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The Cycloid and the Tautochrone Problem

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The Cycloid and the Tautochrone Problem THE CYCLOID AND THE TAUTOCHRONE PROBLEM Laura Panizzi August 27, 2012 Abstract In this dissertation we will first introduce historically the invention of the Pendulum by Christiaan Huygens, in particular the cycloidal one. Then we will discuss mathematically the cycloid curve, related to the Tautochrone Problem and we’ll compare the circular and cycloidal pendulum. Finally we’ll introduce Abel’s Integral Equation as another way to attack and solve the Tautochrone Problem. 1HistoricalIntroduction Christiaan Huygens (14 April 1629 - 8 July 1695), was a prominent Dutch mathematician, astronomer, physicist and horologist. His work included early telescopic studies elucidating the nature of the rings of Saturn and the discovery of its moon Titan, the invention of the pendulum clock and other investigations in timekeeping, and studies of both optics and the cen- trifugal force. In 1657 patented his invention of the pendulum clock and in 1673 published his mathematical analysis of pendulums, Horologium Oscillatorium sive de motu pendulorum,hisgreatestworkonhorology.Ithadbeenobservedby Marin Mersenne and others that pendulums are not quite isochronous, that is, their period depends on their width of swing, wide swings taking longer than narrow swings. Huygens analysed this problem by finding the shape of the curve down which a mass will slide under the influence of gravity in the same amount of time, regardless of its starting point; the so-called 1 Tautochrone Problem.Bygeometricalmethodswhichwereanearlyuseof calculus, he showed that this curve is a Cycloid. ”On a cycloid whose axis is erected on the perpendicular and whose vertex is located at the bottom, the times of descent, in which a body arrives at the lowest point at the vertex after hav- ing departed from any point on the cycloid, are equal to each other...”1 2TheCycloid The cycloid is the locus of a point on the rim of a circle of radius R rolling without slipping along a straight line. It was first studied by Nicola Cusano and it was named by Galileo in 1599. It is also the solution to the Tautochrone Problem. The cycloid is represented by a parametric expression that contains the radius R of the circle and the rotation angle θ.Thefollowingformularepresents two cycloids generated by a circle of radius R,onetroughtheoriginandthe other trough π,whichconsistofthepoints(x, y), with x = R(θ sin θ) ± ± θ [ π,π] C ≡ y = R(1 cos θ) ∈ − ! − For a given θ,thecircle’scentreliesatx = Rθ, y = R. 2.1 Arc Length The arc length of a curve defined parametrically by x = f(t)andy = g(t) is θ s(θ)= ds where ds = dx2 + dy2 0 " # For the cycloid we have dx = R(1 cos θ)dθ ± !dy = R sin θdθ 1Blackwell, Richard J. (1986). Christiaan Huygens’ The Pendulum Clock.Ames,Iowa: Iowa State University Press. ISBN 0-8138-0933-9. Part II, Proposition XXV, p. 69 2 so θ θ ds =2R cos dθ s(θ)=4R sin 2 ⇒ 2 2.2 Critical Points From the study of the first derivative we obtain θ + dy sin θ tan 2 ( ) = = θ C dx 1 cos θ cot ( −) ± ! 2 C + we have the maxima for in θ =0+kπ and minima for − in θ = kπ. While from the study ofC the second derivative we get2 C 2 1 > 0(+) d y 1 4cos2 θ 2 C 2 = 2 = 1 dx ±(1 cos θ) − < 0( ) 4sin2 θ − ± 2 C + has always a positive curvature and − anegativeone. C C x = πR x =0 x = πR A" θ = π − ; O θ =0 ; A θ = π ⇔ − !y =2R ⇔ !y =0 ⇔ !y =2R For + the circle rolls over the line y =0 C For − the circle rolls under the line y =2R C 2Let us recall 2 2 2 1 cos 2α cos α sin α =cos2α sin α = − 2 − 2 1+cos 2α !2cosα sin α = sin 2α ⇒ !cos α = 2 3 Figure 1: Plot of Cycloid with R = 1 and π<θ<π, + (continuous) and − C − (dashed) C 3CircularPendulumvs.CycloidalPendu- lum Now we consider the equation of motion for a particle mass constrained to move in a vertical curve without friction. The equation is: d2s F (s)=m t dt2 where Ft(s) is the tangential force and s the curvilinear abscissa. Gravity is the only force acting upon the mass, so dy F (s)= mg cos φ;cosφ = . t − ds With this equation one can describe the motion of two type of pendulums: the Circular Pendulum, first studied by Galileo Galilei in 1581, and the Cycloidal Pendulum, studied by Huygens in 1673. 3.1 Cycloidal Pendulum Starting from the previous equation for a cycloidal curve, generated by a circle of radius R x = R(θ sin θ) ± π θ π y = R(1 cos θ) − ≤ ≤ ! − 4 we have θ dy = R sin θdθ and ds =2R cos dθ 2 then we obtain dy sin θ θ cos φ = = θ = sin ds 2cos 2 2 and the equation of motion becomes d2s θ + g sin =0 dt2 2 also noting that θ θ s s(θ)=4R sin = sin = 2 ⇒ 2 4R we finally have d2s g + s =0 dt2 4R The result is an harmonic motion s = s(θ)withperiod ∀ 4R T =2π ' g that does NOT depend on the amplitude of the oscillation. 3.2 Circular Pendulum In the case of a circular curve of radius L x = L sin α π π α y = L(1 cos α) − 2 ≤ ≤ 2 ! − we have dy = L sin αdα and ds = Ldα 5 Figure 2: Cycloidal pendulum: T = T0,CircularpendulumT T0, T0 = L ≥ 2π g with L =4R ( then we obtain dy cos φ = = sin α ds and the equation of motion becomes d2s + g sin α =0 dt2 d2α g + sin α =0 dt2 L In this case there is an harmonic motion only for sin α α with period ≈ L T 2π . * ' g 4AbelandTautochrone Niels Henrik Abel attacked a generalized version of the Tautochrone Problem (Abel’s mechanical problem), namely, to find, given a function T (y)that 6 specifies the total time of descent for a given starting height, an equation for the curve that yields such result. The Tautochrone Problem is a special case of Abel’s mechanical problem when T (y) is a constant. For the principle of conservation of energy, since the particle is frictionless, 2 1.8 1.6 y 1.4 0 1.2 y 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 x Figure 3: Tautochrone curve and thus loses no energy to heat, its kinetic energy at any point is exactly equal to the difference in potential energy from its starting point. Formally: 1 ds 2 m = mg(y y) 2 dt 0 − ) * ds where dt is the velocity of the particle, s the distance measured along the curve and mg(y y) the gravitational potential energy gained in falling from 0 − an initial height y0 to a height y. ds = 2g(y y) dt ± 0 − # ds dt = ± 2g(y y) 0 − 1 ds dt = # dy − 2g(y y) dy 0 − # 7 In the last equation, we’ve anticipated writing the distance remaining along the curve as a function of height (s(y)) and recognized that the distance remaining must decrease as time increases (thus the minus sign). Now we integrate from y = y0 to y =0togetthetotaltimerequiredforthe particle to fall: y=0 1 y0 1 ds T (y )= dt = dy 0 √2g √y y dy "y=y0 "0 0 − This is called Abel’s integral equation and allows us to compute the total ds time required for a particle to fall along a given curve (for which dy would be easy to calculate). Abel’s mechanical problem is the opposite one: given ds T (y0)wewishtofind dy ,fromwhichanequationforthecurvewouldfollow in a straightforward manner. ds To proceed, we note that the integral on the right is the convolution of dy 1 with √y and thus take the Laplace transform of both sides: 1 1 ds [T (y0)] = L √2g L √y L dy + , + , 1 1 Since = √πz− 2 ,wenowhaveanexpressionfortheLaplacetransform L √y ds of dy in- terms. of T (y0)’s Laplace transform: ds 2g 1 = z 2 [T (y )] L dy π L 0 + , / For the tautochrone problem, T (y0)=T0 is constant. Since the Laplace 1 transform of 1 is z ,weproceed: ds 2g 1 = T z− 2 L dy π 0 + , / Making use again of the Laplace transform above, we anti-transform and conclude: ds √2g 1 = T0 . dy π √y 8 Now if we let ds dx 2 2gt2 = 1+ and b = dy dy π2 ' ) * we obtain dx b y b y = − from which x = − dy + c dy y y ' " ' with the substitutions R y = b sin2 φ, b y = b cos2 φb= and dy =2b sin cos φdφ − 2 R cos2 φ x = R sin cos φdφ= R cos2 φdφ+ c R sin2 φ " ' " Recalling 1+cos2φ cos2 φ = with 2φ = θ 2 we finally have x = R(θ sin θ) ± . y = R(1 cos θ) ! − References [1] R. Gorenflo, S. Vessella, Abel Integral Equation -AnalysisandApplica- tion, Springer-Verlag Berlin Heidelber, 1991 [2] F. Mainardi, Lecture Notes on Mathematical Physics 9.
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