elliptic f;s handout

Elliptic Functions sn, cn, dn, as Trigonometry W. Schwalm, Physics, Univ. N. Dakota Background : Jacobi discovered that rather than studying elliptic integrals themselves, it is simpler to think of them as inverses for some functions liketrig functions. For instance, recall that sin −1(x)=/integraldisplayx 0dx √ 1−x2, but that it is easier to study sin( x) than the inverse sine. The resulting elliptic functions satisfy non-linear D Es that arise in many applications. Here we develop the Jacobi elliptic func tions as a form of trigonometric func- tions, but using an ellipse rather than a circle. These notes evolved from alecture by William M. Kinnersley, circa 1975. The approach ought to be in some classic text, but I have not found it. /g20/g52 /g85 θ /g68/g51/g91/g92 Figure 1: ellipse featured in construction. Trigonometry of the ellipse : The ellipse equation is /parenleftbiggx a/parenrightbigg2 +/parenleftbiggy b/parenrightbigg2 =1, but we normalize the ellipse by choosing b=1s ot h a t , /parenleftbiggx a/parenrightbigg2 +y2=1. (1) 1 Also of course, x2+y2=r2. (2) The eccentricity of an ellipse with general a, bis b2 a2=1−/epsilon12,or/epsilon1=/radicalBigg 1−b2 a2, so that /epsilon1=0f o rac i r c l e , /epsilon1= 1 for a parabola. Since b= 1, the eccentricity is /epsilon1≡k=/radicalBigg 1−1 a2, which is the modulus of the corresponding elliptic functions. Thus 0 ≤k≤1, andk= 1 should give ordinary trigonometry. The next and very important thing to define is the argument uof the elliptic functions. The uis the thing the elliptic functions are functions of .I n t h e case of trig functions, the argument would be the angle θ, but here uis a bit more complicated. u≡/integraldisplayQ Prd θ, (3) where PandQare as shown in Fig. 1. Notice that uis not an angle. It is not arc length and it is not area either. However, ubecomes the angle θor arc length in the limit a→1, ork→0 when the ellipse becomes a circle. With the argument and modulus of the elliptic functions defined, the func- tions themselves are just ratios, just as in the case of trigonometry. sn(u,k)= y, (4) cn(u,k)= x/a, (5) dn(u,k)= r/a. (6) The first two generalize the sine and cosine, and the third comes about because the radius is not constant on an ellipse. When k→0, so that a=1 , these become just y, x,and +1, since r→1 also. This connects the elliptic functions to sin θ,c o sθand +1. There are several notational points to mention here. First, one often omits the modulus kin writing the elliptic functions and just writes snu=s n ( u,k),and so on. 2 Corresponding to a given modulus kthere is a complementary modulus k/prime such that k/prime=√ 1−k2. There are also other notations. For example, a modern invention is to use m=k2so that fewer square roots appear. Then one defines sn(u|m)≡sn(u,k),where m=k2. In fact there are twelve Jacobi elliptic functions, defined using a simple con- vention nsu=1 snuncu=1 cnundu=1 dnu scu=snu cnudcu=dnu cnucsu=cnu snu dsu=dnu snusdu=snu dnucdu=cnu dnu and these all satisfy certain nonlinear differential equations, as we shall see. From Eq.(1) we have cn2u+s n2u=1, (7) which generalizes cos2θ+s i n2θ= 1. An then from Eq.(2), dn2u+k2sn2u=1. (8) The differential relations now follow essentially from Eqs(1) and (2), just as the differentials of the sine and cosine follow from the Pytagorean formula. From θ=t a n−1/parenleftbiggy x/parenrightbigg , one has dθ=1 r2(xdy−yd x). But du=rd θ=1 r(xdy−yd x). Also, from Eq.(1), xdx a2+yd y=0, 3 so one can replace either dy=−x a2ydx, or dx=−a2y xdy. The corresponding substitutions for duare therefore du=1 r/parenleftBigg −x2 a2y−y/parenrightBigg dx, or du=1 r/parenleftBigg x+a2y2 x/parenrightBigg dy. With these substitutions we get the following formulas for differentiating elliptic functions (with respect to the argument u,n o tk), d dusnu=c nudnu, (9) d ducnu=−snudnu, (10) d dudnu=−k2snucnu. (11) Equations (9) and (10) relate in obvious ways to the trigonometric limit, while Eq.(11) is new. It reduces to an identity when k→0. The elliptic functions satisfy differential equations that we find by starting with a solution and working backward. Apparently the modulus kshould enter the DE as a parameter. d dusnu=c nudnu=√ 1−sn2u√ 1−k2sn2u, so ify(u)=s n u,t h e n /parenleftBiggdy du/parenrightBigg2 =( 1−y2)(1−k2y2). (12) If I solve for u(y), u=c+/integraldisplaydy √ 1−y2√ 1−k2y2, 4 which I recognize as an elliptic integral of the first kind, F(y,k). Thus, as I mentioned earlier, the elliptic functions are the inverse functions for the elliptic integrals. On the other hand, If I differentiate Eq.(12) again with respect to uIg e t y/prime/prime+( 1+ k2)y−2k2y3=0. (13) This relates to a nonlinear duffing-type o scillator. In fact, all twelve of the Jacobi elliptic functions satisfy nonlinear first order DEs like Eq.(12), and also nonlinear second order DEs like Eq.(13). Moreover, you will find thatthe squares of the elliptic function s satisfy equations of the form (y /prime)2+αy2+βy3=0, and of the form y/prime/prime+γy+δy2=0. One can thus solve all such equations exactly, in closed form, in terms of elliptic functions. Different function s cover different parameter ranges. Elliptic functions open up a window of solvable nonlinear (polynomial) DEs, all of which relate to physical problems and physical phenomena. I do not know of other types of solutions of this quality for any nonlinear dynamical problems. Homework : Perform the same construction starting from a hyperbola, x2 a2−y2=1 rather than from the ellipse in Fig.(1). Thus define the “Jacobi hyperbolic functions,” sn( u,k),=y,ch(u,k)=x/aand dh( u,k)=r/aand derive their properties. You should find that, ch2u−sh2u=1 andd dushu=c hudhu and then compute all the other properties, including the first and second order DEs these functions satisfy. (By the way, these functions are not dis- cussed in the literature, since they are related to elliptic functions with com- plex arguments, just as hyperbolic sines and cosines relate to sines and cosines of complex argument. Using the DEs, can you show this relationship?) 5