23 Weierstrass Elliptic and Modular Functions Applications 23.20 Mathematical Applications 23.22 Methods of Computation

§23.21 Physical Applications

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Keywords:: Weierstrass elliptic functions, applications, classical dynamics, modular functions, physical, quantum field theory, statistical mechanics, string theory
Permalink:: http://dlmf.nist.gov/23.21
See also:: Annotations for Ch.23

§23.21(i) Classical Dynamics
§23.21(ii) Nonlinear Evolution Equations
§23.21(iii) Ellipsoidal Coordinates
§23.21(iv) Modular Functions

§23.21(i) Classical Dynamics

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Permalink:: http://dlmf.nist.gov/23.21.i
See also:: Annotations for §23.21 and Ch.23

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 $\wp$ 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).

§23.21(ii) Nonlinear Evolution Equations

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Keywords:: KdV equation, Korteweg–de Vries equation, Weierstrass elliptic functions, nonlinear, nonlinear evolution equations, partial differential equations, solitons
Permalink:: http://dlmf.nist.gov/23.21.ii
See also:: Annotations for §23.21 and Ch.23

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

§23.21(iii) Ellipsoidal Coordinates

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Keywords:: Laplacian, Weierstrass elliptic functions, coordinate systems, ellipsoidal, ellipsoidal coordinates
Notes:: See Jones (1964, pp. 31–33).
Referenced by:: §23.1
Permalink:: http://dlmf.nist.gov/23.21.iii
See also:: Annotations for §23.21 and Ch.23

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

23.21.1 $\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,$

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Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, : real part of , : imaginary part of , : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: TeX, pMML, png
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

where 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

23.21.2 $(\eta-\zeta)(\zeta-\xi)(\xi-\eta)\nabla^{2}=(\zeta-\eta)f(\xi)f^{\prime}(\xi)% \frac{\partial}{\partial\xi}+(\xi-\zeta)f(\eta)f^{\prime}(\eta)\frac{\partial}% {\partial\eta}+(\eta-\xi)f(\zeta)f^{\prime}(\zeta)\frac{\partial}{\partial% \zeta},$

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: TeX, pMML, png
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

where

23.21.3 $f(\rho)=2\left((\rho-e_{1})(\rho-e_{2})(\rho-e_{3})\right)^{1/2}.$

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Symbols:: $e_{\NVar{j}}$ : Weierstrass lattice roots, $\mathbb{L}$ : lattice, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: TeX, pMML, png
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

Another form is obtained by identifying $e_{1}$ , $e_{2}$ , $e_{3}$ as lattice roots (§23.3(i)), and setting

23.21.4

$\xi=\wp\left(u\right),$

$\eta=\wp\left(v\right),$

$\zeta=\wp\left(w\right).$

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\mathbb{L}$ : lattice, $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate, : change of variable, : change of variable and : change of variable
Permalink:: http://dlmf.nist.gov/23.21.E4
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

Then

23.21.5 $\left(\wp\left(v\right)-\wp\left(w\right)\right)\left(\wp\left(w\right)-\wp% \left(u\right)\right)\left(\wp\left(u\right)-\wp\left(v\right)\right)\nabla^{2% }=\left(\wp\left(w\right)-\wp\left(v\right)\right)\frac{{\partial}^{2}}{{% \partial u}^{2}}+\left(\wp\left(u\right)-\wp\left(w\right)\right)\frac{{% \partial}^{2}}{{\partial v}^{2}}+\left(\wp\left(v\right)-\wp\left(u\right)% \right)\frac{{\partial}^{2}}{{\partial w}^{2}}.$

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Symbols:: $\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of with respect to , $\,\partial\NVar{x}$ : partial differential of , $\mathbb{L}$ : lattice, : change of variable, : change of variable and : change of variable
Permalink:: http://dlmf.nist.gov/23.21.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §23.21(iii), §23.21 and Ch.23

§23.21(iv) Modular Functions

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Keywords:: Weierstrass elliptic functions, applications, modular functions, physical
Permalink:: http://dlmf.nist.gov/23.21.iv
See also:: Annotations for §23.21 and Ch.23

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).

23.20 Mathematical Applications 23.22 Methods of Computation

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§23.21 Physical Applications

Contents

§23.21(i) Classical Dynamics

§23.21(ii) Nonlinear Evolution Equations

§23.21(iii) Ellipsoidal Coordinates

§23.21(iv) Modular Functions