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Swinging in Imaginary Time More on the Not-So-Simple Pendulum

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Swinging in Imaginary Time More on the Not-So-Simple Pendulum GENERAL ARTICLE Swinging in Imaginary Time More on the Not-So-Simple Pendulum Cihan Saclioglu When the small angle approximation is not made, the exact solution of the simple pendulum is a Jacobian elliptic function with one real and one imaginary period. Far from being a physically meaningless mathematical curiosity, the second period represents the imaginary time the pendu- lum takes to swing upwards and tunnel through Cihan Saclioglu teaches the potential barrier in the semi-classical WKB physics at Sabanci approximation1 in quantum mechanics. The tun- University, Istanbul, neling here provides a simple illustration of simi- Turkey. His research lar phenomena in Yang{Mills theories describing interests include solutions of Yang–Mills, Einstein weak and strong interactions. and Seiberg–Witten equations and group 1. Introduction theoretical aspects of string theory. Consider a point mass m at the end of a rigid massless stick of length l. The acceleration due to gravity is g and the oscillatory motion is con¯ned to a vertical plane. Denoting the angular displacement of the stick from the 1When the exact quantum me- vertical by Á and taking the gravitational potential to chanical calculation of the tun- be zero at the bottom, the constant total energy E is neling probability for an arbitrary given by potential U(x) is impracticable, the Wentzel–Kramers–Brillouin 1 dÁ 2 approach, invented indepen- E = ml2 + mgl(1 cos Á) dently by all three authors in the 2 dt ¡ µ ¶ same year as the Schrödinger = mgl(1 cos Á0); (1) equation, provides an approxi- ¡ mate answer. 2 where Á0 is the maximum angle. Isolating (dÁ=dt) and taking its square root gives Á as an implicit function of the time t through Keywords Pendulum, Jacobi elliptic func- l Á dÁ tions, tunneling, WKB, instan- t = tons. 2g (cos Á cos Á )1=2 s Z0 ¡ 0 104 RESONANCE February 2010 GENERAL ARTICLE l Á dÁ = : (2) 2 2 1=2 s4g 0 sin (Á =2) sin (Á=2) Z 0 ¡ ¡ ¢ Following the treatment in Landau{Lifshitz [1], we de- ¯ne sin » = sin(Á=2)= sin(Á0=2), in terms of which the period T becomes l Á0 dÁ T = 2 1=2 sg 0 sin2(Á =2) sin2(Á=2) Z 0 ¡ ¼ l 2 ¡ d» ¢ = 4 : (3) 2 2 1=2 sg 0 (1 k sin ») Z ¡ This can be used to de¯ne K(k), the complete elliptic integral of the ¯rst kind, via l T 4 K(k) : (4) ´ sg 2 2 When k = (sin(Á0=2)) can be neglected, one recovers the period 2¼ l=g of the simple small angle pendulum. Note that the standard elliptic function notation K(k) p is a bit misleading: K is really a function NOT of k, but of k2. Hence for k2 1, we can consistently keep k as a small non-zero qua¿ntity while neglecting k2. The substitution y = sin » shows that (3) can also be written as ¼ 2 d» K(k) = (1 k2 sin2 »)1=2 Z0 ¡ 1 dy = : (5) ((1 y2)(1 k2y2))1=2 Z0 ¡ ¡ RESONANCE February 2010 105 GENERAL ARTICLE 2. Elliptic Functions and Integrals Let us convert the de¯nite integral (5) into an inde¯nite one by replacing the upper limit 1 by y. Also introducing the dimensionless time variable x = t g=l, this yields y dy p x = 0 : (6) ((1 y 2)(1 k2y 2))1=2 Z0 ¡ 0 ¡ 0 The right-hand side of (6) is known as the Legendre ellip- tic integral of the ¯rst kind and is denoted by F (arcsin y; k). The inverse of F, implicitly de¯ned by (6), is the Jacobian elliptic function y = snx [2]. Assembling all this, the exact time dependence of the angle is found to be g Á Á(t) = 2 arcsin sin t sin 0 ; (7) l 2 µ µr ¶ ¶ This is an appropriate which becomes the familiar point to mention an essential difference g Á(t) = Á0 sin t (8) between trigonometric l µr ¶ and elliptic functions: in the limit sin Á Á . Considered as 0 ¼ 0 functions of the This is an appropriate point to mention an essential complex variable di®erence between trigonometric and elliptic functions: z = x + iy, the former Considered as functions of the complex variable z = are bounded and have x + iy, the former are bounded and have a real period a real period along the along the x-axis, but blow up exponentially as y goes to x-axis, but blow up positive or negative in¯nity, while the latter have an ad- exponentially as y ditional imaginary period along the y-axis. We will have goes to positive or to state this and a few other properties of elliptic func- negative infinity, while tions without proof, as the required complex analysis is the latter have an a bit involved. We refer the interested reader to Math- additional imaginary ews and Walker [2], whose treatment we mostly follow. period along the y- For snx, we have already met the real period K(k) in axis. (4); the imaginary period 2K0 is de¯ned via 106 RESONANCE February 2010 GENERAL ARTICLE Box 1. Elliptic Functions A function f(z) is said to be doubly periodic in the complex z-plane with periods !1 and !2 if f(z + n!1 + m!2) = f(z) for any pair of integers n; m. Double periodicity along two independent directions requires the ratio !1=!2 to have an imaginary part. Thus the function maps a single parallelogram cell ¦ de¯ned by !1 and !2 to the entire z-plane. If the singularities of f(z) consist of poles, the resulting meromorphic doubly-periodic function is called an elliptic function. By Liouville's theorem, f(z) must have poles in ¦ if it is not to be a constant. Since the parallel sides of the parallelogram @¦ bounding ¦ give equal and opposite contributions, and f(z) is meromorphic, @¦ f(z)dz = 0 by Cauchy's theorem. This means the residues of the poles in ¦ must add up to zero. The simplest way to get vanishing total residue is either to choose two simple poles wHith opposite residues, or one double pole in ¦. These correspond to the Jacobi and the Weierstrass elliptic functions, respectively. The latter, denoted by }, is easier to represent explicitly (in the form of an in¯nite double sum over all lattice points), but our present problem of a planar pendulum involves the Jacobi elliptic functions (interestingly, the Weierstrass ones appear in the treatment of the spherical pendulum!). The lattice is generated by !1 = K and !2 = iK(k0) that are de¯ned in (5) and (9). The double periodicity properties follow by integrating (6) in the complex y-plane along di®erent contours: sn(2K y) = sny is proven by choosing a path that goes from 0 to y to 1 along the real axis, enci¡rcles 1 in a clockwise sense and comes back to y on a path going left just below the real axis. Using sn( y) = sny which follows from (6), we have sn(2K +y) = sny. Shifting the argument aga¡in by a¡nother 2K, we get sn(4K +y) = +sny. Extending the¡contour from 1 to 1=k, clockwise encircling 1=k and coming back to y on a path just below the real axis yields sn(y + 2iK0) = sny on account of (9) and the fact that for 1 < x < 1=k, p1 x2 = ipx2 1. ¡ § ¡ Historically, the study of elliptic functions originated around the end of the seventeenth century in the study of arc lengths of ellipses and a curve called Bernoulli's lemniscate, leading to the integral ®dx/p1 x4. Count Fagnano presented an addition theorem for 0 ¡ this integral in 1750. Euler extended the theorem to integrals with p1 + ax2 + bx4 in R the denominator. Legendre's studies of more general integrands such as E + F= Q(x), where E and F are rational functions and Q is a cubic or quartic polynomial culminated in his 1832 Treatise on Elliptic functions and Euler integrals. In 1828 Abel andpJacobi almost simultaneously de¯ned new functions by inverting such integrals and showed that they were doubly-periodic. The terminology was also changed: these functions are now called elliptic, while Legendre's elliptic functions are referred to as elliptic integrals. Jacobi's elliptic functions can also be expressed as the ratios of quadratic combinations of Jacobi £- functions that are quasiperiodic rather than periodic in the complex plane. The so-called Ramanujan £-function is a further generalization of Jacobi £-functions, but it is not due to Ramanujan himself. Ramanujan introduced what he called \mock £-functions", which are nowadays regarded as a class of `modular forms', de¯ned by their transformation properties under modular transformations of the complex plane. Such a transformation takes z to (az + b)=(cz + d), where a; b; c; d are integers and ab cd = 1. For more on Ramanujan's remarkable work in this area we refer the reader to ¡the book Ramanujan's lost notebook, George E Andrews and Bruce C Berndt, Part I, Springer, New York, 2005. RESONANCE February 2010 107 GENERAL ARTICLE 1=k dy K0 = : (9) ((y2 1)(1 k2y2))1=2 Z1 ¡ ¡ The double-periodicity property means that sn(z + 4K) = sn(z + 2iK0) = snz : (10) 2 2 It can be shown that K0(k) = K(k0) if k0 = 1 k by going over to the variable z = (1 k2y2)=(1 ¡k2) in (9). This `duality relation' will be us¡eful in rela¡ting the p pendulum's behavior near the top and bottom points.
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