Exact Solution for the Nonlinear Pendulum (Solu¸CÃoexata Do PêndulonÃolinear)

Home , Harmonic oscillator, Pendulum (mathematics), Simple harmonic motion

Revista Brasileira de Ensino de F´ısica, v. 29, n. 4, p. 645-648, (2007) www.sbﬁsica.org.br

Notas e Discuss˜oes

Exact solution for the nonlinear pendulum (Solu¸cãoexata do pêndulonãolinear)

A. Beléndez1, C. Pascual, D.I. Méndez,T. Beléndezand C. Neipp

Departamento de F´ısica, Ingenier´ıade Sistemas y Teor´ıade la Se˜nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007

This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is diﬀerent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75π (135◦) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ω in terms of the complete elliptic integral of the ﬁrst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics. Keywords: simple pendulum, large-angle period, angular displacement.

Este artigo aborda a oscila¸cãonão-linearde um pêndulosimples e apresenta nãoapenas a fórmula exata do per´ıodo mas tambéma dependencia temporal do deslocamento angular para amplitudes das oscila¸cõese a freqüênciaangular para pequenas oscila¸cões. O deslocamento angular éescrito em termos da fun¸cãoel´ıptica de Jacobi sn(u;m) usando as seguintes condi¸cõesiniciais: o deslocamento angular inicial édiferente de zero enquanto que a velocidade angular inicial ézero. Os deslocamentos angulares sãoplotados usando Mathematica, um dispon´ıvel programa simbólicode computador que nos permite plotar facilmente a fun¸cãoobtida. Como ver- emos, mesmo para amplitudes tãograndes quanto 0,75π (135o) époss´ıvel usar a expressãopara o deslocamento angular mas considerando a expressãoexata para a freqüênciaangular w em termos da integral el´ıpticacompleta de primeira espécie.Conclu´ımosque, para amplitudes menores que 135o, o movimento periódicoexibido por um pêndulosimples épraticamente harmônico,mas suas oscila¸cõesnãosãoisócronas(o per´ıodo éuma fun¸cãoda amplitude inicial). Acreditamos que o presente estudo possa tornar-se um exerc´ıcioconveniente e frut´ıferopara o ensino e para uma melhor compreensãodo pêndulonão-linearem cursos avan¸cadosde mecânicaclássicana gradua¸cão. Palavras-chave: pêndulosimples, per´ıodo a grandes ângulos,deslocamento angular.

Perhaps one of the nonlinear systems most studied ferential equation with a nonlinear term (the sine of an and analyzed is the simple pendulum [1-12], which is angle). It is possible to find the integral expression for the most popular textbook example of a nonlinear sys- the period of the pendulum and to express it in terms of tem and is studied not only in advanced but also in elliptic functions. Although it is possible in many cases introductory university courses of classical mechanics. to replace the nonlinear differential equation by a corre- The periodic motion exhibited by a simple pendulum sponding linear differential equation that approximates is harmonic only for small angle oscillations [1]. Be- the original equation, such linearization is not always yond this limit, the equation of motion is nonlinear: feasible. In such cases, the actual nonlinear differential the simple harmonic motion is unsatisfactory to model equation must be directly dealt with. the oscillation motion for large amplitudes and in such In this note we obtain analytical exact formulas for cases the period depends on amplitude. Application of the period of a simple pendulum and we present some Newton’s second law to this physical system gives a dif- figures in which the angular displacement is plotted as 1E-mail: [email protected].

a function of the time for different initial amplitudes. where we use the trigonometric relation Moreover, we stress that one may show to the stu- µ ¶ θ dents the advantage of using available symbolic com- cos θ = 1 − 2 sin2 . (8) puter programs such as Mathematica. 2 As we can point out previously, one of the simplest Now let nonlinear oscillating systems is the simple pendulum. This system consists of a particle of mass m attached y = sin(θ/2) (9) to the end of a light inextensible rod, with the motion and taking place in a vertical plane. The differential equation modelling the free undamped simple pendulum is 2 k = sin (θ0/2). (10) 2 d θ 2 From Eqs. (3), (9) and (10) we have 2 + ω0sinθ = 0, (1) dt √ where θ is the angular displacement, t is the time and y(0) = k. (11) ω0 is defined as It is easy to obtain the value of dθ/dt as a function r of dy/dt as follows. Firstly, from Eq. (10), we have g ω0 = . (2) l dy dy dθ 1 dθ = = cos(θ/2) (12) Here l is the length of the pendulum and g is the ac- dt dθ dt 2 dt celeration due to gravity. Because of the presence of and secondly the trigonometric function sinθ, Eq. (1) is a nonlinear differential equation. µ ¶ µ ¶ µ ¶ dy 2 1 θ dθ 2 We consider that the oscillations of the pendulum = cos2 = are subjected to the initial conditions dt 4 2 dt µ ¶ µ ¶ µ ¶ 1 £ ¤ dθ 2 1 ¡ ¢ dθ 2 dθ 1 − sin2(θ/2) = 1 − y2 .(13) θ(0) = θ0 = 0 (3) 4 dt 4 dt dt t=0 Then, we have where θ0 is the amplitude of the oscillation. The system oscillates between symmetric limits [-θ ,+θ ]. The µ ¶2 µ ¶2 0 0 dθ 4 dy periodic solution θ(t) of Eq. (1) and the angular fre- = 2 . (14) quency ω (also with the period T = 2π/ω) depends on dt 1 − y dt the amplitude θ0. Substituting Eqs. (9), (10) and (14) into Eq. (7), one Equation (1), although straightforward in appear- get ance, is in fact rather difficult to solve because of the µ ¶ 4 dy 2 ¡ ¢ nonlinearity of the term sinθ. In order to obtain the = 4ω2 k − y2 , (15) exact solution of Eq. (1), this equation is multiplied by 1 − y2 dt 0 dθ/dt, so that it becomes which can be rewritten as follows

dθ d2θ dθ µ ¶2 µ ¶ + ω2sinθ = 0 (4) dy y2 2 0 = ω2k(1 − y2) 1 − . (16) dt dt dt dt 0 k which can be written as We do deﬁne new variables τ and z as " µ ¶ # d 1 dθ 2 y − ω2cosθ = 0 (5) τ = ω0t and z = √ . (17) dt 2 dt 0 k Equation (5), which corresponds to the conservation Then Eq. (16) becomes of the mechanical energy, is immediately integrable, µ ¶ dz 2 taking into account initial conditions in Eq. (3). From = (1 − z2)(1 − kz2), (18) Eq. (5) we can obtain dτ where 0 < k < 1, and µ ¶2 dθ 2 µ ¶ = 2ω0 (cos θ − cos θ0) (6) dz dt z(0) = 1 = 0. (19) dτ which can be written as follows τ=0 Solving Eq. (18) for dτ gives µ ¶ · µ ¶ µ ¶¸ dθ 2 θ θ dz = 4ω2 sin2 0 − sin2 , (7) dτ = ±p . (20) dt 0 2 2 (1 − z2)(1 − kz2) Exact solution for the nonlinear pendulum 647

The time τ to go from the point (1,0) to the point ½ · µ ¶ (z,dz/dτ) in the lower half –plane of the graph of dz/dτ θ θ θ(t) = 2 arcsin sin 0 sn K sin2 0 − as a function of z is 2 2 ¸¾ Z z dζ 2 θ0 τ = − p (21) ω0t; sin (31) 2 2 2 1 (1 − ζ )(1 − kζ ) In Figs. 1-6 we have plotted the exact angular dis- Equation (21) can be rewritten as follows placement θ as a function of ω0t (Eq. (31)) for different values of the initial amplitude θ0. Plots have been R 1 dζ obtained using the Mathematica program. In these fig- τ = 0 p − (1 − ζ2)(1 − kζ2) ures we have also© included£ ¡ the angular¢ displacement¤ª θ0 2 θ0 2 θ0 θ(t) = 2 arcsin sin sn K sin − ω0t ; sin . R z dζ 2 2 2 p , (22) As we can see, for amplitudes θ < 0.75π (135◦) it 0 2 2 0 (1 − ζ )(1 − kζ ) would be possible to use the approximate expression which allows us to obtain τ as a function of z and k as ½ · µ ¶ θ θ θ(t) = 2 arcsin sin 0 sn K sin2 0 − τ(z) = K(k) − F (arcsin z; k), (23) 2 2 ¸¾ 2 θ0 where K(m) and F (ϕ;m) are the complete and the in- ω0t ; sin (32) complete elliptical integral of the first kind, defined as 2 follows [13] for the angular displacement of the pendulum and Z Eq. (32) corresponds to a simple harmonic oscillator, 1 dz K(m) = p (24) but with the following exact expression for the angular 2 2 0 (1 − z )(1 − mz ) frequency Z ϕ dz πω0 F (ϕ; m) = p (25) ω(θ0) = ¡ 2 ¢. (33) 2 2 0 (1 − z )(1 − mz ) 2K sin [θ0/2] However, for larger initial amplitudes it is easy to see and z = sinϕ. that the angular displacement does not correspond to The period of oscillation T is four times the time a simple harmonic oscillator and the exact expression taken by the pendulum to swing from θ = 0 (z = 0) to must be considered. θ = θ0 (z = 1). Therefore

4τ(0) 4 2 T = 4t(0) = = K(k) = T0K(k), (26) ω0 ω0 π where s 2π l T0 = = 2π (27) ω0 g is the period of the pendulum for small oscillations. Equation (23) can be written as follows Figure 1 - Angular displacement θ as a function of ω0t for θ = 0.1π. Equation (31), bold line. Equation (32), thin line. F (arcsin z; k) = K(k) − τ (28) 0 which can be written in terms of the Jacobi elliptic function sn(u;m) [14]

z = sn(K(k) − τ; k). (29) In terms of the original variables and taking into account Eqs. (9), (10), (17) and (26), Eq. (29) becomes

· µ ¶ ¸ θ θ θ sin(θ/2) = sin 0 sn K sin2 0 − ω t ; sin2 0 2 2 0 2 (30) Figure 2 - Angular displacement θ as a function of ω0t for which allows us to express θ as a function of t as θ0 = 0.25π. Equation (31), bold line. Equation (32), thin line. 648 Bel´endez at al.

We can conclude that for amplitudes as high as 135◦ the eﬀect of the nonlinearity is seen only in the fact that the angular frequency of the oscillation ω depends on the amplitude θ0 of the motion. For these amplitudes the harmonic function provides an excellent approxima- tion to the periodic solution of Eq. (1) and the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the amplitude of oscillations). Finally, we point out that in the process of plotting the exact displacement one may show to the students the Figure 3 - Angular displacement θ as a function of ω0t for advantage of using available symbolic computer pro- θ = 0.5π. Equation (31), bold line. Equation (32), thin line. 0 grams such as Mathematica.

Acknowledgments

This work was supported by the “Ministerio de Edu- caci´ony Ciencia”, Spain, under project FIS2005-05881- C02-02, and by the “Generalitat Valenciana”, Spain, under project ACOMP/2007/020.

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