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

§23.21 Physical Applications

Keywords:: Weierstrass elliptic functions, classical dynamics, modular functions, quantum field theory, statistical mechanics, string theory
Permalink:: http://dlmf.nist.gov/23.21

§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

Permalink:: http://dlmf.nist.gov/23.21.i

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 $\mathop{\wp\/}\nolimits$ 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

Keywords:: KdV equation, Korteweg–de Vries equation, Weierstrass elliptic functions, nonlinear evolution equations, partial differential equations, solitons
Permalink:: http://dlmf.nist.gov/23.21.ii

Airault et al. (1977) applies the function $\mathop{\wp\/}\nolimits$ 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

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

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

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

Symbols:: $e_{j}$ : zeros, : real part of , : imaginary part of , : complex and $\rho$ : roots
Permalink:: http://dlmf.nist.gov/23.21.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\xi$ : ellipsoidal coordinate, $\eta$ : ellipsoidal coordinate, $\zeta$ : ellipsoidal coordinate and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E2
Encodings:: TeX, pMML, png

where

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

Symbols:: $e_{j}$ : zeros, $\rho$ : roots and $f(\rho)$ : function
Permalink:: http://dlmf.nist.gov/23.21.E3
Encodings:: TeX, pMML, png

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

23.21.4

$\xi=\mathop{\wp\/}\nolimits\!\left(u\right),$

$\eta=\mathop{\wp\/}\nolimits\!\left(v\right),$

$\zeta=\mathop{\wp\/}\nolimits\!\left(w\right).$

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -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

Then

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

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\frac{\partial f}{\partial x}$ : partial derivative of with respect to , $\mathbb{L}$ : lattice, : change of variable, : change of variable and : change of variable
Permalink:: http://dlmf.nist.gov/23.21.E5
Encodings:: TeX, pMML, png

§23.21(iv) Modular Functions

Keywords:: Weierstrass elliptic functions, modular functions
Permalink:: http://dlmf.nist.gov/23.21.iv

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 23.20 Mathematical Applications 23.22 Methods of Computation

§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