffiffiffiffiffiffi sheared relative to each other for tant role in chemistry while the Poschl-Teller€ potential U0 < E < 0. exhibits bizarre properties when treated quantum mechani- À Because these two potentials are sheared (for cally (e.g., for appropriate values of the constant U0 this U0 < E < 0), we can determine the period of oscillation potential becomes reflectionless8 for all values of E). Àusing either potential (they will both give the same answer). Our purpose in this subsection is to show that these two Let us perform the calculation explicitly for the Morse poten- potentials are sheared relative to each other, so that they tial. Using Eq. (1) and making the same change of variables ax have the same period of oscillation for negative mechanical as before, u eÀ , we get ¼ energies. With this goal, let us start by determining the turn- x ing points of a particle moving in the Morse potential with a þ dx sM E p2m ðÞ¼ 2ax ax negative mechanical energy E. These points can be obtained x E U0 eÀ 2eÀ ax ð À À ðÞÀ by setting E UM x . Conveniently substituting u eÀ , ffiffiffiffiffiffi u 2m þ du this equation leads¼ toð Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: (18) a2 2 ¼ U0 u E=U0 u 2u 1 rffiffiffiffiffiffiffiffiffiffi ð À ðÞþ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Making another change of variables w u E=U0 ¼ð À þ U0=E then leads to À Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w 2m þ dw sM E a2 2 ðÞ¼ U0 w w U0=E 1 rffiffiffiffiffiffiffiffiffiffi ð À À ðÞþ p 2m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a : (19) ¼ rffiffiffiffiffiffiE We leave for the interested reader the task of checking Fig. 1. The Morse (solid) and Poschl-Teller€ (dashed) potentials. explicitly through an analogous calculation that the period

919 Am. J. Phys., Vol. 84, No. 12, December 2016 Terra, de Melo e Souza, and Farina 919

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 146.164.138.88 On: Tue, 22 Nov 2016 17:05:18 associated to the Poschl-Teller€ potential (also for period is simply twice the time spent by the particle to move U0 < E < 0) is indeed the same as found here. from s1 to s2 along the track. Denoting by v the scalar veloc- À ity of the particle at position s along the track, we have C. General prescription for sheared potentials s2 ds 2 s2 ds After working through these examples, one might wonder s H 2 : (21) ðÞ v whether there is a general prescription for constructing ¼ s1 ¼ sgffiffiffi s1 H ys ð ð À ðÞ sheared potential wells. We can always begin with the sym- pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi metric potential well U(x) associated with a given period Note that because s1 and s2 depend on H, the period s is a function, which has xR U xL U . Then, in order to function only of H, uniquely determined by knowledge of shear this potential we justð displaceÞ¼À ð theÞ function horizontally the track y(s). Furthermore, this equation is written by d U to obtain new functions x~R U and x~L U for the completely in terms of variables with a clear geometrical right-sideð Þ and the left-side branches4,6ð Þ ð Þ meaning. We now wish to tackle the inverse problem, which is to x~R U xR U d U find the function y(s) that constrains the motion to a given ð Þ¼ ð Þþ ð Þ (20) periodic behavior s H . We apply the same mathematical x~L U xL U d U :  ð Þ¼ ð Þþ ð Þ procedure as given byð LandauÞ and Lifshitz to this new case. Consequently, the difference between the corresponding We invert the function y(s) by means of the piecewise func- turning points is preserved for each value of U: tion split in sR y and sL y for the right and left branches of the track, respectively,ð Þ andð Þ integrate using a trick analogous D U x~R U x~L U xR U xL U . Both x~R and x~L mustð Þ¼ be single-valued.ð ÞÀ ð Þ¼ If thisð conditionÞÀ ð Þ is met, calculating to that presented in Sec. II, to get the inverse piecewise function will give a new potential y s U~ x sheared from U(x). 1 g H dH ð Þ sR y sL y p ðÞ : (22) ðÞÀ ðÞ¼ 2 0 py H rffiffiffi ð À III. SHEARING TRACKS The reader may note that this resultffiffiffiffiffiffiffiffiffiffiffiffi is similar to Eq. (4), with Though not very common, sheared potential wells are x s; U y, and E H, but we emphasize that it has a very well established. Inspired by the surprising properties of different! meaning! and! interpretation. these potential wells, we now turn our attention to a different Equation (22) shows that knowledge of the period function problem that (as far as the authors know) has not yet been s H does not uniquely determine the shape of a track, but ð Þ investigated. Instead of the one-dimensional motion driven rather its length L y sR y sL y below every height by a given potential well U(x), we consider the two- y > 0. Two differentð tracksÞ¼ yð(sÞÀ) and yð~ Þs will lead to motions ð Þ dimensional movement of a particle sliding along a friction- with the same period function s H if sR y sL y ð Þ ð ÞÀ ð Þ¼ less track under a constant gravitational force. As construct- s~R y s~L y for every y. We call these tracks length- ing tracks is generally more feasible than supplying the shearedð ÞÀ relativeð Þ to one another in analogy with sheared conditions for obtaining an arbitrary potential well, this tech- potentials. Concretely, this means that we can begin by con- nique may prove useful for actual visualization of the structing a track that has a particular functional form by motions under study. using, say, measuring tape against a wall, and measure the We state the problem as follows: Let a particle be subject period of oscillations of a small marble moving on it. Then solely to a uniform gravitational field, so that the gravita- we carefully re-shape the tape so as to ensure that the length tional potential energy is U(y) mgy, and be constrained to of the tape under every horizontal line remains the same. move along a smooth track contained¼ in the xy-plane. Due to This transformation will generate a new track, related to the this constraint, the motion has only one degree of freedom. first by length-shearing, and oscillations along this second The shape of the track, described by the function track will have the same period as the first one for every f : x y f x , determines the net force on the particle at maximum height H. This procedure is noteworthy, because a each7! point.¼ Forð Þ convenience, we choose a coordinate system similar strategy is not generally possible with sheared so that the single minimum of the track coincides with the potentials. origin of the axes. Though the shape of the track uniquely A comment is in order here. Suppose we consider only the determines the period of the motion as a function of its motions of a particle that moves down a frictionless track energy, s E , it is not obvious whether the knowledge of characterized by a function f : x y f x and denote by s E uniquelyð Þ determines the shape of the track along which s H the time spent by the particle7! to¼ reachð Þ the origin at theð Þ motion takes place. In other words, it is natural to ask yð 0Þ after being released from rest at an arbitrary height what tracks give rise to oscillations with a given period func- y ¼ H. Again, given the track f(x), the period s H is tion s : E s E . uniquely¼ determined. However, in this case, knowledgeð Þ of As will7! becomeð Þ evident later, it is convenient to use the s H would also uniquely determine the shape of the track arclength coordinate s to parametrize the track instead of the (noteð Þ that we are not talking about periodic motions here but cartesian coordinate x, and henceforth we denote the shape only downward motions). This problem was first solved by of the track by the function y : s y s , with s 0 at the ori- Abel9 in 1826. In 2010, Munoz~ and Fernandez-Anaya10 dis- gin (x y 0). Setting the maximum7! ð Þ height H¼achieved by cussed Abel’s result for particular curves in which the corre- the particle¼ ¼ also sets the mechanical energy E mgH of the sponding periods are proportional to a fractional power of H. ¼ 11 system, and the turning points s1 and s2 are given by the The same authors (with additional collaborators) also pre- roots of the equation y(s) H. From the conservation of sent an illuminating set of direct and inverse problems mechanical energy, it is straightforward¼ to obtain the corre- involving beads on a frictionless rigid wire in a paper from sponding period of oscillation s H for an arbitrary H; this 2011. ð Þ

920 Am. J. Phys., Vol. 84, No. 12, December 2016 Terra, de Melo e Souza, and Farina 920

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 146.164.138.88 On: Tue, 22 Nov 2016 17:05:18 IV. ROUND-TRIP TAUTOCHRONES pendulum clock, did not succeed as maritime chronometers. However, his legacy on the developments of curves, evo- A. Brief historical survey lutes, and involutes, which had their origins in his study of Before we apply our previous result to the tautochrone clocks, can still be seen in many different areas from differ- 14 problem, it is worth saying a few words about the tauto- ential geometry to quasicrystals. chrone curve, as it played an important role in the history of classical mechanics of the 17th century. At that time, mea- B. Obtaining the tautochrone curve suring latitude was relatively easy whereas measuring longi- tude, which was of vital importance for sea navigations, was Equation (22) shows us how to construct tracks on which particles oscillate with a given period s H . If we impose a difficult task because it demanded a very accurate measure- ð Þ ment of time.12 The pendulum clock, constructed by the that the track be symmetric, then it is uniquely determined. great Dutch physicist, mathematician, and astronomer We shall now apply the previously discussed techniques to Christiaan Huygens in 1658 (Galileo had already attempted obtain the tracks on which the period does not depend on the to construct a pendulum clock but he never completed it and energy; that is, solutions to the tautochrone problem. As the patent for this invention was given to Huygens), these tracks must have upward and downward branches improved the accuracy of time measurements by at least one allowing for periodic motion, we call them round-trip order of magnitude, but this was not enough to guarantee a tautochrones. safe measurement of longitude. Choosing a suitable energy-independent period function s E pj, where j is a positive constant with dimensions With the purpose of improving maritime chronometers, ð Þ¼ Huygens started to look for an isochronous pendulum, since of time squared, and demanding that the track be symmetric ffiffiffi he knew that the simple pendulum was isochronous only for allows us to compute Eq. (22) exactly small amplitudes. A maritime chronometer constructed with j y an isochronous pendulum would not change the period of its 1 g dH sy p oscillations even if the corresponding amplitudes changed ðÞ¼ 2 2 0 py H rffiffiffiffiffiffi ð (23) due to a rough sea. Huygens knew that if he put lateral 1 gj À 2pyffiffiffiffiffiffiffiffiffiffiffiffi; obstacles of appropriate shape near a simple pendulum he ¼ 2p 2 could achieve his purpose but, unfortunately, he was not able rffiffiffiffiffiffi ÀÁffiffiffi to find empirically the exact shape of such lateral obstacles. giving Destiny then came to Huygens’ aid. Blaise Pascal, the famous French physicist, mathematician, and philosopher, 2p2 y s2: (24) who had abandoned science after a religious epiphany in ¼ jg 1654, had an unbearable toothache in 1658 that seemed to resist any alleviating efforts. In a desperate attempt to forget The above result is in perfect analogy with the harmonic the pain, Pascal decided to focus on mathematics, particu- oscillator. As expected, it also corresponds to the arclength larly on some problems dealing with the cycloid that the parametrization of a cycloid whose generating circle has French priest Mersenne had passed to him. Soon the pain dis- radius r jg= 4p 2, namely, y 1=8r s2. The period of appeared completely and Pascal interpreted this fact as a oscillation¼ in termsð Þ of r and g is readily¼ð givenÞ by divine sign for him to go on thinking about problems involv- ing the cycloid. He ended up solving many of Mersenne’s r s r 4p : (25) problems and even formulated a few new ones. But instead ðÞ¼ g of publishing these problems, he decided to propose a contest rffiffiffi composed of six problems involving the cycloid. Although the cycloid is indeed the single symmetric round- Many important scientists of that time were encouraged to trip tautochrone, we may length-shear it in infinitely many participate in that contest, including Huygens. And once he ways. We will now present some unconventional solutions to had become an expert on the cycloid, he decided to check the tautochrone problem, given by asymmetric tracks. and see, if by any chance, the cycloid would help solve the isochronous pendulum problem. Fortunately, Huygens found C. Half-cycloid branches that the cycloid was a tautochrone—a curve on which a par- ticle, sliding without friction and under the action of a con- Let us begin with a symmetric cycloid whose generating stant gravitational force, would have a period that is circle has radius r. We first attempt an asymmetric length- independent of the height from which it was released. But he sheared solution using cycloids with different generating still needed to find the shape of the lateral obstacles that radii. We consider for the left-side and right-side branches, would make a pendulum describe a cycloidal trajectory. In respectively, r~L and r~R, in analogy to what was presented in other words, he had to find out what we now call the evolute Sec. II A for coefficients of power-law potentials. of the cycloid. Again he tried the cycloid and once more he For both tracks to give rise to oscillations with equal peri- was successful—the evolute of the cycloid is the cycloid ods, they must be length-sheared, so the total length of the (shifted and out of phase)! In this particular case, Huygens track must be the same below every height y. The relation was very lucky; it is not common that a curve is the evolute between the radii follows directly from applying the of itself. More details on this history can be found in constraint Gindikin’s book.13 Among other things, Huygens worked on trying to sR y sL y s~R y s~L y : (26) improve clocks for almost four decades, but his cycloidal ð ÞÀ ð Þ¼ ð ÞÀ ð Þ pendulum clock, as well as his isochronous conical From the relation y 1=8r s2, it follows immediately that ¼ð Þ

921 Am. J. Phys., Vol. 84, No. 12, December 2016 Terra, de Melo e Souza, and Farina 921

This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 146.164.138.88 On: Tue, 22 Nov 2016 17:05:18 Fig. 3. Cycloid (dashed) and an isoperiodical asymmetric tautochrone hav- Fig. 2. Symmetric cycloid (dashed) and an isochronous asymmetric track ing a semicubical parabola for its left branch (solid). In this example, we made from different cycloids (solid). have set r 1/8 and a 2. ¼ ¼ Since we chose an arbitrary function for the left branch of 2pr r~R r~L : (27) ¼ þ the tautochrone, the time it spends on each branch will We presentffiffi onepffiffiffiffiffi suchp solutionffiffiffiffiffi in Fig. 2. depend on the maximum height. However, the geometrical Alternatively, we may obtain such a curve by imposing condition given by length-shearing guarantees that the differ- that the periods of oscillations along both the symmetric and ence of time the moving particle spends on the left branch the asymmetric tracks be the same (provided the maximum (with respect to the cycloid) will be exactly compensated for height of the motion is contained in the lower branch of the on the right branch. curve). The interested reader may easily verify, by means of Eq. (25), that the condition obtained is the same as in V. FINAL REMARKS Eq. (27). In the scope of one-dimensional systems in classical mechanics, we have reviewed the result that oscillations with D. Completing tautochrones a given period as a function of the energy correspond not to Equation (22) allows for a very broad range of solutions one, but to an infinite family of potential wells, which satisfy for round-trip tautochrones that are not restricted to branches the shearing condition. Particularly, isochronous oscillations of cycloids. We may obtain such a track using a prescription are not a property associated uniquely to the quadratic poten- similar to the one presented in Sec. II C, namely, by applying tial well (harmonic potential) but to any potential well that is (to the original cycloidal track) equations analogous to those sheared from it. written in Eq. (20), but now with a length-shearing function Then we have extended this result to the motions of a par- d y , instead of the shearing function d U . With this proce- ticle moving along frictionless tracks contained in a vertical dureð Þ in mind, if we choose an arbitraryð functionÞ for the left plane and subject to a constant gravitational force. We have ~ branch of the track, say yL s , we will obtain the correspond- shown that by imposing a new condition—length-shearing— ~ ð Þ ing yR function for the right branch that completes the tauto- to the tracks, their corresponding periods, as functions of the chrone track. maximum height achieved by the particle, will be the same. Since any track constructed in this way will be length- Applying the length-shearing condition to a cycloid, the orig- sheared from the original cycloidal track (which corresponds inal tautochrone discovered by Huygens in the 17th century, to the single symmetric tautochrone), we see that this proce- we have shown that it is possible to obtain an infinite family dure provides a method for generating as many tautochrone of solutions to the tautochrone problem. tracks as we want. All tracks constructed in this way will The actual construction of tracks can be accomplished share the property of isochronous motion—all of them will using a 3-D printer and is far more practical than providing lead to periodic motions that are independent of the maxi- the conditions for a certain potential energy U(x) to drive the mum height. motion in a given period. Also, it is worth mentioning that An explicit example is appropriate here. Let us set the the techniques employed in this work are accessible to ~ left branch of the track to the semi-cubical parabola yL undergraduate students. Hence the information presented a x 3=2 with a > 0. We compute the arclength in order to¼ here can be useful for enriching courses on both theoretical parametrizeðÀ Þ it conveniently and experimental university-level classical mechanics. Finally, for pedagogical reasons, we would like to outline 3=2 8 9a4=3 the most important aspects of the present work: s~ y y2=3 1 1 : (28) L a2 • The study of sheared potentials, although being a well ðÞ¼ 27 "#4 þ À established problem, still holds a few surprises, such as We may set the period of oscillation by adjusting the radius r the shearing relation between the Morse and Poschl-Teller€ of the original cycloid, 8ry s2, in accordance with Eq. potentials. (25). So by imposing that the¼ new track satisfies the length- • There is a new condition that is analogous to shearing in shearing relation to this cycloid, the right-side branch of the one-dimensional potential wells, called length-shearing, track is given by the equation which can be applied to frictionless tracks contained in vertical planes. • s~R y 2 8ry s~L y : (29) There is not one, but an infinite number of tautochrones. ð Þ¼ þ ð Þ • We can make a tautochrone by choosing any monotonic A solution to thepffiffiffiffiffiffiffi track having a semi-cubical parabola for its function y : x y x for the shape of one of its branches, left branch is shown in Fig. 3. and then computing7! ð Þ the corresponding complementary

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This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 146.164.138.88 On: Tue, 22 Nov 2016 17:05:18 branch simply by imposing that they are length-sheared to 4E. T. Osypowsky and M. G. Olsson, “Isochronous motion in classical the cycloid. mechanics,” Am. J. Phys. 55(8), 720–725 (1987). 5M. Asorey, J. F. Carinena,~ G. Marmo, and A. Perelomov, “Isoperiodic We leave for the interested reader the task of exploring classical systems and their quantum counterparts,” Ann. Phys. 322(6), new examples of tracks exhibiting different properties 1444–1465 (2007); e-print arXiv:0707.4465 [hep-th] 6  regarding the periods of their respective driven motions, fur- C. Anton and J. L. Brun, “Isochronous oscillations: Potentials derived from a parabola by shearing,” Am. J. Phys. 76(6), 537–540 ther applying length-shearing. (2008). 7J. F. Carinena,~ C. Farina, and C. Sigaud, “Scale invariance and the Bohr- ACKNOWLEDGMENTS Wilson-Sommerfeld quantization for power law one-dimensional potential wells,” Am. J. Phys. 61(8), 712–717 (1993). The authors thank Felipe Rosa for his very fruitful insights. 8J. Lekner, “Reflectionless eigenstates of the sech2 potential,” Am. J. Phys. C.F. is also indebted to some members of the Theoretical 75, 1151–1157 (2007). Physics Department of University of Zaragoza, particularly to 9N. H. Abel, “Auflosung€ einer mechanischen Aufgabe,” J. Reine Angew. ~ Math. 1, 153–157 (1826). M. Asorey, J. Esteves, F. Falceto, L. J. Boya, and J. Carinena 10 ~ for enlightening discussions. This work was partially R. Munoz and G. Fernandez-Anaya, “On a tautochrone-related family of paths,” Rev. Mex. Fıs. E. 56(2), 227–233 (2010). supported by the Brazilian agencies CNPq and FAPERJ. 11R. Munoz,~ G. Gonzalez-Garcıa, E. Izquierdo-De La Cruz, and G. Fernandez-Anaya, “Scleronomic holonomic constraints and conservative a)Electronic mail: [email protected] nonlinear oscillators,” Eur. J. Phys. 32(3), 803–818 (2011). b)Electronic mail: [email protected] 12Dava Sobel, Longitude: The True Story of a Lone Genius Who Solved the c)Electronic mail: [email protected] Greatest Scientific Problem of His Time, Reprint edition (Walker, London, 1Craig F. Bohren and Donald R. Huffman, Absorption and Scattering of 2007). Light by Small Particles (John Wiley and Sons, New York, 1983). 13Simon Gindikin, Tales of Physicists and Mathematicians, 2nd ed. 2L. Landau and E. Lifshitz, Mechanics, 2nd ed. (Pergamon Press, New (Springer, New York, 2007). York, 1969). 14V. I. Arnold, Huygens and Barrow, Newton and Hooke: Pioneers in 3A. B. Pippard, The Physics of Vibration, 1st ed. (Cambridge U.P., New Mathematical Analysis and Catastrophe Theory from Evolvents to York, 1979). Quasicrystals, 1st ed. (Birkh€auser, Basel, 1990).

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