On a Tautochrone-Related Family of Paths

ENSENANZA˜ REVISTA MEXICANA DE FISICA´ E 56 (2) 227–233 DICIEMBRE 2010

On a tautochrone-related family of paths

R. Munoz˜ Universidad Autonoma´ de la Ciudad de Mexico,´ Centro Historico,´ Fray Servando Teresa de Mier 92 y 99, Col. Obrera, Del. Cuauhtemoc,´ Mexico´ D.F., 06080, Mexico,´ e-mail: [email protected] G. Fernandez-Anaya´ Universidad Iberoamericana, Departamento de F´ısica y Matematicas,´ Av. Prolongacion´ Paseo de la Reforma 880, Col. Lomas de Santa Fe, Del. Alvaro Obregon, Mexico´ D.F., 01219, Mexico,´ e-mail: [email protected] Recibido el 26 de julio de 2010; aceptado el 30 de agosto de 2010

An alternative approach to the properties of the tautochrone and brachistochrone curves is used to introduce a family of curves complying with relations where the time of descent is proportional to a fractional power of the height difference. These curves are classiﬁed acording with their symmetries. Further properties of these curves are studied.

Keywords: Analytical mechanics; Huygens’s isochrone curve; Abel’s mechanical problem.

Utilizamos un tratamiento alternativo de las curvas tautocrona y braquistocrona para introducir una familia de curvas que cumplen con relaciones en las que el tiempo de descenso es directamente proporcional a la altura descendida, elevada a un valor fraccionario. Las mencionadas curvas son clasiﬁcadas de acuerdo con sus simetr´ıas. Se estudian otras propiedades de dichas curvas.

Descriptores: Mecanica´ anal´ıtica; curva isocrona de Huygens; problema mecanico´ de Abel.

PACS: 01.55.+b; 02.30Xx; 02.30.Hq; 02.30.Em

1. Introduction cloid. This is in stark contrast with the inclined plane, where the time of descent is proportional to the square root of the In 1658 Christiaan Huygens made public his discovery of height difference: the tautochrone, that is: the path which a point-like parti- T ∝ (∆y)1/2 (2) cle must follow so that the period of its motion comes out to be exactly independent of its amplitude. This is in con- It is then natural to ask for the possible values of an exponent trast with Galileo’s pendulum, where the period is indepen- β such that one can construct a path that complies with dent only in the small amplitude approximation. A thought- provoking, insightful, description of Huygens’s work can be T ∝ (∆y)β (3) found in Ref. 1. For a discussion in simpler terms terms see [2] and [3]. The rest of this article deals with the solution of such ques- Some forty years after Huygens’s work, Johann Bernoulli tion, and with its consequences. In this endeavour the found the brachistochrone, i.e. the path of quickest descent brachistochrone will appear more than once. In order to ad- from a fixed starting point to a fixed end point. As is well dress the question we have just posed, use will be made of a known, this was later to become one of the seminal problems generalization of Abel’s elegant alternative deduction of the in variational calculus. A sketch of Bernoulli’s deduction is tautochrone [4], which, as Abel himself commented, can be included in Ref. 4, while [5] contains a brief account of the cast in the language of fractional integro-differential opera- discovery, as well as pointers for the deduction of the curve tors [6]. Albeit a long and noble history in mathematics (dat- through variational calculus. ing back to Leibniz), [7] fractional analysis has only begun Both curves are related, as Bernoulli himself noticed. to be used by physicist in the last thirty years or so. We thus Huygens’s tautochrone is the complete cycle of an inverted hope that this article will serve, if nothing more, to call atten- cycloid, while a brachistochrone is a segment of an inverted tion on this exciting subject (some examples of the applica- cycloid, with the initial point at an apex. tions of fractional analysis in physics may be found in Refs. 6 As can be deduced from the tautochrone property (i.e. to 11). that for this path the period is independent of the amplitude), Some recent generalizations of the tautochrone include: the time T it takes for a point-like particle to descend along the tautochrone with friction [12], the relativistic tau- an inverted cycloid to a minimum of the path is independent tochrone [13], the tautochrone in rotating frames of Ref. 14 of ∆y, the height difference from the starting point to the and the tautochrone under an arbitrary potential [15]. Finally, minimum: 0 for other applications of the calculus of variations in classi- T ∝ (∆y) (1) cal mechanics and other branches of physics one may recom- and depends only on the parameters that define the given cy- mend [16] and [17]. 228 R. MUNOZ˜ AND G. FERNANDEZ-ANAYA´

The rest of this article is organized as follows: in Sec. 2 we review the properties of the tautochrone and brachistochrone curves under a new light. In Sec. 3 we introduce a family of curves complying with relation (3) for −1/2 < β ≤ 1/2, and classify them according to their symmetries. In Sec. 4 we study further (probably counterintu- itive) properties of these paths. Sec. 5 is reserved for conclusions.

2. Tautochrone vs. brachistochrone

Mathematicians carefully distinguish between curves (with a given parametrization) and paths (which are the images of curves.) We are interested in obtaining paths along which a point-like particle would descend, but we will do so by find- FIGURE 1. Some examples of brachistochrones. The thick curve is ing curves which have those paths as images. A given curve also a half tautochrone. describes a unique path, but there are an infinite number of curves that correspond to a given path. In the present context, On the other side, in a brachistochrone the total horizon- this distinction is quite unimportant: we will make clear each tal length and the maximum difference of height must be in- time that the same path gets two different parametrizations. dependent as there is a brachistochrone connecting any two Huygens’s tautochrone can be parametrized as follows: chosen points xi and xf . ¡ ¢ The brachistochrone curve between (x , y ) and (x , y ) can x(θ) = a θ − sin θ + x − aπ i i f f 2 L 2 be parametrized as follows: ¡ ¢ 0 ≤ θ ≤ 2π a (4) ¡ ¢ a y(θ) = 2 1 + cos θ + yL x(θ) = 2 θ − sin θ + xi ¡ ¢ 0 ≤ θ ≤ θf a (5) the x-axis lying along the horizontal, with the positive y-axis y(θ) = 2 1 + cos θ + yi − a in the upward direction. As already said, this describes a family of inverted cy- cloids, and the presence of parameters yL and xL allows us to where a and 0 < θf ≤ 2π are such that translate the tautochrone’s lowest lying point, (x , y ), any- L L a where we want to (one can take a tautochrone from Antwerp xf = x(θf ) = (θf − sin θf ) + xi to Paris and it will remain a tautochrone). 2 Now, parameter a > 0 allows us to choose the maximum and difference of height for a given tautochrone, that is: the difference of height y(0) − y(π) = a from apex (x(0), y(0)) a yf = y(θf ) = (cos θf + 1) + yi − a. to the lowest lying point (x(π), y(π)) = (xL, yL). But once 2 this maximum difference of height is fixed, the total horizontal length of the tautochrone is also fixed, its value given by Thus, an extra parameter is needed to describe the family of brachistochrones. x(2π) − x(0) = aπ. Equation (4) indicates us that we can divide the tau- Let us remark: in a tautochrone the total difference of height tochrone into a left half, (x+(θ), y+(θ)), and its reflection and the total horizontal length are not independent. through the vertical axis x = xf , (x−(θ), y−(θ)), giving rise to a new parametrization:

¡ ¢ a aπ x±(θ)= ± 2 θ− sin θ +xL ∓ 2 ¡ ¢ 0 ≤ θ ≤ π (6) a y±(θ)= 2 1+ cos θ +yL

Each one of these halves is a brachistochrone (its fixed points being an apex and the lowest lying point) of a very independent of any parameter (the lowest lying point is com- special kind: the ratio of the maximum height difference to mon to both curves, which represents no problem.) the total horizontal length has a fixed value: 2 If, on the other hand, on takes a brachistochrone (5) and (y (0) − y (π))/|x (0) − x (π)| = , (7) ± ± ± ± π joins it with its reflection through the x = xL axis, one sim-

Rev. Mex. F´ıs. E 56 (2) (2010) 227–233 ON A TAUTOCHRONE-RELATED FAMILY OF PATHS 229 ply does not get a tautochrone in the general case. A similar result is easily proven for the tautochrone. Perhaps a better way of understanding these curves is by Thus, if you build a scale model of a brachistochrone (tau- noticing that both the tautochrone and the brachistochrone tochrone), the result will also be a brachistochrone (tau- properties are invariant under scale transformations of the tochrone). In the next section we will generalize this results type (x, y) → (x0, y0) = (κx, κy) for κ > 0. for a wider family of curves. Indeed, in the frictionless motion of a particle along any Summing up, we have pointed out the differences be- given path σ, the conservation of the total mechanical en- tween tautochrones and brachistochrones, while making clear ergy dictates that T , the time it takes the particle to travel that the tautochrone and brachistocrone properties are both from the starting point x = (x , y ) ∈ σ to the end point i i i preserved by scale transformations. From Eqs. (4) and (5) xf = (xf , yf ) ∈ σ, is given by Z it is easily proven that (as would be expected) both proper- 1 ds ties are also preserved by translations and reflections through Tσ = √ √ (8) 2g y any vertical axis. If this seems a bit trivial, consider that nei- σ p ther of this properties is preserved by a general rotation in the (where ds = (dx)2 + (dy)2 is the differential of arch- x − y plane nor by a general compression or expansion of the length along σ) granted that the particle starts its motion with curve (i.e. by transformations of the type (x, y) → (x0, y0) = zero initial velocity and moves under the influence of a grav- (kx, y)). In other words: a tilted tautochrone (brachis- itational field g = (0, −g). Consider now a scale transforma- tochrone) or a compressed tautochrone (brachistochrone) is 0 0 0 0 tion x → x = κx κ > 0, that sends xi to xi = (xi, yi) no longer a tautochrone (brachistochrone). In Sec. 3 we will 0 0 0 and sends (xf , yf ) to xf = (xf , yf ). There is a one-to-one generalize this to a wider family of curves, but before going mapping between each curve connecting xi with xf and each on there are a couple of facts regarding the tautochrone that 0 0 curve connecting xi with xf , given by we need to point out. ¡ ¢ y(x) → y0(x0) = κy(x) = κy κ−1x0 First, the period of motion in a given tautochrone is easy 0 0 to calculate [2], and the result is: (in symbolic terms σ → σ = κσ). According with (8) Tσ, 0 0 the time it will take a particle to travel from xi to xf along 0 r σ , is related to Tσ through: a 4T = 4π , (9) 0 1/2 g Tσ = κ Tσ so that if a particular curve σ minimizes T , the corresponding curve σ0 = κσ will minimize T 0, that is: we will frequently make use of this result.

0 1/2 Finally, consider, for a ﬁxed value H > 0, and a ﬁxed point δTσ = κ δTσ = 0. (xL, yL), the uni-parametric family of curves {S+,φ}|0≤φ<π given by:

¡ ¢ H πH x+,φ(θ) = 1+cos φ θ − sin θ + xL − 1+cos φ ¡ ¢ φ ≤ θ ≤ π H (10) y+,φ(θ) = 1+cos φ 1 + cos θ + yL

Notice that:

- all this curves start at a same height

yi = y+,φ(φ) = H + yL

(x , y ) = (x (π), y (π)) and they all end at point L L φ φ . FIGURE 2. Some S+,φ paths for H=1. The thick curve is the half tautochrone S+,φ=0. Dotted: φ=π/5, dot-dashed: φ=2π/5, - The curve S+,φ is a segment of a left half tautochrone dashed: φ=3π/5, and thin: φ=4π/5. with apex at In so many words {S+,φ}|0≤φ<π is just the family of all πH 2H left half tautochrone segments with a ﬁxed total height differ- (x , y ) : =(x − , +y ) M,φ M,φ L 1+ cos φ 1 + cos φ L ence H, ending at a ﬁxed lowest lying point (xL, yL). (But they are not segments of the same tautochrone!) We note in passing that, with the exception of S+,φ=0,

Rev. Mex. F´ıs. E 56 (2) (2010) 227–233 230 R. MUNOZ˜ AND G. FERNANDEZ-ANAYA´ none of this segments is a brachistochrone. Indeed, for each Suppose, further more, that the particle is constrained to point (x+,φ(θ), y+,φ(θ)) lying in a given segment S+,φ we move, without friction, along a curve σ univocally described can easily find, from (5), the brachistochrone that connects it by the function with (xL, yL); and this is not, in general case, S+,φ itself. x = x (y) for all y ∈ [y , y ] ⊆ R More to the point: each one of the S+,φ curves, being a σ L M segment of a tautochrone, has what we will call the indepen- where function x (y) is continuously differentiable in dence of height property. That is, if a particle moving along σ 0 0 (yL, yM ), and such that S+,φ has zero initial velocity at some point (x , y ) ∈ S+,φ, then it will reach (xL, yL) in a time Tφ, given by: lim xσ(y) = xσ(yL), + s y→yL H T = π lim xσ(y) = xσ(yM ), φ − g(1 + cos φ) y→yM independently of (x0, y0), according with (9). The family of and that the limit paths just described can be extended to include the reflections dx lim σ (y) of the S+,φ, giving us what we will call the {S±,φ}|0<φ≤π y→y− dy family. These are, up to a translation, the only paths with M the independence of height property with a given total height exists. difference H (the unicity of the solution of Abel’s equation, If the particle has zero initial velocity at an initial “height” which we will see later, warrants this). As Tφ is the same for yi ∈ (yL, yM ], then the conservation of the total mechani- S−,φ as it is for S+,φ we can then state that: cal energy of the system dictates that the particle will reach Of all the paths with the height independence property, with height yL in a time Tσ(yi) > 0 given by a total height difference H, there is one (up to translations yi Z dsσ and reflections) with minimum T, given by: dy Tσ(yi) = − √ 1 dy (12) s 2g(yi − y) 2 H yL Tφ=0 = π (11) 2g where s µ ¶2 this path is S+,φ=0. dx ds (y) = − 1 + σ dy Granted, S+,φ=0 happens to be a brachistochrone, but the σ dy property just stated is logically independent of the brachistochrone property because: is the differential of the arch length traveled by the particle from height yi to height y while moving on σ. - The family {S±,φ}|0≤φ<π is not a good sample of all Equation (12) can be written in the slightly more concise the curves connecting two given points: we have not form:

presented an alternative proof of the brachistochrone. q ³ 1 ´ π 2 dsσ Tσ(yi) = − 2g IyL (yi) (13) - On the other side, we will find other families of curves dyi with an element that minimizes the time of descent for with the use of the Riemann-Liouville (right hand) fractional that given family; and this element cannot be, in the integral, defined as [6-7,18]: general case, a brachistochrone. Zt 1 f(τ) If we consider now the invariance under scale transforma- Iαf(t) := dτ (14) a Γ(α) (t − τ)1−α tions, we will see that the half tautochrone is really unique (up a to an arbitrary combination of translations, reflections and scale transformations.) We will generalize this result in the for t > a and α > 0. Here, Γ stands for the gamma function following sections. (the necessary information about the properties of the gamma function may be found in Ref. 19). We now examine the implications of imposing on curve 3. A family of curves σ a condition of the type

β Consider a classical, non-relativistic, point-like particle of Tσ(yi) = K(yi − yL) (15) mass m moving on the x-y-plane with potential energy for all y ∈ (yL, yM ] for a and ﬁxed β ∈ R and a real constant U(x, y) = mgy K > 0 (it is implicit that σ depends on the values of β and K, and that the admissible values of K may be restricted by for some g ∈ R+. the value of β).

Rev. Mex. F´ıs. E 56 (2) (2010) 227–233 ON A TAUTOCHRONE-RELATED FAMILY OF PATHS 231

Let us note in passing that for β = 0 (15) essentially co- where, by definition: incides with Abel’s equation for the tautochrone: Ãr ! 2 2g Γ(β + 1) 1−2β ³ ´ d 1 ds h := K (21) I 2 σ (y) = 0 (16) π Γ(β + 1/2) dy yL dy In order to retain the physical interpretation of Eq. (20) the while β = 1/2 corresponds to the trivial case of the friction- integrand on the r.h.s. must remain real-valued for values of less inclined plane. y arbitrarily close to yL, and so the expression is only valid To be more precise, for β = 1/2 and a given final position for β ≤ 1/2 . Moreover, for any acceptable value of β (i.e. (xL, yL), the solutions constitute a uni-parametric family of −1/2 < β ≤ 1/2) h must take values in [H, ∞) where H straight lines {lθ}|0<θ<π given by stands for H := yM − yL (22) x = x (y) = (y − y ) cot θ + x 0 < θ < π (17) θ L L Now, in expression (20) not all the symmetries present in our −1 system are explicit. With the use of some algebra, we can for in this case dsθ = −(sin θ) ; and by plugging this last expression directly in (12) we get, after a straightforward cal- rewrite it as: y−yL culation: h Z p r s x (y) = x ± h η2β−1 − 1 dη (23) 2 (y − y )1/2 2S σ L T (y ) = i L = i g sin θ g sin θ 0 Thus, a curve σ is identified by four different parameters: where S is the distance traveled by the particle along the in- β and xL, yL, ± and h. From now on we shall write clined planed, in accordance with the well known result. σβ,xL,yL,±,h in order to identify a specific curve. The free- Turning to the general case, we resort to the well known dom to chose xL and yL means that property (3) is invariant fact [6] that Abel’s (general) integral equation: under translations. In an analogous manner, the freedom to chose the sign ± is the result of invariance under reflections α f(t) = (I0 φ)(t)(t > 0, 0 < α < 1) through a vertical axis. Finally, h is related with the invariance of the property under scale transformations. Indeed, if has as solution one multiplies xL, yL, y and h -all by a same factor κ- then xσ(y) is also multiplied by this factor. Thus, symbolically: d 1−α φ(t) = (I0 f)(t), dt σβ,κxL,κyL,κh = κσβ,xL,yL,h so that for every curve σ complying with (15) we can write and the property is preserved under scale transformations. Thus, the following lemma has been established: √ yZ−yL ds 2g d νβ Lemma 1 There does not exist a continuously differen- σ (y) = − K dν (18) 1/2 tiable bounded curve in the plane, in which a classical non- dy π dy (y − yL − ν) 0 relativistic point-like particle could move under the influence The integral on the r.h.s. of (18) is divergent for β ≤ −1; and of a spatially homogeneous time independent gravitational for β > −1 one can apply the convolution theorem to get field g(x, t) = g, starting with zero initial velocity at position x , complying with a condition r i ds 2g Γ(β + 1) d β σ (y) = − K (y − y )β+1/2 (19) ∆t ∝ (∆y) dy π Γ(β + 3/2) dy L for values of β > 1/2, or β ≤ −1/2 where so that sσ will be finite at yL only if β ≥ −1/2, and the ∆y : =−g · (xi−xL) case β = −1/2 has no geometrical (let alone physical) interpretation, as it would imply that the arch length traveled for a fixed position xL, ∆t being the lapse of time required for the particle to move from x to x along the curve. from (xi, yi) to any given point on the curve would be same, i L irrespective of the end point. Thus, physically acceptable For −1/2 < β ≤ 1/2 the solutions exist, as shown bounded curves exist only for β > −1/2. With this caveat in in (23), and are unique up to an arbitrary combination of mind, from (19) we may conclude reflections, translations and scale transformations. This lemma can easily be extended for charged particles y s Z L µ ¶2 in the presence of spatially homogeneous electrostatic fields, dsσ xσ(y)=xL± −1dy0 but the charge-mass quotient would then appear in our equa- dy0 y tions. Let us note, in passing, that arbitrary combinations of ZyLr ³y0 − y ´2β−1 translations, reflections and scale transformations constitute =x ± L −1dy0, x ∈ R, (20) L h L a group under composition. This is the group that leaves the y tautochrone property unaltered.

Rev. Mex. F´ıs. E 56 (2) (2010) 227–233 232 R. MUNOZ˜ AND G. FERNANDEZ-ANAYA´

FIGURE 4. The function Υ(β) := Γ(β + 1/2)/Γ(β + 1) in the region −1/2 < β ≤ 1/2.

then the particle will reach the origin in a lapse of time TβH (y), given by: µ r ¶ 1/2−β π Γ(β + 1/2) β βH T (y) = H y (27) FIGURE 3. Some σ paths with H=1. The thick path is a half βH 2g Γ(β + 1) tautochrone (β=0). Dot-dashed: β=0.3, dashed: β=0.15, thin: β= − 0.15, and dotted: β= − 0.24. and this is (up to translations and reflections) the swiftest path with a total height difference H and compatible with condition (3). 4. Further properties It is then natural to ask: of all σβH paths which one is the swiftest for a fixed total height difference H? i.e., which one According with definition (21) and the condition established will make a particle descend a height H in a minimum time? for the admissible values of h, Eq. (22), for a given β the Function Γ(β +1/2)/Γ(β +1) is monotonically decreas- physically admissible values of K are restricted by ing in the region of interest (as can be seen in Fig. 4 so that the minimum is achieved at β = 1/2, which corresponds to ∗ K ≥ Kβ (24) a segment of length H of a vertical straight line (quite obvi- ous), and as β approaches the value −1/2, T ∗ diverges: ∗ βH where Kβ stands for ∗ r lim TβH = ∞. π Γ(β + 1/2) β→− 1 − K∗ := H1/2−β 2 β 2g Γ(β + 1) Let us note that there is a fixed ratio between the total height βH Thus, for a fixed β and a fixed total height difference H > 0 difference H and the horizontal length xβ,H (H) in a σ ∗ curve: there is a minimum total time of descent, TβH , compatible with condition (3), this time is given by: 1 Z p r xβH (H) 2β−1 π Γ(β + 1/2) R(β) := = η − 1dη (28) T ∗ = H1/2 (25) H βH 2g Γ(β + 1) 0 and this ratio diverges as β approaches the value −1/2: and it is achieved by a curve (lets call it σβH ) given by the parametric equation: lim R(β) = ∞ . β→− 1 + y/H 2 Z p 2β−1 And so, we have established a second lemma: xβH (y) = H η − 1 dη y ∈ (0,H]. (26) Lemma 2 Each element in a family of paths 0 βH {σ }|−1/2<β<1/2, defined in (26) has a fixed horizontal This curve is unique up to translations and reflections. length to total height difference ratio, independent of H, In other words, if a point-like particle starts its movement given in (28). This ratio is a monotonically decreasing func- at position tion of β. A particle starting from rest at height H would de- ∗ βH ∗ (xβH (H),H) ∈ σβH then it will take a time TβH , as given scend along a given path σ in a time TβH , given by (25), in (25), for the particle to reach the final position (0, 0), but if which is, for fixed H, also a monotonically decreasing func- it starts at any other point of the path, with height y ∈ (0,H), tion of β. Thus, every σβH with 0 < β ≤ 1/2 is shorter

Rev. Mex. F´ıs. E 56 (2) (2010) 227–233 ON A TAUTOCHRONE-RELATED FAMILY OF PATHS 233 and swifter than the half tautochrone, and every σβH with - Highlights the roleˆ of symmetries, even in well-trodden −1/2 < β < 0 is longer and slower than the tautochrone. branches of physics. The path of quickest descent for a given height is a segment of a vertical straight line. As β approaches −1/2 with There are some interesting consequences and questions that fixed height, the total arch-length of the path grows without we have left aside: limit, as does the total time of descent. • These curves are independent of the mass of the parti- This result can not be held against the brachistochrone: cle: m is absent in every expression, starting from (8) first, because we are dealing with paths connecting two given because gravitational mass and inertial mass were can- heights; whereas brachistochrones connect two given fixed celed out even before writing this equation. This is just points. In second place, the vertical straight line segment can a consequence of the equivalence principle. be considered as a kind of collapsed brachistochrone. √ • The eerily ubiquitous constant Γ(1/2) = π, present in all our calculations, just begs the question: are there 5. Conclusions and outlook possible generalizations of our curves in the frame of strong gravitational fields? In the framework of classical mechanics, we have generalized the tautochrone path, generating a family of curves in which • In one-dimensional physics, the harmonic oscillator the time of descent is proportional to a fractional power of the potential is the isochronous potential, i.e., this is the height difference. The material just presented may be used to only potential for which the frequency is strictly inde- complement and enrich any university course exposition of pendent of the amplitude. Can this potential be derived classical mechanics and/or analytical mechanics at an inter- from the Euler-Lagrange equations of the tautochrone mediate or senior level, as it: curve? Can this potential be generalized for some or all of our curves? - Shows us that there are still interesting things to learn in classical mechanics at a fairly elementary level. Acknowledgments - Illustrates the use of mathematical tools such as the Laplace transform and the gamma function in physical The support of SNI-CONACYT (Mexico) is duly acknowl- problems. edged.

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