The Stäckel Conditions and the Eleven Separable Systems

Contents

Orthogonal coordinates and the Laplacian
What separation means, and what it forces
The three systems the treatise uses
The generic system: confocal quadrics
The enumeration

This appendix proves Theorem 14.34 of Partial Differential Equations: the Helmholtz equation Equation (14.25) separates in an orthogonal coordinate system of three-dimensional Euclidean space if and only if the reciprocal squared scale factors are the first column of the inverse of a Stäckel matrix — a matrix each of whose rows depends on one coordinate alone — and the volume element satisfies the accompanying Robertson condition. Both conditions are derived here, not postulated: they are read off by substituting the product ansatz into the Laplacian and demanding that the three separation constants enter the separated ordinary differential equations linearly. The three systems the treatise uses (Example 14.31, Example 14.32, Example 14.33) are then verified against the criterion, and so is the generic member of the family, the ellipsoidal system, from which all the others descend by confluence. The section closes with the enumeration of the eleven systems and with an exact statement of the one input that is quoted rather than proved.

Everything is set in the observed three-dimensional space, signature $3+0$, with the flat Euclidean metric; the classification is a statement about $\R^{3}$ and has no higher-dimensional counterpart in this treatise. Latin indices $i,j,l$ run over $1,2,3$ and are not summed unless a summation sign is written, because the sums that appear below are not all over the same range.

Orthogonal coordinates and the Laplacian

Definition A15.1 (Orthogonal curvilinear coordinates; scale factors).

Let $\vect{x} = \vect{x}(q^{1},q^{2},q^{3})$ be a $C^{2}$ diffeomorphism of an open set of $\R^{3}$ onto another, with

\begin{equation}\tag{A15.1} \pdv{\vect{x}}{q^{i}}\cdot\pdv{\vect{x}}{q^{j}} = 0 \qquad (i \neq j)\ec \qquad h_{i} = \abs{\pdv{\vect{x}}{q^{i}}} > 0\ep \end{equation}

The $h_{i}$ are the scale factors, the line element is $\dd\ell^{2} = \sum_{i}h_{i}^{2}\,(\dd q^{i})^{2}$, and the volume element is $\dd^{3}x = g\,\dd q^{1}\dd q^{2}\dd q^{3}$ with $g = h_{1}h_{2}h_{3}$. Rests on Definition 11.98 and Equation (14.25).

Lemma A15.2 (Laplacian in orthogonal coordinates).

For $\psi \in C^{2}$,

\begin{equation}\tag{A15.2} \nabla^{2}\psi = \frac{1}{g}\sum_{i=1}^{3}\pp_{i} \left(\frac{g}{h_{i}^{2}}\,\pp_{i}\psi\right)\ec \qquad \pp_{i} = \pdv{}{q^{i}}\ep \end{equation}

Rests on Definition A15.1 and Theorem 11.133.

Proof.

Derives Lemma A15.2. Let $\hat{\vect{e}}_{i} = h_{i}^{-1}\pp\vect{x}/\pp q^{i}$; by Equation (A15.1) these form an orthonormal triad at each point. The gradient is characterized by $\dd\psi = \nabla\psi\cdot\dd\vect{x}$; writing $\dd\vect{x} = \sum_{i}h_{i}\,\dd q^{i}\,\hat{\vect{e}}_{i}$ and $\dd\psi = \sum_{i}\pp_{i}\psi\,\dd q^{i}$ and matching coefficients of the independent increments $\dd q^{i}$,

\begin{equation}\tag{A15.3} \nabla\psi = \sum_{i}\frac{1}{h_{i}}\,\pp_{i}\psi\; \hat{\vect{e}}_{i}\ep \end{equation}

For the divergence of $\vect{V} = \sum_{i}V_{i}\hat{\vect{e}}_{i}$ apply the divergence theorem Equation (11.162) to the coordinate box $[q^{1},q^{1}+\dd q^{1}]\times\cdots$. Its face at fixed $q^{1}$ has outward normal $-\hat{\vect{e}}_{1}$ and area $h_{2}h_{3}\,\dd q^{2}\dd q^{3} = (g/h_{1})\,\dd q^{2}\dd q^{3}$, and the opposite face has the same area evaluated at $q^{1}+\dd q^{1}$; the net flux through the pair is therefore $\pp_{1}\bigl((g/h_{1})V_{1}\bigr)\,\dd q^{1}\dd q^{2}\dd q^{3}$ to first order. Summing over the three pairs and dividing by the volume $g\,\dd q^{1}\dd q^{2}\dd q^{3}$,

\begin{equation}\tag{A15.4} \Div\vect{V} = \frac{1}{g}\sum_{i}\pp_{i}\left(\frac{g}{h_{i}}\,V_{i}\right)\ep \end{equation}

Composing Equation (A15.4) with Equation (A15.3), so that $V_{i} = h_{i}^{-1}\pp_{i}\psi$, gives Equation (A15.2).

∎

What separation means, and what it forces

Definition A15.3 (Simple separation of the Helmholtz equation).

The Helmholtz equation Equation (14.25) separates simply in the orthogonal system $(q^{i})$ if there exist nowhere-vanishing functions $f_{i}(q^{i})$ and functions $\phi_{ij}(q^{i})$ ($i,j = 1,2,3$), each row index $i$ carrying dependence on $q^{i}$ alone, with

\begin{equation}\tag{A15.5} S = \det\bigl(\phi_{ij}\bigr) \neq 0\ec \end{equation}

such that for every triple of constants $(\lambda_{1},\lambda_{2},\lambda_{3})$ with $\lambda_{1} = k^{2}$, any solutions $X_{i}$ of the three ordinary differential equations

\begin{equation}\tag{A15.6} \frac{1}{f_{i}}\dv{}{q^{i}} \left(f_{i}\dv{X_{i}}{q^{i}}\right) + \left(\sum_{j=1}^{3}\lambda_{j}\,\phi_{ij}(q^{i})\right)X_{i} = 0 \qquad (i = 1,2,3) \end{equation}

compose into a solution $\psi = X_{1}X_{2}X_{3}$ of Equation (14.25). The matrix $\Phi = (\phi_{ij})$ is the Stäckel matrix and $S$ its Stäckel determinant. Rests on Definition A15.1 and Lemma 14.28.

Three remarks fix the content of Definition A15.3 before it is used. First, the form Equation (A15.6) is not a restriction but the general second-order linear equation in $q^{i}$ with no first-derivative term beyond the one a weight can absorb: any equation $X'' + p X' + \sigma X = 0$ is brought to it by $f = \exp\int p$. Second, the requirement that the constants $\lambda_{j}$ enter linearly is the definition of separation in Stäckel's sense, and it is the requirement that carries the physics: each $\lambda_{j}$ is an eigenvalue of a commuting operator, and the separated problem is a Sturm–Liouville problem in each variable (Ordinary Differential Equations and Sturm–Liouville Theory) exactly because Equation (A15.6) is linear in them. Third, $\lambda_{1} = k^{2}$ singles out the first column of $\Phi$; the remaining two constants are free, and $S\neq0$ is what makes them genuinely independent.

Lemma A15.4 (Cofactor identities).

Let $\Phi = (\phi_{ij})$ be a $3\times3$ matrix with $S = \det\Phi \neq 0$, and let $M_{ij}$ be the cofactor of $\phi_{ij}$, that is $(-1)^{i+j}$ times the determinant of the $2\times2$ matrix obtained by deleting row $i$ and column $j$. Then

\begin{equation}\tag{A15.7} \sum_{i=1}^{3}\phi_{ij}\,M_{i1} = S\,\delta_{j1}\ec \qquad \bigl(\Phi^{-1}\bigr)_{1i} = \frac{M_{i1}}{S}\ec \end{equation}

and, if the $i$-th row of $\Phi$ depends on $q^{i}$ alone, then $M_{i1}$ does not depend on $q^{i}$ at all. Rests on Equation (A15.5).

Proof.

Derives Lemma A15.4. For $j = 1$ the sum in Equation (A15.7) is the Laplace expansion of $\det\Phi$ along the first column. For $j \neq 1$ it is the Laplace expansion along the first column of the matrix obtained from $\Phi$ by overwriting its first column with its $j$-th, since the cofactors $M_{i1}$ do not involve the first column at all; that matrix has two equal columns and hence vanishing determinant. Dividing Equation (A15.7) by $S$ identifies $M_{i1}/S$ as the $(1,i)$ entry of the inverse. The last assertion is immediate: the minor belonging to $M_{i1}$ is built from the two rows other than the $i$-th, which by hypothesis carry no dependence on $q^{i}$.

∎

Theorem A15.5 (Stäckel and Robertson conditions).

The Helmholtz equation separates simply in the orthogonal system $(q^{i})$, in the sense of Definition A15.3, if and only if there is a Stäckel matrix $\Phi = (\phi_{ij}(q^{i}))$ with $S = \det\Phi \neq 0$ such that

\begin{equation}\tag{A15.8} \frac{1}{h_{i}^{2}} = \frac{M_{i1}}{S} \qquad (i = 1,2,3)\ec \end{equation}

and the volume element satisfies the Robertson condition

\begin{equation}\tag{A15.9} \frac{h_{1}h_{2}h_{3}}{S} = f_{1}(q^{1})\,f_{2}(q^{2})\,f_{3}(q^{3}) \end{equation}

for some functions $f_{i}$ of one variable each — the same $f_{i}$ that weight the separated equations Equation (A15.6). Rests on Definition A15.3, Lemma A15.2 and Lemma A15.4.

Sufficiency. Derives Theorem A15.5. Assume Equation (A15.8) and Equation (A15.9), and let the $X_{i}$ solve Equation (A15.6). Put $\psi = X_{1}X_{2}X_{3}$. By Equation (A15.8) and $g = S f_{1}f_{2}f_{3}$,

\begin{equation}\tag{A15.10} \frac{g}{h_{i}^{2}} = \frac{g}{S}\,M_{i1} = f_{1}f_{2}f_{3}\,M_{i1}\ec \end{equation}

and by the last clause of Lemma A15.4 both $M_{i1}$ and the factors $f_{l}$ with $l \neq i$ are independent of $q^{i}$. Hence

\begin{equation}\tag{A15.11} \pp_{i}\left(\frac{g}{h_{i}^{2}}\,\pp_{i}\psi\right) = M_{i1}\left(\prod_{l\neq i}f_{l}X_{l}\right) \dv{}{q^{i}}\!\left(f_{i}\dv{X_{i}}{q^{i}}\right)\ec \end{equation}

because everything except $f_{i}X_{i}'$ passes through $\pp_{i}$ untouched. Insert Equation (A15.11) into Equation (A15.2) and divide by $\psi = X_{1}X_{2}X_{3}$, using $g = Sf_{1}f_{2}f_{3}$ once more:

\begin{equation}\tag{A15.12} \frac{\nabla^{2}\psi}{\psi} = \frac{1}{S f_{1}f_{2}f_{3}}\sum_{i}M_{i1} \frac{\prod_{l\neq i}f_{l}X_{l}}{X_{1}X_{2}X_{3}} \bigl(f_{i}X_{i}'\bigr)' = \frac{1}{S}\sum_{i}M_{i1}\, \frac{1}{f_{i}X_{i}}\bigl(f_{i}X_{i}'\bigr)'\ep \end{equation}

The separated equations Equation (A15.6) say that the last factor equals $-\sum_{j}\lambda_{j}\phi_{ij}$, so by the cofactor identity Equation (A15.7)

\begin{equation}\tag{A15.13} \frac{\nabla^{2}\psi}{\psi} = -\frac{1}{S}\sum_{j}\lambda_{j}\sum_{i}\phi_{ij}M_{i1} = -\frac{1}{S}\sum_{j}\lambda_{j}\,S\,\delta_{j1} = -\lambda_{1} = -k^{2}\ec \end{equation}

that is $\nabla^{2}\psi + k^{2}\psi = 0$. Note where each hypothesis acted: Equation (A15.9) made Equation (A15.11) a one-variable derivative, and Equation (A15.8) collapsed the sum in Equation (A15.13) onto the single constant $\lambda_{1}$, killing the two free constants exactly as separation requires.

∎

Necessity. Derives Theorem A15.5. Assume the equation separates simply and substitute $\psi = X_{1}X_{2}X_{3}$ into Equation (14.25) through Equation (A15.2), dividing by $\psi$:

\begin{equation}\tag{A15.14} \sum_{i}\frac{1}{h_{i}^{2}} \left[\frac{X_{i}''}{X_{i}} + \frac{X_{i}'}{X_{i}}\,\pp_{i}\ln\frac{g}{h_{i}^{2}}\right] + k^{2} = 0\ec \end{equation}

obtained by expanding $g^{-1}\pp_{i}\bigl((g/h_{i}^{2})\pp_{i}\psi\bigr)$ and cancelling the $l \neq i$ factors of $\psi$.

Step 1: the weights. The $i$-th bracket in Equation (A15.14) must be expressible through $q^{i}$ and $X_{i}$ alone — that is what makes the $i$-th separated equation an ordinary differential equation in $q^{i}$ — and $X_{i}$ is an arbitrary solution of one, so $X_{i}''/X_{i}$ and $X_{i}'/X_{i}$ are functionally independent quantities that cannot cancel each other's coefficients. Hence $\pp_{i}\ln(g/h_{i}^{2})$ is a function of $q^{i}$ alone. Call it $(\ln f_{i})'(q^{i})$, which defines $f_{i}>0$ up to a constant; the bracket is then $\bigl(f_{i}X_{i}'\bigr)'/(f_{i}X_{i})$, and Equation (A15.14) reads

\begin{equation}\tag{A15.15} \sum_{i}\frac{E_{i}}{h_{i}^{2}} + k^{2} = 0\ec \qquad E_{i} = \frac{1}{f_{i}X_{i}} \bigl(f_{i}X_{i}'\bigr)'\ec \end{equation}

each $E_{i}$ a function of $q^{i}$ alone.

Step 2: linearity in the constants. By hypothesis the separated equations carry the three constants linearly, so $E_{i} = -\sum_{j}\lambda_{j}\phi_{ij}(q^{i})$ with $\lambda_{1}=k^{2}$ and $\lambda_{2},\lambda_{3}$ free. Substituting into Equation (A15.15),

\begin{equation}\tag{A15.16} \sum_{j}\lambda_{j}\left[\sum_{i}\frac{\phi_{ij}}{h_{i}^{2}} - \delta_{j1}\right] = 0\ec \end{equation}

and since this must hold identically in the three independent constants $\lambda_{j}$, every bracket vanishes: $\sum_{i}\phi_{ij}/h_{i}^{2} = \delta_{j1}$. In matrix language the row vector $v$ with $v_{i} = 1/h_{i}^{2}$ satisfies $v\,\Phi = e_{1}\transpose$, whence $\Phi$ is invertible — so $S\neq0$, which is the independence of the constants — and $v_{i} = \bigl(\Phi^{-1}\bigr)_{1i} = M_{i1}/S$ by Lemma A15.4. That is Equation (A15.8).

Step 3: the Robertson condition. Step 1 gave $\pp_{i}\ln\bigl(g/h_{i}^{2}\bigr) = (\ln f_{i})'(q^{i})$. Insert Equation (A15.8): $g/h_{i}^{2} = (g/S)\,M_{i1}$, and $M_{i1}$ is independent of $q^{i}$ by Lemma A15.4, so its logarithmic $q^{i}$-derivative vanishes and

\begin{equation}\tag{A15.17} \pp_{i}\ln\frac{g}{S} = \bigl(\ln f_{i}\bigr)'(q^{i}) \qquad (i = 1,2,3)\ep \end{equation}

Write $\Lambda = \ln\abs{g/S}$. Then $\pp_{1}\Lambda$ depends on $q^{1}$ alone, so $\Lambda = \ln\abs{f_{1}(q^{1})} + \Lambda_{1}(q^{2},q^{3})$; applying Equation (A15.17) for $i=2$ to this expression gives $\pp_{2}\Lambda_{1} = (\ln f_{2})'(q^{2})$, hence $\Lambda_{1} = \ln\abs{f_{2}(q^{2})} + \Lambda_{2}(q^{3})$, and one more step gives $\Lambda_{2} = \ln\abs{f_{3}(q^{3})}$ up to an additive constant, which is absorbed into any one $f_{i}$. Exponentiating, $\abs{g/S} = \abs{f_{1}f_{2}f_{3}}$; the sign is constant on a connected domain and is absorbed likewise, giving Equation (A15.9).

∎

Remark A15.6 (What the two conditions say).

Equation (A15.8) is a condition on the metric alone and is the substantive one: it says the three functions $1/h_{i}^{2}$, which in general are functions of all three coordinates, are the first column of the inverse of a matrix assembled out of nine functions of one coordinate each. That is an enormous restriction, and it is what makes the list of admissible systems finite. Equation (A15.9) is a condition on the volume element relative to $S$, and is the one that distinguishes the Helmholtz operator from the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy: the latter involves no second derivatives of $\psi$ and so imposes Equation (A15.8) alone. Every system in the list below satisfies both, but the logical order matters — Stäckel's condition governs separation of the classical problem, Robertson's is the extra price of the wave operator, which is why the two subjects are almost but not exactly the same, as Remark 14.35 records.

The three systems the treatise uses

In each case we exhibit $\Phi$, $S$, the cofactors $M_{i1}$ and the weights $f_{i}$, and check Equation (A15.8) and Equation (A15.9) outright. The separation constants are named to match Section 14.2.1.

Example A15.7 (Cartesian).

$h_{1}=h_{2}=h_{3}=1$, $g=1$. With $\lambda_{1}=k^{2}$, $\lambda_{2}=\beta^{2}$, $\lambda_{3}=\gamma^{2}$ the separated equations of Example 14.31 are $X''+(k^{2}-\beta^{2}-\gamma^{2})X=0$, $Y''+\beta^{2}Y=0$, $Z''+\gamma^{2}Z=0$, so $f_{i}=1$ and

\begin{equation}\tag{A15.18} \Phi = \begin{pmatrix} 1 & -1 & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\ec \qquad S = 1\ep \end{equation}

The cofactors of the first column are $M_{11} = +\det\begin{pmatrix}1&0\\0&1\end{pmatrix} = 1$, $M_{21} = -\det\begin{pmatrix}-1&-1\\0&1\end{pmatrix} = 1$ and $M_{31} = +\det\begin{pmatrix}-1&-1\\1&0\end{pmatrix} = 1$, all equal to $1/h_{i}^{2}$; and $g/S = 1 = f_{1}f_{2}f_{3}$. Rests on Theorem A15.5 and Example 14.31.

Example A15.8 (Circular cylindrical).

$(q^{1},q^{2},q^{3}) = (\rho,\varphi,z)$ with $h_{1}=1$, $h_{2}=\rho$, $h_{3}=1$ and $g=\rho$. Set $\lambda_{1}=k^{2}$, $\lambda_{2}=m^{2}$, $\lambda_{3}=\gamma^{2}$; the separated equations of Example 14.32 are Equation (14.32), $\Psi''+m^{2}\Psi=0$ and $Z''+\gamma^{2}Z=0$, so $f_{1}=\rho$, $f_{2}=f_{3}=1$ and

\begin{equation}\tag{A15.19} \Phi = \begin{pmatrix} 1 & -\rho^{-2} & -1\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}\ec \qquad S = 1\ep \end{equation}

Then $M_{11} = 1 = 1/h_{1}^{2}$, $M_{21} = -\det\begin{pmatrix}-\rho^{-2}&-1\\0&1\end{pmatrix} = \rho^{-2} = 1/h_{2}^{2}$ and $M_{31} = +\det\begin{pmatrix}-\rho^{-2}&-1\\1&0\end{pmatrix} = 1 = 1/h_{3}^{2}$; and $g/S = \rho = f_{1}f_{2}f_{3}$. Rests on Theorem A15.5 and Example 14.32.

Example A15.9 (Spherical).

$(q^{1},q^{2},q^{3}) = (r,\theta,\varphi)$ with $h_{1}=1$, $h_{2}=r$, $h_{3}=r\sin\theta$ and $g = r^{2}\sin\theta$. Set $\lambda_{1}=k^{2}$, $\lambda_{2}=l(l+1)$, $\lambda_{3}=m^{2}$; the separated equations of Example 14.33 are Equation (14.34), the associated Legendre equation and $\Psi''+m^{2}\Psi = 0$, so $f_{1}=r^{2}$, $f_{2}=\sin\theta$, $f_{3}=1$ and

\begin{equation}\tag{A15.20} \Phi = \begin{pmatrix} 1 & -r^{-2} & 0\\ 0 & 1 & -\bigl(\sin\theta\bigr)^{-2}\\ 0 & 0 & 1 \end{pmatrix}\ec \qquad S = 1\ep \end{equation}

The first-column cofactors are $M_{11} = 1 = 1/h_{1}^{2}$, $M_{21} = -\det\begin{pmatrix} -r^{-2}&0\\0&1\end{pmatrix} = r^{-2} = 1/h_{2}^{2}$ and $M_{31} = +\det\begin{pmatrix} -r^{-2}&0\\1&-(\sin\theta)^{-2}\end{pmatrix} = r^{-2}\bigl(\sin\theta\bigr)^{-2} = 1/h_{3}^{2}$; and $g/S = r^{2}\sin\theta = f_{1}f_{2}f_{3}$. The three checks reproduce, in one line each, the structure the chapter obtained by hand: the centrifugal term $-l(l+1)/r^{2}$ of Equation (14.34) is the entry $\phi_{12}$, and the term $-m^{2}/\sin^{2}\theta$ of the Legendre equation is $\phi_{23}$. Rests on Theorem A15.5 and Example 14.33.

The generic system: confocal quadrics

The three systems just checked are degenerate members of one family. The generic member is the ellipsoidal system, and it is worth carrying out in full because everything else follows from it by letting parameters coincide or run to infinity.

Definition A15.10 (Ellipsoidal coordinates).

Fix $a_{1}^{2} > a_{2}^{2} > a_{3}^{2} > 0$ and write $x_{1},x_{2},x_{3}$ for Cartesian coordinates and

\begin{equation}\tag{A15.21} P(\theta) = \bigl(\theta+a_{1}^{2}\bigr) \bigl(\theta+a_{2}^{2}\bigr)\bigl(\theta+a_{3}^{2}\bigr)\ep \end{equation}

The confocal family is

\begin{equation}\tag{A15.22} \sum_{p=1}^{3}\frac{x_{p}^{2}}{\theta+a_{p}^{2}} = 1\ep \end{equation}

Through a generic point pass exactly three members of the family, with parameters $\xi_{1} > \xi_{2} > \xi_{3}$ interlacing the poles,

\begin{equation}\tag{A15.23} \xi_{1} > -a_{3}^{2} > \xi_{2} > -a_{2}^{2} > \xi_{3} > -a_{1}^{2}\ec \end{equation}

an ellipsoid, a hyperboloid of one sheet and a hyperboloid of two sheets respectively. These are the ellipsoidal coordinates. Rests on Definition A15.1.

Lemma A15.11 (Cartesian coordinates and scale factors).

With the notation of Definition A15.10,

\begin{equation}\tag{A15.24} x_{p}^{2} = \frac{\prod_{i=1}^{3}\bigl(\xi_{i}+a_{p}^{2}\bigr)} {\prod_{q\neq p}\bigl(a_{p}^{2}-a_{q}^{2}\bigr)}\ec \end{equation}

the system is orthogonal, and

\begin{equation}\tag{A15.25} h_{i}^{2} = \frac{\prod_{l\neq i}\bigl(\xi_{i}-\xi_{l}\bigr)} {4\,P(\xi_{i})}\ > 0\ep \end{equation}

Rests on Definition A15.10 and Equation (A15.1).

Proof.

Derives Lemma A15.11. Multiply Equation (A15.22), in the form $1 - \sum_{p}x_{p}^{2}/(\theta+a_{p}^{2}) = 0$, by $P(\theta)$. The result,

\begin{equation}\tag{A15.26} P(\theta) - \sum_{p}x_{p}^{2}\prod_{q\neq p} \bigl(\theta+a_{q}^{2}\bigr) = \prod_{i}\bigl(\theta-\xi_{i}\bigr)\ec \end{equation}

is a monic cubic in $\theta$ whose roots are by definition $\xi_{1},\xi_{2},\xi_{3}$, which identifies it with the right-hand side. Evaluating Equation (A15.26) at $\theta = -a_{p}^{2}$ kills $P$ and every term of the sum except the $p$-th, giving $-x_{p}^{2}\prod_{q\neq p}(a_{q}^{2}-a_{p}^{2}) = -\prod_{i}(\xi_{i}+a_{p}^{2})$, which is Equation (A15.24) after the two sign changes in the denominator.

Differentiating Equation (A15.24) with respect to $\xi_{i}$ removes one factor from the numerator: $2x_{p}\,\pp_{i}x_{p} = x_{p}^{2}/(\xi_{i}+a_{p}^{2})$, hence

\begin{equation}\tag{A15.27} \pp_{i}x_{p} = \frac{x_{p}}{2\bigl(\xi_{i}+a_{p}^{2}\bigr)}\ep \end{equation}

Divide Equation (A15.26) by $P(\theta)$ and rearrange:

\begin{equation}\tag{A15.28} G(\theta) = \sum_{p}\frac{x_{p}^{2}}{\theta+a_{p}^{2}} = 1 - \frac{N(\theta)}{P(\theta)}\ec \qquad N(\theta) = \prod_{i}\bigl(\theta-\xi_{i}\bigr)\ep \end{equation}

For $i \neq j$ the inner product of the two coordinate tangent vectors is, by Equation (A15.27) and the partial fraction $\bigl[(\xi_{i}+a)(\xi_{j}+a)\bigr]^{-1} = (\xi_{j}-\xi_{i})^{-1} \bigl[(\xi_{i}+a)^{-1}-(\xi_{j}+a)^{-1}\bigr]$,

\begin{equation}\tag{A15.29} \sum_{p}\pp_{i}x_{p}\,\pp_{j}x_{p} = \frac{1}{4}\sum_{p} \frac{x_{p}^{2}}{\bigl(\xi_{i}+a_{p}^{2}\bigr) \bigl(\xi_{j}+a_{p}^{2}\bigr)} = \frac{G(\xi_{i})-G(\xi_{j})}{4\bigl(\xi_{j}-\xi_{i}\bigr)} = 0\ec \end{equation}

because $N(\xi_{i}) = N(\xi_{j}) = 0$ makes $G(\xi_{i}) = G(\xi_{j}) = 1$ by Equation (A15.28): the system is orthogonal. For the scale factors, Equation (A15.27) gives $h_{i}^{2} = \frac14\sum_{p}x_{p}^{2} \bigl(\xi_{i}+a_{p}^{2}\bigr)^{-2} = -\tfrac14 G'(\xi_{i})$, and differentiating Equation (A15.28),

\begin{equation}\tag{A15.30} G'(\theta) = -\frac{N'(\theta)P(\theta)-N(\theta)P'(\theta)} {P(\theta)^{2}}\ec\qquad G'(\xi_{i}) = -\frac{N'(\xi_{i})}{P(\xi_{i})} = -\frac{\prod_{l\neq i}\bigl(\xi_{i}-\xi_{l}\bigr)}{P(\xi_{i})}\ec \end{equation}

using $N(\xi_{i})=0$. That is Equation (A15.25). Positivity follows from Equation (A15.23): $P(\xi_{1})>0$ and $\prod_{l\neq1}(\xi_{1}-\xi_{l})>0$; $P(\xi_{2})<0$ and $\prod_{l\neq2}(\xi_{2}-\xi_{l})<0$; $P(\xi_{3})>0$ and $\prod_{l\neq3}(\xi_{3}-\xi_{l})>0$.

∎

Lemma A15.12 (Lagrange's interpolation identity).

For distinct $\xi_{1},\xi_{2},\xi_{3}$ and an integer $0 \le p \le 2$,

\begin{equation}\tag{A15.31} \sum_{i=1}^{3} \frac{\xi_{i}^{\,p}}{\prod_{l\neq i}\bigl(\xi_{i}-\xi_{l}\bigr)} = \delta_{p2}\ep \end{equation}

Rests on Equation (A15.25).

Proof.

Derives Lemma A15.12. The polynomial $L(\theta) = \sum_{i}\xi_{i}^{\,p}\prod_{l\neq i} \frac{\theta-\xi_{l}}{\xi_{i}-\xi_{l}}$ has degree at most $2$ and agrees with $\theta^{p}$ at the three distinct points $\xi_{1},\xi_{2},\xi_{3}$. Since $\theta^{p}$ also has degree at most $2$, their difference is a polynomial of degree at most $2$ with three roots, hence identically zero: $L(\theta) = \theta^{p}$. Comparing the coefficients of $\theta^{2}$ on the two sides gives Equation (A15.31), the left-hand coefficient being the sum displayed and the right-hand one $\delta_{p2}$.

∎

Proposition A15.13 (The ellipsoidal system separates).

The ellipsoidal system of Definition A15.10 satisfies Equation (A15.8) and Equation (A15.9) with

\begin{equation}\tag{A15.32} \phi_{ij} = \frac{\xi_{i}^{\,3-j}}{4\,P(\xi_{i})}\ec\qquad S = \frac{W}{64\,P(\xi_{1})P(\xi_{2})P(\xi_{3})}\ec\qquad W = \prod_{i<l}\bigl(\xi_{i}-\xi_{l}\bigr)\ec \end{equation}

and, with $\epsilon_{1}=\epsilon_{3}=+1$ and $\epsilon_{2}=-1$ the signs of $P$ on the three intervals of Equation (A15.23),

\begin{equation}\tag{A15.33} f_{1} = -2\sqrt{P(\xi_{1})}\ec\qquad f_{2} = 2\sqrt{-P(\xi_{2})}\ec\qquad f_{3} = 2\sqrt{P(\xi_{3})}\ep \end{equation}

Only the product of the three signs is fixed, by Equation (A15.9); the sign of an individual $f_{i}$ is immaterial because $f_{i}$ occurs twice in Equation (A15.6). The separated equations Equation (A15.6) become the Lamé wave equations

\begin{equation}\tag{A15.34} \frac{1}{\sqrt{\epsilon_{i}P}}\,\dv{}{\xi_{i}} \left(\sqrt{\epsilon_{i}P}\,\dv{X_{i}}{\xi_{i}}\right) + \frac{k^{2}\xi_{i}^{2}+\lambda_{2}\xi_{i}+\lambda_{3}} {4\,P(\xi_{i})}\,X_{i} = 0\ec \end{equation}

with $P$ evaluated at $\xi_{i}$. Rests on Lemma A15.11, Lemma A15.12 and Theorem A15.5.

Proof.

Derives Proposition A15.13. Each row of $\Phi$ in Equation (A15.32) depends on $\xi_{i}$ alone, as a Stäckel matrix must. By Equation (A15.25), $1/h_{i}^{2} = 4P(\xi_{i})/\prod_{l\neq i}(\xi_{i}-\xi_{l})$, so

\begin{equation}\tag{A15.35} \sum_{i}\frac{\phi_{ij}}{h_{i}^{2}} = \sum_{i}\frac{\xi_{i}^{\,3-j}} {\prod_{l\neq i}\bigl(\xi_{i}-\xi_{l}\bigr)} = \delta_{3-j,\,2} = \delta_{j1} \end{equation}

by Lemma A15.12. By Step 2 of the proof of Theorem A15.5 this identity is equivalent to Equation (A15.8), and it also shows $S \neq 0$.

For the determinant, factor $1/\bigl(4P(\xi_{i})\bigr)$ out of the $i$-th row: $S = \bigl[64\prod_{i}P(\xi_{i})\bigr]^{-1}\det\bigl(\xi_{i}^{3-j}\bigr)$. Reversing the column order — one transposition, of columns $1$ and $3$ — turns $\det(\xi_{i}^{3-j})$ into $-\det(\xi_{i}^{j-1})$, and the latter is the Vandermonde determinant $\prod_{i<l}(\xi_{l}-\xi_{i}) = -W$; hence $\det(\xi_{i}^{3-j}) = W$ and $S$ is as stated.

For Robertson, take the product of Equation (A15.25) over $i$. The numerator is $\prod_{i}\prod_{l\neq i}(\xi_{i}-\xi_{l}) = \prod_{i<l}\bigl[-(\xi_{i}-\xi_{l})^{2}\bigr] = -W^{2}$, so $g^{2} = -W^{2}\bigl[64\prod_{i}P(\xi_{i})\bigr]^{-1}$; the sign is consistent because $\prod_{i}P(\xi_{i}) < 0$ by Equation (A15.23), and therefore

\begin{equation}\tag{A15.36} g = \frac{W}{8\prod_{i}\sqrt{\epsilon_{i}P(\xi_{i})}}\ec\qquad S = \frac{-W}{64\prod_{i}\epsilon_{i}P(\xi_{i})}\ec \end{equation}

the second because $\prod_{i}\epsilon_{i} = -1$. Dividing,

\begin{equation}\tag{A15.37} \frac{g}{S} = -8\,\frac{\prod_{i}\epsilon_{i}P(\xi_{i})} {\prod_{i}\sqrt{\epsilon_{i}P(\xi_{i})}} = -8\prod_{i}\sqrt{\epsilon_{i}P(\xi_{i})} = \prod_{i}f_{i}(\xi_{i})\ec \end{equation}

the last step by Equation (A15.33), whose three signs multiply to $-1$: a product of three functions of one variable each, which is exactly Equation (A15.9). Substituting $f_{i}$ and $\sum_{j}\lambda_{j}\phi_{ij} = \bigl(k^{2}\xi_{i}^{2}+\lambda_{2}\xi_{i}+\lambda_{3}\bigr)/ \bigl(4P(\xi_{i})\bigr)$ into Equation (A15.6) gives Equation (A15.34), the constant factor $2$ in $f_{i}$ cancelling between the two occurrences.

∎

Remark A15.14.

For $k = 0$, Equation (A15.34) is Lamé's equation and its polynomial solutions are the ellipsoidal harmonics, the analogue for a triaxial body of the spherical harmonics Equation (13.163). The chapter's three examples are visible in Equation (A15.34) as limits: when two of the $a_{p}^{2}$ coincide one of the three coordinates becomes an angle and the corresponding Lamé equation degenerates to the associated Legendre equation Equation (13.157); when all three coincide the remaining two degenerate to the Legendre equation and to $\Psi''+m^{2}\Psi=0$, and Example A15.9 is recovered.

The enumeration

What remains is to count the systems. The count has two halves, and only one of them is proved here.

Theorem A15.15 (Quoted: separable systems are confocal-quadric systems).

Let $(q^{i})$ be an orthogonal coordinate system of $\R^{3}$ in which the Helmholtz equation separates simply. Then, up to a Euclidean motion and a relabelling and reparametrization of the coordinates, the coordinate surfaces are the members of a confocal family of quadrics Equation (A15.22) or of one of its degenerate limits.

Remark A15.16 (What is quoted here).

Theorem A15.15 is the only statement in this section that is not proved. It is the substantial half of the classification: the assertion that a metric of flat space admitting a Stäckel matrix must have confocal quadrics for its coordinate surfaces. Its proof is a long analysis of the second-order system that Equation (A15.8) imposes on the nine functions $\phi_{ij}$ together with the vanishing of the Riemann tensor of the Euclidean metric written in the coordinates $q^{i}$ — a computation belonging to classical differential geometry rather than to the theory of partial differential equations, and one this treatise does not carry out anywhere. The result and the resulting tables of separated equations, system by system, are those of Morse and Feshbach [Morse:1953]; the differential-geometric analysis behind it is Eisenhart's, whose paper is not among the sources catalogued in this treatise's bibliography, so the attribution here is uncited and the reader who wishes to check it must go to the secondary account just named. Everything else in this section — the two conditions, their necessity and sufficiency, the ellipsoidal verification and the degeneration analysis below — is derived in full.

Granting Theorem A15.15, the enumeration becomes a finite case analysis over the ways the confocal family Equation (A15.22) can degenerate, and that we carry out. The family is fixed by the cubic $P$ of Equation (A15.21), i.e. by the three parameters $a_{1}^{2} > a_{2}^{2} > a_{3}^{2}$; a common shift of all three is a reparametrization $\theta \to \theta+c$ and changes nothing, so only the two differences $a_{1}^{2}-a_{2}^{2}$ and $a_{2}^{2}-a_{3}^{2}$ matter, and a common rescaling fixes the unit of length. The degenerations are then exhausted by three independent binary choices:

Coincidence. Either both differences are nonzero (three distinct foci), or exactly one vanishes (an axis of rotational symmetry appears, and there are two inequivalent ways for this to happen — the focal set degenerates to a segment when $a_{2}^{2}=a_{3}^{2}$ and to a disc when $a_{1}^{2}=a_{2}^{2}$), or both vanish (full rotational symmetry: concentric spheres).
A focus at infinity. One may let one root of $P$ recede, $a_{1}^{2}\to\infty$, while rescaling; the ellipsoids and one family of hyperboloids open up into paraboloids. This can be done to the generic family and to the rotationally symmetric one, but not to the spherical one, where there is nothing left to send away.
Translation. One may instead let the whole configuration become independent of one Cartesian direction, so that the quadrics become cylinders over a two-dimensional confocal family of conics. The two-dimensional families are themselves classified by the same first two choices, one dimension down: confocal conics with two distinct foci, confocal parabolas, concentric circles, or the degenerate family of parallel lines.

Reading the choices off gives the list, in which the first seven are the genuinely three-dimensional families and the last four the cylindrical ones.

System	Coordinate surfaces	Degeneration
Ellipsoidal	three confocal quadrics	none: $a_{1}^{2}>a_{2}^{2}>a_{3}^{2}$
Paraboloidal	elliptic and hyperbolic paraboloids	one focus to infinity
Prolate spheroidal	prolate spheroids, two-sheeted hyperboloids, half-planes	$a_{2}^{2}=a_{3}^{2}$ (focal segment)
Oblate spheroidal	oblate spheroids, one-sheeted hyperboloids, half-planes	$a_{1}^{2}=a_{2}^{2}$ (focal disc)
Parabolic	two families of paraboloids of revolution, half-planes	rotational, one focus to infinity
Spherical	spheres, cones of revolution, half-planes	$a_{1}^{2}=a_{2}^{2}=a_{3}^{2}$
Conical	spheres and two families of elliptic cones	spheres with the focal cone retained
Elliptic cylindrical	confocal elliptic and hyperbolic cylinders, planes	translational, two foci
Parabolic cylindrical	confocal parabolic cylinders, planes	translational, one focus at infinity
Circular cylindrical	circular cylinders, half-planes, planes	translational, coincident foci
Cartesian	three families of parallel planes	translational, all foci at infinity

The eleven orthogonal systems in which the three-dimensional Helmholtz equation separates simply, organized by the degeneration of the confocal quadric family that produces them. That each entry does separate is checked by exhibiting its Stäckel matrix, as done in the text for the Cartesian, cylindrical, spherical and ellipsoidal systems; that the list is complete rests on the one quoted classification theorem of this section.

Proposition A15.17 (The count).

The degeneration scheme just described yields exactly the eleven systems of Table A15.1, and no others. Rests on Proposition A15.13 and Theorem A15.15.

Proof.

Derives Proposition A15.17. Consider first the non-translational families. The coincidence pattern of the three parameters has four cases: all distinct; $a_{2}^{2}=a_{3}^{2}$; $a_{1}^{2}=a_{2}^{2}$; all equal. In the first three cases one may additionally send a focus to infinity or not, which doubles them — but the two rotationally symmetric cases have the same paraboloidal limit, because sending the distinct root to infinity destroys the distinction between a focal segment and a focal disc, leaving the single parabolic system. That gives $1+1$ (ellipsoidal, paraboloidal) plus $2+1$ (prolate, oblate, parabolic), i.e. five. The fully symmetric case admits no focus at infinity, but it does admit two inequivalent coordinate systems on the sphere: the one whose second and third coordinates are the polar angle and the azimuth — spherical — and the one in which the sphere is cut by two families of elliptic cones sharing a vertex, which is the conical system and is the residue of the ellipsoidal system when the quadrics have collapsed but their asymptotic cones have not. That gives seven.

The translational families are cylinders over a two-dimensional confocal family of conics, and the same analysis one dimension down applies to the plane cross-section: two distinct foci give confocal ellipses and hyperbolae (elliptic cylindrical); sending one focus to infinity gives confocal parabolae (parabolic cylindrical); coincident foci give concentric circles and their radii (circular cylindrical); and sending both to infinity gives two families of parallel lines (Cartesian). No further case arises, because a family of conics in the plane is fixed by the two foci alone, and their configuration — distinct, coincident, one at infinity, both at infinity — has exactly these four types. That gives four, and $7+4 = 11$.

∎

Proof of Theorem 14.34. Derives Theorem 14.34. The criterion asserted by the theorem is Theorem A15.5: separation holds if and only if the reciprocal squared scale factors are expressible through a matrix each of whose rows depends on one coordinate alone — explicitly, if and only if $1/h_{i}^{2} = M_{i1}/S$ in Equation (A15.8), together with the Robertson condition Equation (A15.9) on the volume element, which the chapter's statement subsumes under “the Stäckel conditions”. The list of eleven systems is Table A15.1: that each of them separates is verified by exhibiting its Stäckel matrix, done in Examples A15.7, A15.8 and A15.9 for the three the treatise uses and in Proposition A15.13 for the generic one, from which the remaining entries follow by the confluences of Proposition A15.17; and that there are no others is Proposition A15.17 together with the quoted Theorem A15.15.

∎

Remark A15.18.

The Stäckel Conditions and the Eleven Separable Systems discharges the proof obligation of Theorem 14.34 in Partial Differential Equations. Two consequences are used elsewhere in the treatise. The first is negative and is the practical one recorded in Remark 14.35: a boundary that is not a coordinate surface of one of the eleven systems admits no separation, which is why the boundary chooses the coordinates and not the operator. The second is that Equation (A15.8) alone — without Robertson's Equation (A15.9) — is the condition for the Hamilton–Jacobi equation of Hamilton–Jacobi Theory and the Optical–Mechanical Analogy to separate, so the separation constants $\lambda_{2},\lambda_{3}$ of Equation (A15.6) are the quantum counterparts of the classical integrals of motion of an integrable system; the spherical case (Example A15.9), where they are $l(l+1)$ and $m^{2}$, is the one worked out physically in The Hydrogen Atom.