Staeckel Conditions for Separability

By the middle of the nineteenth century, classical mechanics had accumulated a trophy cabinet of "miraculously" solvable problems: Kepler orbits in spherical coordinates, Euler's two-fixed-center problem in elliptic coordinates, Jacobi's geodesics on the ellipsoid, the charged particle in a Coulomb field plus a uniform field in parabolic coordinates. Each solution rested on finding a coordinate system in which the Hamilton-Jacobi equation fell apart into one-dimensional pieces — and each discovery looked like luck, a chart stumbled upon by genius. In 1891 Paul Stäckel ended the era of luck. He proved that for orthogonal coordinates there is a complete algebraic criterion — necessary and sufficient — for additive separation: a nonsingular matrix $\Phi$ must exist whose $i$ -th row depends on the coordinate $q_i$ alone, reproducing the kinetic metric through its inverse's first row and the potential as a matching single-variable sum. The construction is startlingly rigid. Every separation constant enters linearly; inverting the Stäckel matrix hands you $n$ uncoupled equations $p_i^2=2\sum_j\phi_{ij}(q_i)\alpha_j-2u_i(q_i)$ , each a quadrature in its own coordinate. And the payoff is more than technique: a Stäckel system automatically carries $n$ independent, Poisson-commuting integrals quadratic in the momenta — the hypotheses of the Liouville-Arnold theorem verified wholesale. Orthogonal separability does not merely suggest integrability; it certifies it, constants, tori, and all. This article builds the matrix machinery, proves both the separation and the involution theorems, and then runs the test concretely on spherical coordinates — watching $E$ , $L^2$ , and $L_z^2$ emerge as the linear constants the theorem demands.

Key takeaways

The Stäckel conditions are the complete algebraic criterion — necessary and sufficient — for additive separation of the time-independent Hamilton-Jacobi equation in orthogonal coordinates.
The certificate is a nonsingular $n\times n$ Stäckel matrix $\phi_{ij}(q_i)$ , each row a function of its own coordinate only, whose inverse's first row reproduces the kinetic metric coefficients: $1/m_i=(\Phi^{-1})_{1i}$ .
The potential must match the same structure, $V=\sum_i(\Phi^{-1})_{1i}\,u_i(q_i)$ — a "Stäckel-weighted" sum of single-variable functions.
When both conditions hold, the momenta obey $p_i^2=2\sum_j\phi_{ij}(q_i)\,\alpha_j-2u_i(q_i)$ : every separation constant enters linearly, and each equation is a one-coordinate quadrature.
The construction supplies $n$ functionally independent integrals $F_j$ , quadratic in momenta and in involution — so every orthogonally separable system is Liouville integrable.
Key idea: orthogonal separability ⇔ a Stäckel matrix exists — an $n\times n$ matrix whose $i$ -th row depends only on $q_i$ , expressing the inverse metric and the potential as matrix-weighted sums of single-variable functions; the separation constants are then read off linearly by inverting the matrix.

The core insight

Orthogonal separability ⇔ a Stäckel matrix exists: an $n\times n$ matrix whose $i$ -th row depends only on $q_i$ , so that the inverse metric and the potential are linear combinations (via that matrix) of single-variable functions — the separation constants are then read off by inverting the matrix.

Where this lives today

Eisenhart's geometric reading — a Stäckel structure is a family of commuting Killing tensors on the configuration manifold — is the version that escaped classical mechanics and conquered relativity. The Kerr spacetime of a rotating black hole is the showpiece: its metric admits a nontrivial Killing tensor, the geodesic Hamilton-Jacobi equation separates in Boyer-Lindquist coordinates (Carter, 1968), and the resulting Carter constant is a textbook Stäckel separation constant — quadratic in momenta, commuting with everything, corresponding to no isometry whatsoever. Every Kerr ray-tracing code behind the Event Horizon Telescope images, and every extreme mass-ratio inspiral waveform model being built for the LISA mission, runs on quadratures that exist because Kerr passes the (curved-spacetime) Stäckel test. On the mathematical side the subject is alive as the classification of separable and superintegrable systems: cataloguing, chart family by chart family and manifold by manifold, which potentials admit Stäckel matrices, which admit extra integrals beyond the $n$ Liouville ones, and how these structures deform and quantize. The eleven separable charts of Euclidean 3-space have long been settled; their analogues on spheres, hyperbolic spaces, and black-hole spacetimes — and the algebras their integrals generate — remain an active research industry in 2026.

The physical picture

The article on Separation of Variables in Hamilton-Jacobi Theory showed the move: isolate one coordinate's dependence in a self-contained block, freeze the block at a constant, repeat. The question it left open is when the isolating rearrangement exists. Stäckel's answer is a piece of bookkeeping machinery. Picture the $n$ separated equations stacked as a vector: the $i$ -th line may involve only $q_i$ , yet all $n$ constants $\alpha_1,\dots,\alpha_n$ are allowed to appear in it. The object that routes constants into single-coordinate lines without ever smuggling in a foreign coordinate is precisely a matrix whose $i$ -th row is built from functions of $q_i$ alone. That is the Stäckel matrix — a wiring diagram between the space of conserved constants and the space of coordinates.

Figure

The Stäckel matrix as a wiring diagram — each row speaks one coordinate, each column carries one constant

Why does the whole theory insist on orthogonal coordinates? Because orthogonality makes the kinetic energy a diagonal quadratic form, $T=\tfrac12\sum_i p_i^2/m_i(q)$ , with no $p_ip_j$ cross terms. Squared momenta are exactly the objects the separated equations control, one per line; an off-diagonal metric would tie $p_i$ to $p_j$ inside a single term and no row-structured matrix could untangle them. (Separation with cross terms is not impossible — but it needs the genuinely different Levi-Civita machinery, outside Stäckel's theorem.) The conditions themselves come in a matched pair, and it helps to hear them as two verses of the same demand. The metric condition says the inverse metric coefficients $1/m_i(q)$ — which in general depend on all the coordinates, as $1/mr^2$ does in spherical coordinates — must be manufacturable as the first row of $\Phi^{-1}$ . The potential condition says $V$ must be a sum of single-variable functions weighted by those same first-row entries. Kinetic and potential energy must be "Stäckel-shaped" together, with one and the same matrix: that shared rigidity is what makes the criterion genuinely restrictive, and genuinely decisive.

Figure

Three charts that pass the test — the separable coordinate families of the plane and space

One more reading, the deepest: the constants $\alpha_1,\dots,\alpha_n$ delivered by the matrix are coordinates on the space of invariant tori. The energy $E=\alpha_1$ is one of them; the rest are the "hidden" conserved quantities — total angular momentum, a Laplace-Runge-Lenz component, the two-center quadratic invariant — that make integrability work. Stäckel's theorem manufactures them wholesale, quadratic in momenta and mutually commuting, which is why it slots directly beneath the Liouville-Arnold theorem as the practical certificate of integrability.

The conditions, precisely

Everything happens inside the time-independent Hamilton-Jacobi equation for a natural Hamiltonian in orthogonal coordinates. The kinetic metric is diagonal, so the Hamiltonian reads

H=\frac12\sum_{i=1}^{n}\frac{p_i^2}{m_i(q)}+V(q)=E \qquad (1)

where: $m_i(q)$ — the (generally coordinate-dependent) metric coefficients; in spherical coordinates, for instance, $m_r=m$ , $m_\theta=mr^2$ , $m_\phi=mr^2\sin^2\theta$ ; $V$ — the potential; $E$ — the conserved energy. The goal, as always, is an additively separated characteristic function, $W=\sum_iW_i(q_i)$ with $p_i=\mathrm{d}W_i/\mathrm{d}q_i$ . The certificate has three parts. First, the matrix itself:

\Phi=[\phi_{ij}],\qquad \phi_{ij}=\phi_{ij}(q_i),\qquad \det\Phi\neq0 \qquad (2)

where: the defining property sits in the middle: the entries of row $i$ are functions of the single coordinate $q_i$ . Nonsingularity is not decoration — it is what will make the $n$ separation constants functionally independent (Derivation B). Note the asymmetry: rows are single-variable, but the inverse matrix $\Phi^{-1}$ has entries depending on all coordinates at once, which is exactly how single-variable ingredients can conspire to build the fully coordinate-mixed functions $1/m_i(q)$ . Second, the metric condition — the inverse metric must be the first row of $\Phi^{-1}$ :

\frac{1}{m_i(q)}=(\Phi^{-1})_{1i} \qquad (3)

where: the right side is the $(1,i)$ entry of the inverse Stäckel matrix. This is a strong structural demand on the chart: most orthogonal coordinate systems on most manifolds admit no such matrix, which is why the separable charts form a short, classifiable list (eleven families in Euclidean 3-space). Third, the potential condition — $V$ must decompose over the same weights:

V(q)=\sum_{i=1}^{n}(\Phi^{-1})_{1i}\,u_i(q_i) \qquad (4)

where: $u_i$ — a single-variable "potential vector". Read (3) and (4) together: kinetic and potential terms must both be linear in the same first-row weights. A chart can pass the metric test and still fail on the potential — Cartesian coordinates pass (3) trivially for any flat metric, but (4) then demands $V=\sum_iu_i(q_i)$ , an additive potential, which is why $V\propto xy$ refuses to separate in Cartesian axes (Worked Example 2). When both conditions hold, substituting the ansatz collapses the PDE into the vector of separated equations — the payoff formula of the subject:

\left(\frac{\mathrm{d}W_i}{\mathrm{d}q_i}\right)^2=p_i^2=2\sum_{j=1}^{n}\phi_{ij}(q_i)\,\alpha_j-2\,u_i(q_i) \qquad (5)

where: $\alpha_j$ — the separation constants, with $\alpha_1=E$ by convention. Two features deserve a slow look. Every constant enters linearly — no products, no squares of the $\alpha$ 's — because the constants arrive by inverting a linear system; if your problem's natural constants appear squared (as $L_z^2$ does in the spherical $\theta$ -equation), the Stäckel constants are those squares themselves. And every quantity on the right depends on $q_i$ alone, so each momentum is an explicit single-variable function and $W_i$ follows by one quadrature. Finally, the same matrix hands over the integrals of motion in closed form:

F_j(q,p)=\sum_{i=1}^{n}(\Phi^{-1})_{ji}\left(\frac{p_i^2}{2}+u_i(q_i)\right),\qquad F_1=H,\qquad \{F_j,F_k\}=0 \qquad (6)

where: $F_j$ — the $j$ -th integral, quadratic in the momenta, whose on-shell value is precisely $\alpha_j$ . The involution property $\{F_j,F_k\}=0$ is proved in Derivation B; together with independence (from $\det\Phi\neq0$ ) it delivers Liouville integrability outright.

Which equation answers which question

you want	use	notes
to test a chart for separability	conditions (2)+(3)+(4)	metric and potential must pass together
the separated momenta	eq. (5)	constants enter linearly; one coordinate per line
the conserved quantities explicitly	eq. (6)	quadratic in momenta; $F_j=\alpha_j$ on shell
the constants from initial data	invert (5): $\alpha=\Phi^{-1}\big(\tfrac12p^2+u\big)$	evaluate at any single phase-space point
geodesic/free-particle separability	(3) alone with all $u_i=0$	a purely metric statement
non-orthogonal or magnetic problems	Levi-Civita conditions; Killing-tensor methods	outside the standard Stäckel theorem

Derivation AFrom the Stäckel matrix to the separated equations

We prove sufficiency: if the metric condition and the potential condition hold, the additive ansatz separates the Hamilton-Jacobi equation, with the separated momenta given by equation (5). The strategy is constructive linear algebra. We substitute the ansatz, rewrite the equation as the first component of an $n$ -component vector identity, and then posit the full vector system — the other $n-1$ components define the remaining separation constants. The single algebraic fact doing all the work is $\Phi^{-1}\Phi=\mathbb{1}$ ; watch how the row structure of $\Phi$ keeps every posited line confined to one coordinate, which is the entire point of the construction.

Step 1. Write the time-independent Hamilton-Jacobi equation in orthogonal coordinates and insert the additive ansatz $W=\sum_iW_i(q_i)$ :

\frac12\sum_i\frac{1}{m_i(q)}\left(\frac{\mathrm{d}W_i}{\mathrm{d}q_i}\right)^2+V(q)=E

Why the ansatz simplifies the derivatives: each $\partial W/\partial q_i$ collapses to the ordinary derivative $W_i'(q_i)$ , a single-variable object — the coordinate mixing now lives only in $1/m_i(q)$ and $V(q)$ .

Step 2. Replace the mixed-coordinate data by their Stäckel expressions, $1/m_i=(\Phi^{-1})_{1i}$ and $V=\sum_i(\Phi^{-1})_{1i}u_i$ , and collect the common weights:

\sum_i(\Phi^{-1})_{1i}\left[\frac12\,W_i'^2+u_i(q_i)\right]=E

Why this regrouping matters: the bracketed quantity is a function of $q_i$ alone — all the coordinate mixing has been swept into the first-row weights $(\Phi^{-1})_{1i}$ , which multiply single-variable brackets.

Step 3. Posit the full vector system that this equation is the first row of. Demand constants $\alpha_1,\dots,\alpha_n$ such that, for every $i$ ,

\frac12\,W_i'^2+u_i(q_i)=\sum_j\phi_{ij}(q_i)\,\alpha_j

Why each line is legitimate: the right side involves only row $i$ of $\Phi$ — functions of $q_i$ — and constants; the left side involves only $q_i$ . Each line is a self-consistent single-variable ODE, the block-freezing move of separation theory written for all coordinates simultaneously.

Step 4. Verify consistency by contracting with the first row of the inverse: multiply line $i$ by $(\Phi^{-1})_{1i}$ and sum,

\sum_i(\Phi^{-1})_{1i}\left[\frac12W_i'^2+u_i\right]=\sum_j\Big(\sum_i(\Phi^{-1})_{1i}\,\phi_{ij}\Big)\alpha_j=\sum_j\delta_{1j}\,\alpha_j=\alpha_1

Why this closes the loop: comparing with Step 2 forces $\alpha_1=E$ — the posited system reproduces the original Hamilton-Jacobi equation exactly, with the energy as the first constant. The other rows of $\Phi^{-1}$ contract the same system to $\alpha_j$ , defining the remaining constants (that contraction is equation (6)).

Step 5. Solve each line of Step 3 for the squared derivative to obtain the separated equations $W_i'^2=2\sum_j\phi_{ij}(q_i)\alpha_j-2u_i(q_i)$ — equation (5). Each right side is an explicit function of its own coordinate, so $W_i=\int W_i'\,\mathrm{d}q_i$ is a quadrature and the construction is complete. The only algebra used is $\Phi^{-1}\Phi=\mathbb{1}$ . Necessity — that separation forces this structure — is Stäckel's harder converse, established in the classical literature.

Go deeper · Derivation B · 2 minThe Stäckel integrals commute — separability certifies integrability

Derivation BThe Stäckel integrals commute — separability certifies integrability

We show that the $n$ quantities $F_j=\sum_i(\Phi^{-1})_{ji}\big(\tfrac12p_i^2+u_i\big)$ are functionally independent integrals of motion in involution, which by the Liouville-Arnold theorem places the dynamics on invariant tori. The strategy: first read off conservation directly from the separated equations, then establish independence from $\det\Phi\neq0$ , and finally sketch the bracket cancellation, whose engine is — once again — the single-variable row structure. Watch what actually kills the brackets: it is not symmetry or luck but the fact that row $i$ of $\Phi$ and the function $u_i$ can be differentiated only by $q_i$ , paired against a diagonal metric that ties each $\partial/\partial p_i$ to the same index.

Step 1. Conservation, by construction. The separated equations (5) say $\tfrac12p_i^2+u_i=\sum_k\phi_{ik}\alpha_k$ along the motion. Contract with $(\Phi^{-1})_{ji}$ :

F_j=\sum_i(\Phi^{-1})_{ji}\Big(\frac{p_i^2}{2}+u_i\Big)=\sum_k\Big(\sum_i(\Phi^{-1})_{ji}\phi_{ik}\Big)\alpha_k=\alpha_j

Why this proves conservation: the $\alpha_j$ are constants of the particular trajectory, so each $F_j$ holds a frozen value along the flow; and $F_1$ recovers the Hamiltonian itself via the metric and potential conditions.

Step 2. Functional independence. The momentum gradients $\partial F_j/\partial p_i=(\Phi^{-1})_{ji}\,p_i$ assemble, across $j$ , into the rows of $\Phi^{-1}$ scaled by the $p_i$ — and $\det\Phi^{-1}\neq0$ . Why nonsingularity is exactly what is needed: away from the measure-zero set where momenta vanish, the $n$ gradient vectors are linearly independent, so no $F_j$ is a function of the others.

Step 3. The bracket, organized index by index. For any pair $(j,k)$ ,

\{F_j,F_k\}=\sum_i\left(\frac{\partial F_j}{\partial q_i}\frac{\partial F_k}{\partial p_i}-\frac{\partial F_k}{\partial q_i}\frac{\partial F_j}{\partial p_i}\right)

with $\partial F_k/\partial p_i=(\Phi^{-1})_{ki}p_i$ and $\partial F_j/\partial q_i$ acting on the $(\Phi^{-1})_{ji}$ weights and on $u_i$ . Why everything stays index-diagonal: the diagonal metric means each momentum derivative selects one index $i$ , and the $q_i$ -derivative can strike only row $i$ of $\Phi$ and the function $u_i$ — no term ever couples two different coordinate indices in one factor.

Step 4. Cancellation. Write $\partial_{q_i}\Phi^{-1}=-\Phi^{-1}(\partial_{q_i}\Phi)\Phi^{-1}$ , where $\partial_{q_i}\Phi$ has only its $i$ -th row nonzero. Substituting, every term of $\{F_j,F_k\}$ acquires the form (single-variable derivative of row $i$ ) × (inverse-matrix weights) × $p_i$ , symmetric under $j\leftrightarrow k$ except for the antisymmetrized pairing — and the antisymmetrization cancels the survivors pairwise. Hence $\{F_j,F_k\}=0$ for all $j,k$ .

Step 5. Conclude. $n$ independent integrals in involution on a $2n$ -dimensional phase space: the Liouville-Arnold hypotheses hold, so bounded motion lies on invariant $n$ -tori, with the $\alpha_j$ (equivalently the $F_j$ values) labelling the torus. Separability is thereby a sufficient certificate of Liouville integrability — the practical one, since it also hands over the quadratures. The cancellation can be checked by direct expansion for $n=2$ — the two-fixed-center problem in elliptic coordinates is the natural test case.

Go deeper · Derivation C · 2 minRunning the test on spherical coordinates

Derivation CRunning the test on spherical coordinates

We verify the Stäckel conditions concretely for a particle of mass $m$ in a central potential $V(r)$ in spherical coordinates $(r,\theta,\phi)$ — exhibiting the matrix, checking both conditions by explicit inversion, and reading off the constants. The strategy is reverse-engineering: we know the separated momenta from the direct separation (done in Separation of Variables in Hamilton-Jacobi Theory), so we read the matrix rows off those equations, then confirm that this candidate matrix reproduces the metric and the potential as the theorem demands. Watch the one subtlety: to keep every constant linear, the Stäckel constants must be chosen as $\alpha_2=L^2$ and $\alpha_3=L_z^2$ — the squares of the familiar momenta.

Step 1. Recall the separated momentum equations for the central-force problem:

p_r^2=2m\big(E-V(r)\big)-\frac{L^2}{r^2},\qquad p_\theta^2=L^2-\frac{L_z^2}{\sin^2\theta},\qquad p_\phi^2=L_z^2

Why squares everywhere: the Stäckel form (5) governs $p_i^2$ , and the natural constants enter these equations through $L^2$ and $L_z^2$ — so linearity dictates the constant vector $\alpha=(E,\,L^2,\,L_z^2)$ .

Step 2. Match against $p_i^2=2\sum_j\phi_{ij}(q_i)\alpha_j-2u_i(q_i)$ , row by row, reading off coefficients of each $\alpha_j$ :

\Phi=\begin{pmatrix} m & -\dfrac{1}{2r^2} & 0\\[2mm] 0 & \dfrac12 & -\dfrac{1}{2\sin^2\theta}\\[2mm] 0 & 0 & \dfrac12 \end{pmatrix},\qquad u=\begin{pmatrix} m\,V(r)\\ 0\\ 0\end{pmatrix}

where: row 1 depends only on $r$ , row 2 only on $\theta$ , row 3 on nothing at all (constants are allowed — $\phi$ is cyclic). $\det\Phi=m/4\neq0$ . Why the row test passes: each equation of Step 1 involves exactly one coordinate on its right-hand side, which is what let us read single-variable rows off it.

Step 3. Check the metric condition. The first row $c_i=(\Phi^{-1})_{1i}$ solves $\sum_ic_i\phi_{ij}=\delta_{1j}$ . Column by column: $j=1$ gives $c_r\,m=1$ , so $c_r=1/m$ ; $j=2$ gives $-c_r/2r^2+c_\theta/2=0$ , so $c_\theta=1/mr^2$ ; $j=3$ gives $-c_\theta/2\sin^2\theta+c_\phi/2=0$ , so $c_\phi=1/mr^2\sin^2\theta$ :

(\Phi^{-1})_{1i}=\left(\frac1m,\;\frac{1}{mr^2},\;\frac{1}{mr^2\sin^2\theta}\right)=\frac{1}{m_i}

Why this is the punchline of the check: three single-variable rows have conspired, through inversion, to manufacture the fully coordinate-mixed spherical metric coefficients — the mechanism promised abstractly by condition (3), seen working.

Step 4. Check the potential condition: $\sum_i(\Phi^{-1})_{1i}u_i=\frac1m\cdot mV(r)+0+0=V(r)$ — passed, for any central potential. Why the generality: the potential vector had the whole first slot to itself; centrality, not the specific form of $V$ , is what the condition tests here.

Step 5. Confirm the certificate's dividends. Contracting with the other rows of $\Phi^{-1}$ (equation (6)) returns the integrals $F_1=H$ , $F_2=p_\theta^2+p_\phi^2/\sin^2\theta=L^2$ , $F_3=p_\phi^2=L_z^2$ — independent, quadratic in momenta, and in involution, exactly the involutive set of the central-force problem. As a consistency check: substituting Step 3's weights and Step 1's momenta back into $\tfrac12\sum_ip_i^2/m_i+V$ returns $E$ identically.

Limiting cases

Case	Result	Meaning
free particle / geodesics	all $u_i=0$ ; only condition (3) remains	separability becomes a property of the metric alone — Jacobi's ellipsoid geodesics live here
Cartesian chart, flat metric	$(\Phi^{-1})_{1i}=$ const; condition (4) $\Rightarrow V=\sum_iu_i(q_i)$	in Cartesian axes only additive potentials separate
cyclic coordinate $q_k$	row $k$ constant, $u_k=0$ , $p_k^2=\alpha_k$	the trivial separation, absorbed as one constant row of the matrix
$n=1$	$\Phi=(\phi_{11})$ , any nonzero function	every one-dimensional system passes — energy conservation is the separation
Liouville systems	$H=\dfrac{\sum_i\big(\tfrac12p_i^2+v_i(q_i)\big)}{\sum_iw_i(q_i)}$	the classic 1846 special case — always Stäckel; historically the first general family

Common trapIf a system is integrable, the Stäckel test will certify it

No — the theorem's scope is precise, and overreading it causes real errors. The Stäckel conditions characterize orthogonal additive separability of natural Hamiltonians ( $T+V$ , diagonal metric, no magnetic terms). Integrable systems exist that pass no such test in any orthogonal chart: the Toda lattice's integrals are cubic and higher in momenta — no quadratic-integral machinery can produce them — and a charged particle in a magnetic field has momentum-linear terms that the standard conditions do not even parse (the Levi-Civita conditions and the Killing-tensor methods of the Benenti and Kalnins-Miller school handle some of these). The converse trap is subtler: passing the metric condition (3) does not certify a chart — the potential condition (4) must pass with the same matrix, which is why Cartesian coordinates, always metrically fine, still refuse any non-additive potential. Stäckel separability is a sufficient certificate of integrability, never a necessary one, and both halves of the certificate must be checked.

Worked examples

1 · A full Stäckel audit in the plane (confidence builder)

A particle of mass $m=2\ \mathrm{kg}$ moves in the plane in the potential $V=-k/r$ with $k=8\ \mathrm{J\,m}$ . At the instant of interest it is at $r=2\ \mathrm{m}$ with radial velocity $\dot r=1\ \mathrm{m/s}$ and transverse velocity $r\dot\phi=2\ \mathrm{m/s}$ . Exhibit the Stäckel matrix in polar coordinates, evaluate both separation constants numerically from the given data, and verify the radial separated equation.

Step 1. In polar coordinates $H=\frac{1}{2m}\big(p_r^2+p_\phi^2/r^2\big)+V(r)$ . Matching $p_r^2=2m(E-V)-\alpha_2/r^2$ and $p_\phi^2=\alpha_2$ (with $\alpha_2=L_z^2$ ) gives the $2\times2$ Stäckel data $\Phi=\begin{pmatrix}m & -1/2r^2\\ 0 & 1/2\end{pmatrix}$ , $u=(mV(r),\,0)$ , with $\det\Phi=m/2\neq0$ and first inverse row $(1/m,\,1/mr^2)$ — the polar metric coefficients, as condition (3) demands.

Step 2. Momenta from the data: $p_r=m\dot r=2\times1=\mathbf{2\ kg\,m/s}$ ; $p_\phi=mr(r\dot\phi)=2\times2\times2=\mathbf{8\ kg\,m^2/s}$ .

Step 3. First constant (energy): $E=\frac{1}{2m}\big(p_r^2+p_\phi^2/r^2\big)+V=\frac{1}{4}\big(4+64/4\big)-\frac{8}{2}=\frac{20}{4}-4=\mathbf{1\ J}$ .

Step 4. Second constant from the matrix route, $\alpha=\Phi^{-1}\big(\tfrac12p^2+u\big)$ : the second row of $\Phi^{-1}$ is $(0,\,2)$ , so $\alpha_2=2\times\tfrac12p_\phi^2=p_\phi^2=\mathbf{64\ (kg\,m^2/s)^2}$ — the squared angular momentum, delivered by pure linear algebra.

Step 5. Verify the radial separated equation at the given point: $2m(E-V)-\alpha_2/r^2=2\times2\times\big(1-(-4)\big)-64/4=20-16=\mathbf{4}$ , which equals $p_r^2=2^2$ exactly. The certificate checks out on the numbers: the frozen constants, the matrix, and the instantaneous momenta are one consistent structure.

2 · Diagnosing and curing a separation failure (exam level)

A particle of mass $m=0.5\ \mathrm{kg}$ moves in the coupled planar potential $V=\tfrac12k(x^2+y^2)+\lambda xy$ with $k=13\ \mathrm{N/m}$ and $\lambda=5\ \mathrm{N/m}$ . Show with the Stäckel conditions that Cartesian axes $(x,y)$ cannot separate this problem, find a chart that does, and compute the two fundamental frequencies.

Step 1. Audit the Cartesian chart. The metric is flat and constant, so condition (3) holds with a constant matrix and the potential condition (4) collapses to the demand that $V$ be additive: $V=u_x(x)+u_y(y)$ . Apply the mixed-derivative test: $\partial^2V/\partial x\partial y=\lambda=5\neq0$ , while any additive potential has $\partial^2(u_x+u_y)/\partial x\partial y=0$ identically. The Cartesian chart fails the potential condition — no choice of $u_x,u_y$ exists.

Step 2. Cure by rotating the chart. In axes $x'=(x+y)/\sqrt2$ , $y'=(x-y)/\sqrt2$ (still orthogonal, metric still flat), use $x^2+y^2=x'^2+y'^2$ and $xy=\tfrac12(x'^2-y'^2)$ : the potential becomes $V=\tfrac12(k+\lambda)x'^2+\tfrac12(k-\lambda)y'^2$ — additive. Both Stäckel conditions now pass, with $u_{x'}(x')$ and $u_{y'}(y')$ as the separated blocks.

Step 3. Evaluate the effective stiffnesses: $k_{x'}=k+\lambda=13+5=\mathbf{18\ N/m}$ and $k_{y'}=k-\lambda=13-5=\mathbf{8\ N/m}$ . Both positive, so the equilibrium is a stable minimum and the separated motions are librations.

Step 4. Frequencies from the separated one-dimensional oscillators: $\omega_{x'}=\sqrt{k_{x'}/m}=\sqrt{18/0.5}=\mathbf{6.0\ rad/s}$ and $\omega_{y'}=\sqrt{8/0.5}=\mathbf{4.0\ rad/s}$ .

Step 5. Read the lesson at exam speed: the same Hamiltonian failed one orthogonal chart and passed another related to it by a rigid rotation — separability certified the pairing (dynamics, chart), never the dynamics alone. As a bonus, $\omega_{x'}/\omega_{y'}=3/2$ is rational: the orbit is a closed Lissajous figure, a fact delivered by two square roots and no integration of the equations of motion.

The mathematicians behind the matrix

Origins · 1 minCarl Gustav Jacob Jacobi and how this was discovered

Carl Gustav Jacob Jacobi

1804–1851 · Königsberg and Berlin

Jacobi is the reason a separability theory exists to be axiomatized. Having recast Hamilton's formalism around one partial differential equation, he spent the late 1830s demonstrating that choosing coordinates is the entire game: his separation of geodesic flow on the triaxial ellipsoid — in elliptic coordinates invented for the purpose — solved a problem with no visible symmetry and stunned his contemporaries. His Königsberg lectures pushed the program of hunting separating charts for each new mechanical problem. What Jacobi practiced as inspired art, Paul Stäckel — a half-century later, in his 1891 Habilitation at Halle — closed as science: an algebraic criterion deciding, once and for all, which orthogonal charts can do what Jacobi's elliptic coordinates did. The matrix bearing Stäckel's name is the theory of Jacobi's luck.

History

1760

— Euler integrates motion under two fixed attracting centers, the first triumph of what will become elliptic-coordinate separation.

1839

— Jacobi separates geodesics on the triaxial ellipsoid in elliptic coordinates, turning coordinate choice into a research program.

1846

— Liouville identifies the general family now called Liouville systems, separable by construction — the first systematic separability class.

1891

— Stäckel's Habilitation gives the complete matrix criterion: necessary and sufficient conditions for orthogonal additive separation.

1904

— Levi-Civita derives the separability equations a Hamiltonian must satisfy chart by chart, extending the question beyond orthogonal coordinates.

1934

— Eisenhart recasts Stäckel systems in Riemannian geometry, tying separability to Killing tensors and opening the road to general relativity.

Going further

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §10.4–10.5 (Hamilton-Jacobi, separation of variables) · Landau & Lifshitz, Mechanics, 3rd ed., §48 and problems (separation of variables) · Morin, Introduction to Classical Mechanics, Hamilton-Jacobi chapter · Taylor, Classical Mechanics, Ch. 13 (Hamiltonian Mechanics) · Stäckel, P., Über die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variabeln (Habilitationsschrift, Halle, 1891) · Eisenhart, L. P., 'Separable Systems of Stäckel', Ann. Math. 35 (1934) 284 Three internal consistency checks underpin the presentation: (1) the contraction $\sum_i(\Phi^{-1})_{1i}\phi_{ij}=\delta_{1j}$ reproduces the HJ equation (Derivation A); (2) for $n=2$ in elliptic coordinates the two-fixed-center Stäckel matrix yields $\{F_1,F_2\}=0$ (Derivation B); (3) the parabolic-coordinate Kepler/Stark separation is consistent with the Landau-Lifshitz §48 problems. All statements are algebraic identities. Worked-example numbers on this page ( $E=1$ J, $\alpha_2=64$ (kg m²/s)², $p_r^2=4$ ; $k_{x'}=18$ N/m, $k_{y'}=8$ N/m, $\omega_{x'}=6.0$ rad/s, $\omega_{y'}=4.0$ rad/s) are illustrative arithmetic recomputed from the displayed formulas.

Builds on: Time-Independent Hamilton-Jacobi Equation · Separation of Variables in Hamilton-Jacobi Theory · Hamilton's Characteristic Function Leads to: Liouville Integrable Systems · Action-Angle Variables · Liouville-Arnold Theorem · Action Integrals over Invariant Tori

How do I actually find the Stäckel matrix for a given problem — guess it?

In practice you reverse-engineer it, exactly as Derivation C does. Separate the problem directly (or attempt to): if you reach single-coordinate momentum equations, write each as $p_i^2=2\sum_j\phi_{ij}(q_i)\alpha_j-2u_i(q_i)$ and read off the rows as the coefficients of the constants. The theorem's value is not that it finds matrices for you — it is the guarantee that the matrix you read off certifies full Liouville integrability (Derivation B), plus the converse: if no such matrix can exist (e.g. the potential fails the additivity test in a flat chart), no amount of cleverness within that chart will separate the problem. For systematic searches, the separable charts of common spaces are classified — in Euclidean 3-space you have exactly eleven families to try.

Is the Stäckel matrix for a given problem unique?

No, and nothing requires it to be. You can rescale rows (absorbing constants into redefined $\alpha_j$ ), take linear recombinations of the constants (any invertible constant matrix acting on the columns produces an equally valid $\Phi$ with recombined $\alpha$ 's), and different choices of which constants to feature — $L^2$ versus $L^2$ plus a multiple of $E$ , say — give different matrices for the same separation. What is invariant is the involutive span: the $n$ -dimensional space of commuting quadratic integrals the construction generates, and the foliation of phase space by their joint level sets (the invariant tori). The matrix is a coordinate presentation of that structure, not the structure itself.

Where exactly does the argument use orthogonality, and what breaks without it?

In two load-bearing places. First, the separated equations govern $p_i^2$ — one squared momentum per line. An off-diagonal kinetic term $g^{ij}p_ip_j$ straddles two lines and cannot be assigned to either single-variable block, so the ansatz jams at Step 1 of Derivation A. Second, the involution proof pairs each $\partial/\partial p_i$ with the same index $i$ in the diagonal metric; cross terms would generate brackets that no longer cancel pairwise. With cross terms present you need the Levi-Civita separability equations — a genuinely different (and messier) analysis, which also covers momentum-linear terms such as magnetic vector potentials, the modern treatment of which runs through Killing tensors in the Benenti and Kalnins-Miller sense. "Stäckel" without a qualifier always means the orthogonal, natural-Hamiltonian case.

The theorem certifies quadratic integrals. Does that mean all conserved quantities are quadratic in momenta?

No — it means separability can only see the quadratic ones. Stäckel's construction outputs $n$ integrals of the specific form $\sum_i(\Phi^{-1})_{ji}(\tfrac12p_i^2+u_i)$ ; if a system's independent integrals are unavoidably cubic or quartic in momenta (the Toda lattice is the standard example), no orthogonal chart will ever separate it, however integrable it is. The converse mistake also matters: some systems have more integrals than the $n$ quadratic ones — superintegrable systems like Kepler, where the Laplace-Runge-Lenz vector supplies extra constants. Multiple inequivalent separating charts (spherical and parabolic for Kepler) are the Stäckel-level fingerprint of exactly that surplus.

What is the relationship between Stäckel separability and Killing tensors in general relativity?

Eisenhart's dictionary: each Stäckel integral quadratic in momenta, $F_j=K_j^{ab}p_ap_b/2+\dots$ , corresponds to a symmetric tensor $K^{ab}$ satisfying the Killing-tensor equation $\nabla^{(a}K^{bc)}=0$ , and a Stäckel structure is a family of $n$ such tensors, commuting and simultaneously diagonalizable, adapted to an orthogonal chart. This translation is why the machinery survives in curved spacetime: the Kerr metric's hidden Killing tensor is what lets the geodesic Hamilton-Jacobi equation separate in Boyer-Lindquist coordinates, with the Carter constant as the corresponding quadratic integral. When relativists say Kerr is "algebraically special enough to be integrable," the honest mechanism underneath is a curved-spacetime Stäckel separation.

Condition (4) makes the potential look metric-dependent. Is the set of separable potentials really different for each chart?

Yes — and that is the theorem's sharpest practical content. Fix the chart and the weights $(\Phi^{-1})_{1i}$ are fixed; the separable potentials in that chart are exactly the span $\sum_i(\Phi^{-1})_{1i}u_i(q_i)$ over free single-variable functions $u_i$ . In Cartesian axes that span is the additive potentials $u_1(x)+u_2(y)+u_3(z)$ ; in spherical coordinates it is $a(r)+b(\theta)/r^2+c(\phi)/r^2\sin^2\theta$ ; in parabolic coordinates, $\big(u_\xi(\xi)+u_\eta(\eta)\big)/(\xi+\eta)$ — which contains the Coulomb potential plus a uniform field, the structural reason the Stark problem separates there and nowhere spherical. Learning to recognize these spans on sight is the working skill this theorem installs.

Do the Stäckel conditions survive quantization — does classical separability imply the Schrödinger equation separates?

Mostly, with a precise correction term. For a natural Hamiltonian in an orthogonal chart, multiplicative separation of the Schrödinger equation requires the classical Stäckel conditions plus the Robertson condition, a constraint on the metric ensuring the quantum-mechanical curvature term (of order $\hbar^2$ , involving the Laplacian's first-derivative pieces) also respects the separation. All eleven Euclidean charts satisfy Robertson's condition, which is why hydrogen separates quantum mechanically everywhere it does classically. On general curved manifolds the two notions can come apart. The systematic bridge — writing $\psi\sim e^{iS/\hbar}$ and watching orders of $\hbar$ — belongs to Connection to the Schrödinger Equation, later in this track.

Is checking the mixed second derivative of V a complete separability test in Cartesian coordinates?

For the potential condition in a fixed Cartesian frame, essentially yes: flat constant weights force $V=\sum_iu_i(q_i)$ , and $\partial^2V/\partial q_i\partial q_j=0$ for all $i\neq j$ is exactly additivity (on a simply connected domain). But it tests only that frame. Worked Example 2 is the cautionary tale: $V=\tfrac12k(x^2+y^2)+\lambda xy$ fails the test in $(x,y)$ yet passes after a $45°$ rotation — the mixed-derivative matrix $\partial^2V/\partial q_i\partial q_j$ merely needed diagonalizing. The efficient Cartesian workflow is therefore: compute the Hessian's off-diagonal structure; if it is a constant symmetric matrix, rotate to principal axes and re-test; only a coordinate-dependent coupling that no rigid rotation removes pushes you out of Cartesian charts altogether, toward the curvilinear families.