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Is the Tautochrone Curve Unique? Pedro Terra,A) Reinaldo De Melo E Souza,B) and C

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Is the Tautochrone Curve Unique? Pedro Terra,A) Reinaldo De Melo E Souza,B) and C 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 .
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