§23.21 Physical Applications

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 $(1-x^2)(1-k^2{x}^2)$. The Weierstrass function $\mathrm{\wp }$ plays a similar role for cubic potentials in canonical form $g_3+g_{2}x-4x^3$. See, for example, Lawden (1989, Chapter 7) and Whittaker (1964, Chapters 4–6).

Airault et al. (1977) applies the function $\mathrm{\wp }$ 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).

Ellipsoidal coordinates $(\xi ,\eta ,\zeta )$ may be defined as the three roots $\rho$ of the equation

where $x,y,z$ are the corresponding Cartesian coordinates and $e_1$, $e_2$, $e_3$ are constants. The Laplacian operator ${\nabla }^2$ (§1.5(ii)) is given by

where

Another form is obtained by identifying $e_1$, $e_2$, $e_3$ as lattice roots (§23.3(i)), and setting

Then

See also §29.18(ii).

Physical applications of modular functions include:

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  Quantum field theory. See Witten (1987).

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

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