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23 Weierstrass Elliptic and Modular FunctionsApplications

§23.21 Physical Applications

Contents

§23.21(i) Classical Dynamics

In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form (1-x2)(1-k2x2). The Weierstrass function plays a similar role for cubic potentials in canonical form g3+g2x-4x3. See, for example, Lawden (1989, Chapter 7) and Whittaker (1964, Chapters 4–6).

§23.21(ii) Nonlinear Evolution Equations

Airault et al. (1977) applies the function to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1).

§23.21(iii) Ellipsoidal Coordinates

Ellipsoidal coordinates (ξ,η,ζ) may be defined as the three roots ρ of the equation

23.21.1 x2ρ-e1+y2ρ-e2+z2ρ-e3=1,

where x,y,z are the corresponding Cartesian coordinates and e1, e2, e3 are constants. The Laplacian operator 21.5(ii)) is given by

23.21.2 (η-ζ)(ζ-ξ)(ξ-η)2=(ζ-η)f(ξ)f(ξ)ξ+(ξ-ζ)f(η)f(η)η+(η-ξ)f(ζ)f(ζ)ζ,

where

23.21.3 f(ρ)=2((ρ-e1)(ρ-e2)(ρ-e3))1/2.

Another form is obtained by identifying e1, e2, e3 as lattice roots (§23.3(i)), and setting

23.21.4 ξ =(u),
η =(v),
ζ =(w).

Then

23.21.5 ((v)-(w))((w)-(u))((u)-(v))2=((w)-(v))2u2+((u)-(w))2v2+((v)-(u))2w2.

See also §29.18(ii).

§23.21(iv) Modular Functions

Physical applications of modular functions include:

  • Quantum field theory. See Witten (1987).

  • Statistical mechanics. See Baxter (1982, p. 434) and Itzykson and Drouffe (1989, §9.3).

  • String theory. See Green et al. (1988a, §8.2) and Polchinski (1998, §7.2).