Elliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota

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1 Elliptic Functions sn, cn, n, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Backgroun: Jacobi iscovere that rather than stuying elliptic integrals themselves, it is simpler to think of them as inverses for some functions like trig functions. For instance, recall that sin 1 () =, 1 2 but that it is easier to stuy sin() than the inverse sine. The resulting elliptic functions satisfy non-linear DEs that arise in many applications. Here we evelop the Jacobi elliptic functions as a form of trigonometric functions, but using an ellipse rather than a circle. These notes evolve from a lecture by William M. Kinnersley, circa The approach ought to be in some classic tet, but I have not foun it. 0 θ Figure 1: ellipse feature in construction. Trigonometry of the ellipse: The ellipse equation is ( a ) 2 + ( y b ) 2 =1, but we normalize the ellipse by choosing b =1sothat, ( ) 2 + y 2 =1. (1) a 1

2 Also of course, 2 + y 2 = r 2. (2) The eccentricity of an ellipse with general a, b is b 2 a =1 2 ɛ2, or ɛ = 1 b2 a, 2 so that ɛ =0foracircle,ɛ = 1 for a parabola. Since b = 1, the eccentricity is ɛ k = 1 1 a, 2 which is the moulus of the corresponing elliptic functions. Thus 0 k 1, an k = 1 shoul give orinary trigonometry. The net an very important thing to efine is the argument u of the elliptic functions. The u is the thing the elliptic functions are functions of. Inthe case of trig functions, the argument woul be the angle θ, but here u is a bit more complicate. Q u rθ, (3) P where P an Q are as shown in Fig. 1. Notice that u is not an angle. It is not arc length an it is not area either. However, u becomes the angle θ or arc length in the limit a 1, or k 0 when the ellipse becomes a circle. With the argument an moulus of the elliptic functions efine, the functions themselves are just ratios, just as in the case of trigonometry. sn(u, k) = y, (4) cn(u, k) = /a, (5) n(u, k) = r/a. (6) The first two generalize the sine an cosine, an the thir comes about because the raius is not constant on an ellipse. When k 0, so that a =1, these become just y,, an +1, since r 1 also. This connects the elliptic functions to sin θ, cosθ an +1. There are several notational points to mention here. First, one often omits the moulus k in writing the elliptic functions an just writes sn u =sn(u, k), an so on. 2

3 Corresponing to a given moulus k there is a complementary moulus k such that k = 1 k 2. There are also other notations. For eample, a moern invention is to use m = k 2 so that fewer square roots appear. Then one efines sn(u m) sn(u, k), where m = k 2. In fact there are twelve Jacobi elliptic functions, efine using a simple convention ns u = sn 1 nc u = 1 u cn n u = 1 u n u sc u = sn u cn u s u = n u sn u c u = n u cn u s u = sn u n u cs u = cn u sn u c u = cn u n u an these all satisfy certain nonlinear ifferential equations, as we shall see. From Eq.(1) we have cn 2 u +sn 2 u =1, (7) which generalizes cos 2 θ +sin 2 θ = 1. An then from Eq.(2), n 2 u + k 2 sn 2 u =1. (8) The ifferential relations now follow essentially from Eqs(1) an (2), just as the ifferentials of the sine an cosine follow from the Pytagorean formula. From ( ) y θ =tan 1, one has But Also, from Eq.(1), θ = 1 (y y). r2 u = rθ = 1 (y y). r a 2 + yy =0, 3

4 so one can replace either y = a 2 y, or = a2 y y. The corresponing substitutions for u are therefore u = 1 ( ) 2 r a 2 y y, or u = 1 ( + a2 y 2 ) y. r With these substitutions we get the following formulas for ifferentiating elliptic functions (with respect to the argument u, notk), sn u =cnu n u, (9) u cn u = sn u n u, (10) u u n u = k2 sn u cn u. (11) Equations (9) an (10) relate in obvious ways to the trigonometric limit, while Eq.(11) is new. It reuces to an ientity when k 0. The elliptic functions satisfy ifferential equations that we fin by starting with a solution an working backwar. Apparently the moulus k shoul enter the DE as a parameter. u sn u =cnu n u = 1 sn 2 u 1 k 2 sn 2 u, so if y(u) =snu, then ( ) 2 y =(1 y 2 )(1 k 2 y 2 ). (12) u If I solve for u(y), u = c + y 1 y 2 1 k 2 y 2, 4

5 which I recognize as an elliptic integral of the first kin, F (y, k). Thus, as I mentione earlier, the elliptic functions are the inverse functions for the elliptic integrals. On the other han, If I ifferentiate Eq.(12) again with respect to u Iget y +(1+k 2 ) y 2 k 2 y 3 =0. (13) This relates to a nonlinear uffing-type oscillator. In fact, all twelve of the Jacobi elliptic functions satisfy nonlinear first orer DEs like Eq.(12), an also nonlinear secon orer DEs like Eq.(13). Moreover, you will fin that the squares of the elliptic functions satisfy equations of the form (y ) 2 + αy 2 + βy 3 =0, an of the form y + γy+ δy 2 =0. One can thus solve all such equations eactly, in close form, in terms of elliptic functions. Different functions cover ifferent parameter ranges. Elliptic functions open up a winow of solvable nonlinear (polynomial) DEs, all of which relate to physical problems an physical phenomena. I o not know of other types of solutions of this quality for any nonlinear ynamical problems. Homework: Perform the same construction starting from a hyperbola, 2 a 2 y2 =1 rather than from the ellipse in Fig.(1). Thus efine the Jacobi hyperbolic functions, sn(u, k), = y, ch(u, k) = /a an h(u, k) = r/a an erive their properties. You shoul fin that, ch 2 u sh 2 u =1 an sh u =chu h u u an then compute all the other properties, incluing the first an secon orer DEs these functions satisfy. (By the way, these functions are not iscusse in the literature, since they are relate to elliptic functions with comple arguments, just as hyperbolic sines an cosines relate to sines an cosines of comple argument. Using the DEs, can you show this relationship?) 5