Scope of the Hamilton-Jacobi and integrability statements: locality, the fibre hypotheses, global obstructions and the frequency conventions

All Hamilton-Jacobi statements of this page are local by construction. The Hamilton-Jacobi equation of a declared potential presentation: the class of action-dimension functions $S$ with $S_t+H(t,q,S_q)=0$ in one momentum chart fixes the equation attached to one declared momentum chart of one potential presentation, with the branch Hamiltonian of The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions; changing the chart or the potential representative changes the equation by the recorded rules; the inverse Legendre map is unique on each fixed hyperregular chart. Thus the equation is part of the datum and not an intrinsic object of the model. A complete integral of the model's Hamilton-Jacobi equation: nondegenerate mixed derivative, momentum map and Jacobi constants requires the nondegeneracy $\det S_{q\alpha }\mathrm{\ne }0$ at the base point only, and A complete integral of the model's Hamilton-Jacobi equation produces chart Hamiltonian characteristics, Jacobi constants and local canonical $(\beta ,\alpha )$ labels concludes only what that nondegeneracy and the local theory give: characteristic curves near the base point, the constancy of the Jacobi constants along them, a local converse and local canonical labels $(\beta ,\alpha )$. No solution is extended across a conjugate time or a caustic; the nondegeneracy hypothesis alone supplies no conclusion at excluded points. The companion harmonic example exhibits refocusing at conjugate times, where the mixed derivative is unbounded and the label-to-position derivative vanishes. Its Kepler example separately concerns simple turning points of a radial action integral. In particular the page claims no global Hamilton-Jacobi solution, no uniqueness of the family and no global existence of the endpoint action.

First integrals of the model's Hamiltonian and families in involution: the bracket criterion and differential independence and Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_1=H$, pairwise in involution and independent on the regular locus define the model properties and the datum. Two features are essential and logically separate: the members of the datum must be in involution, and their differentials must be independent on the regular locus; neither condition implies the other. The striking conclusions are conditional on strictly more than involution and independence: Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice requires a regular value with nonempty compact connected fibre and a saturated neighbourhood on which the integral map is a proper submersion with connected fibres, and it consumes the Axiom of Countable Choice through the published lattice lemma, as its statement records; Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H=h(I)$ and linear motion on the tori additionally produces the period-one action-angle coordinates of Action and angle coordinates and the linear Hamiltonian motion on the tori, under the same hypotheses; identification with Cartesian model motions additionally uses $d\equiv 0$; for a generalized-coordinate immersion the Hamiltonian curves correspond to the substituted Lagrange equations, and identifying them with ambient motions requires the additional reconstruction hypotheses. Compactness cannot be dropped (a free-particle regular level is a noncompact manifold, not a torus), regularity cannot be dropped (the zero level of the isotropic oscillator is a point), and the conclusions are local over a ball of regular values — no global product structure or single-valued global action-angle chart is asserted.

Global action-angle coordinates can genuinely fail even for completely integrable Hamiltonian data on a connected regular region with compact connected fibres. The two published mathematical suppliers have distinct roles. Period-lattice monodromy obstructs global action–angle coordinates gives the conditional obstruction: transporting an ordered integer basis of the period lattice around a base loop yields an invertible integer matrix, and a nonidentity matrix rules out a global action-angle system. That implication alone does not exhibit nontrivial transport.

The witness is the regular torus bundle in the refutation of Liouville–Arnold gives global action–angle coordinates on the entire manifold. Its spherical-pendulum datum, in the source's dimensionless mathematical normalization, is the smooth map $F=(H,J):T^{\ast }{S}^2\to {\mathbb{R}}^2$, where $T^{\ast }{S}^2=\{(q,p)\in {\mathbb{R}}^3\times {\mathbb{R}}^3:\mathrm{\mid }q{\mathrm{\mid }}^2=1,q\cdot p=0\}$ (identifying cotangent vectors using the Euclidean inner product), with the restriction of $dx\wedge d{p}_x+dy\wedge d{p}_y+dz\wedge d{p}_z$, $q=(x,y,z)$, $H(q,p)=\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}2+z$ and $J(q,p)=x{p}_y-y{p}_x$. Over a sufficiently small punctured disk $B$ about the excluded value $(H,J)=(1,0)$, $F^{-1}(B)$ is connected and all fibres are regular compact connected two-tori. Section 3.1 of Martynchuk--Broer--Efstathiou computes transport around the puncture as $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ in its chosen cycle basis; the published refutation uses this nonidentity transport and the obstruction to exclude global action-angle coordinates there. This records a mathematical global counterexample, without identifying its whole cotangent bundle with one momentum chart of this page's force presentation or asserting an empirical pendulum observation. The local-over-a-ball conclusions remain intact.

The angles of Action and angle coordinates and of Action-angle coordinates of a completely integrable presentation: period-one angles, action coordinates, $H=h(I)$ and linear motion on the tori are period-one, with $\omega ={\sum }_{i}d{\theta }_i\wedge d{I}_i$ and action coordinates of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-1}$; the numbers $\mathrm{\partial }h\mathrm{/}\mathrm{\partial }{I}_i$ are then frequencies in cycles per unit time, not angular rates. The radian convention is recovered by the explicit rescaling ${\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}=2\pi \theta$, $I^{\mathrm{r}\mathrm{a}\mathrm{d}}=I\mathrm{/}(2\pi )$ of Period-one to radian action-angle rescaling: ${\theta }^{\mathrm{r}\mathrm{a}\mathrm{d}}=2\pi \theta$ and $I^{\mathrm{r}\mathrm{a}\mathrm{d}}=I\mathrm{/}(2\pi )$ preserve $\omega$, under which the form is unchanged and the angular rates are twice $\pi$ times the cyclic frequencies; the two conventions are never mixed in this page. The sources use the radian action $J=\frac{1}{2\pi }\oint pdq$; the period-one action of a mechanical one-degree-of-freedom model is its $2\pi$-multiple, a convention recorded in the companion examples.

Adiabatic invariance of the action of a slowly varying one-frequency model: an order-$\epsilon$ drift bound on the $O(1\mathrm{/}\epsilon )$ time scale is conditional on a uniform fixed canonical chart for the parameter-dependent Hamiltonian, with $C^2$ data on open extension domains and $\mathrm{\mid }{\mathrm{\partial }}_{I}h\mathrm{\mid }\ge {\omega }_0$ and on the trapping of the solution in the compact chart region; it gives an order-$\epsilon$ drift bound on the $O(1\mathrm{/}\epsilon )$ horizon and no exact invariance. Parameter paths that leave the chart, cross a separatrix or make the frequency vanish are outside the statement, as are the averaged statements of the sources, which the in-run mathematics A one-frequency parameter-dependent action-angle branch, its transformed Hamiltonian and its drift generator and the drift theorem replace by an exact chain-rule estimate, with the source limitations Scope of the action-angle and complete-integrability items: hypotheses, choice, singular and noncompact fibres, and the published proof gap recorded.

The definitions stipulate datum classes and conventions relative to the adopted presentation. The physics theorems establish conditional deductions from the adopted postulates, declared hypotheses and cited mathematics; remarks record their scope. None of these items reports an empirical observation. No item reports a measurement, attaches a measured value to a model parameter or compares a prediction with data. Where the sources motivate the statements by physical examples (oscillators, Kepler motion, adiabatic transport), the page treats those examples as model instances, and any quantitative use of a parameter value would be an empirical premise with its own conditions and uncertainty, to be recorded when such a use is made.