= Solution
For a regular <Lagrangian> define the <canonical momentum> components $p_j=\partial L/\partial\dot q_j$. Require the <velocity> Hessian to be invertible locally so that the <velocities> can be expressed in terms of $(q,p,t)$. The <Legendre transform in mechanics> defines
$$
\boxed{H(q,p,t)=\sum_jp_j\dot q_j-L(q,\dot q,t).}
$$
The explicit <time> argument cannot generally be omitted when the given <Lagrangian> depends on <time>. Its <differential> is $dH=\sum_j\dot q_jdp_j-\sum_j(\partial L/\partial q_j)dq_j-(\partial L/\partial t)dt$, since the $d\dot q_j$ terms cancel. The <Euler-Lagrange equations> give $\partial L/\partial q_j=\dot p_j$, and comparison yields <Hamilton's equations>:
$$
\boxed{\dot q_j=\frac{\partial H}{\partial p_j},\qquad\dot p_j=-\frac{\partial H}{\partial q_j},\qquad\partial_tH=-\partial_tL.}
$$
For the time-independent stellar <gravitational potential> below, $H$ is conserved. Its sign convention differs from the binding <energy> in question 2: here $E=H=v^2/2-\psi$ is the ordinary specific orbital <energy>.
The prescribed nondegenerate <prolate spheroidal coordinates> require $0<\alpha<\beta$; equality collapses the coordinate family. Their physical <velocity> components are $v_j=h_j\dot q_j$, and $h_\phi=R$. Per unit stellar <mass> the <Lagrangian> is $L=\tfrac12\sum_jh_j^2\dot q_j^2+\psi$, so $p_j=h_j^2\dot q_j=h_jv_j$. Its <Hamiltonian> is
$$
\boxed{H=\frac12\left(\frac{p_\lambda^2}{h_\lambda^2}+\frac{p_\mu^2}{h_\mu^2}+\frac{p_\phi^2}{h_\phi^2}\right)-\psi.}
$$
This uses <acceleration> $+\nabla\psi$, as specified, rather than importing the opposite <gravitational potential> convention.
For the <Stäckel potential>, use the <Hamilton-Jacobi equation> with principal function $S=W_\lambda(\lambda)+W_\mu(\mu)+l\phi-Et$. Axisymmetry makes $p_\phi=l=Rv_\phi$ constant. Write $d=\lambda-\mu$ and $A(\tau)=(\tau+\alpha)(\tau+\beta)$. Multiplying $H=E$ by $d$ and using
$$
\frac{d}{(\lambda+\alpha)(\mu+\alpha)}=\frac1{\mu+\alpha}-\frac1{\lambda+\alpha}
$$
separates it as $F(\lambda)-E\lambda=F(\mu)-E\mu$, where $F(\tau)=2A(\tau)(W_\tau')^2-(\alpha-\beta)l^2/[2(\tau+\alpha)]-G(\tau)$. Each side depends on its own coordinate only and therefore equals a constant, chosen as $-I$. Here $G(\tau)$ is the generating function of the <gravitational potential>, not the gravitational constant. Hence
$$
2(\tau+\alpha)(\tau+\beta)p_\tau^2-\frac{(\alpha-\beta)l^2}{2(\tau+\alpha)}-G(\tau)-E\tau=-I,\qquad\tau=\lambda,\mu.
$$
This proves the conserved <axisymmetric Stäckel third integral>. Eliminating $E$ between the two expressions and using $p_j=h_jv_j$ gives
$$
\boxed{I=\frac12\left[\mu v_\lambda^2+\lambda v_\mu^2+(\lambda+\mu+\alpha)v_\phi^2\right]-\frac{\mu G(\lambda)-\lambda G(\mu)}{\lambda-\mu}.}
$$
At regular generic phase-space points the three separation constants $E,l,I$ are independent; special circular or degenerate <orbits> can reduce their local independence. Thus the model is an <integrable Hamiltonian system>.
For the intended regular, phase-mixed steady model, <Jeans theorem> permits a <galactic distribution function> $f=F(E,l,I)$. At any fixed point, $E$ and $I$ depend on $v_\lambda$ and $v_\mu$ only through their squares, and $l$ depends only on $v_\phi$. Thus $f$ is even separately in $v_\lambda$ and $v_\mu$, whether or not it is even in $v_\phi$. Odd <integration> proves <velocity-ellipsoid alignment in a Stäckel potential>:
$$
\boxed{\langle v_\lambda v_\mu\rangle=\langle v_\lambda v_\phi\rangle=\langle v_\mu v_\phi\rangle=0.}
$$
The argument allows azimuthal streaming: it does not require zero mean $v_\phi$. The means in the two meridional directions vanish, so the off-diagonal <velocity> <covariance> also vanishes. <Second moments> must exist for this conclusion.
There is a hypothesis behind the printed universal wording. On a generic nonresonant regular <invariant torus>, a continuous steady distribution is constant along a dense <orbit> and hence depends only on the torus <integrals>; this justifies the phase-mixed form. Bare stationarity alone is not sufficient in a degenerate resonant <gravitational potential> with additional <integrals>. For example $H=\tfrac12(v^2+\omega^2r^2)$ is a Stäckel <Hamiltonian>: choose $G(\tau)=-\tfrac12\omega^2[\tau^2+(\alpha+\beta)\tau]$, up to a constant vacuum shift. The additional conserved quantity $K_{xy}=v_xv_y+\omega^2xy$ gives a positive normalizable steady tracer $f\propto\exp(-aH+\epsilon K_{xy})$ for $a>|\epsilon|>0$. Its local $\langle v_xv_y\rangle=\epsilon/(a^2-\epsilon^2)\ne0$. At azimuth zero and a regular point where the $\lambda$ direction has a radial component, this gives nonzero $\langle v_\lambda v_\phi\rangle$. This tracer need not itself source the harmonic <gravitational potential>. It exhibits why the stronger, ordinary phase-mixed three-integral assumption is necessary instead of claiming that every imaginable steady distribution has the asserted alignment.
The geometry follows by multiplying the coordinate-defining equation and evaluating at $\tau=-\alpha,-\beta$:
$$
R^2=\frac{(\lambda+\alpha)(\mu+\alpha)}{\alpha-\beta},\qquad z^2=\frac{(\lambda+\beta)(\mu+\beta)}{\beta-\alpha}.
$$
Constant $\lambda$ surfaces are <prolate spheroids>, with equatorial semiaxis $\sqrt{\lambda+\alpha}$ and polar semiaxis $\sqrt{\lambda+\beta}$. Constant interior $\mu$ surfaces are two-sheeted hyperboloids. Their common foci lie at $z=\pm\sqrt{\beta-\alpha}$. The <orbit volume in an axisymmetric Stäckel potential> is determined by
$$
p_\tau^2=\frac{E\tau+G(\tau)-I+(\alpha-\beta)l^2/[2(\tau+\alpha)]}{2(\tau+\alpha)(\tau+\beta)}\geq0.
$$
Its allowed coordinate intervals have turning boundaries or coordinate endpoints. A typical bound <orbit> with $l\ne0$ oscillates between inner and outer spheroids, oscillates vertically between hyperboloidal limits, and circulates about the symmetry axis with $\dot\phi=l/R^2$. The equatorial fold $\mu=-\beta$ joins the positive and negative $z$ branches; it is not necessarily a physical turning point in $z$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-320-orbit-volume.png]
{title=Geometric volume swept out by an orbit}
{description=Original schematic coordinate-bounded <orbit> <volume>. The meridional section and cutaway azimuthal sweep show inner/outer confocal spheroids and upper/lower hyperboloidal boundaries. Illustrative intervals are $1\leq\lambda\leq4$, $-4\leq\mu\leq-2$ with $\alpha=1$, $\beta=4$. These are geometric examples, not a numerically integrated <orbit> in a specified <gravitational potential>.}
For three incommensurate frequencies, the regular trajectory densely samples its projected allowed <volume>, with nonuniform occupation determined by its speeds. It does not follow one coordinate surface, and it need not close. Resonant <orbits> may be closed or lie in a smaller projected set; $l=0$ confines azimuthal motion to a meridional <plane> (with the usual axis coordinate qualification), and circular or equatorial limits are also lower-dimensional. Smooth Stäckel separation removes the generic chaotic behavior of nonintegrable potentials; it does not assert that every <orbit> fills a three-dimensional <volume>.
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