# §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 $(1-x^{2})(1-k^{2}x^{2})$. 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

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

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

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

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

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

where

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

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

 23.21.4 $\displaystyle\xi$ $\displaystyle=\mathop{\wp\/}\nolimits\!\left(u\right),$ $\displaystyle\eta$ $\displaystyle=\mathop{\wp\/}\nolimits\!\left(v\right),$ $\displaystyle\zeta$ $\displaystyle=\mathop{\wp\/}\nolimits\!\left(w\right).$

Then

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