Scope of the symplectic and canonical methods for a potential presentation: locality of generating functions, projections and determinant tests, absent global claims, and empirical status

Locality in time. Flow preservation, bracket preservation and volume preservation are asserted on the actual time-slice domains supplied by the dependence theorem, not on the whole phase domain (The Hamiltonian evolution of a potential presentation preserves the two-form, the bracket and the determinant on its actual time-slice domains, also when the Hamiltonian depends on time, Liouville's theorem for a potential presentation: the phase-volume form ${\omega }^{3N}\mathrm{/}(3N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains). No completeness of the model's Hamiltonian flow is claimed, and the maximal existence of the model's motions remains the business of the model's own initial-value theory (Hamilton's equations of the mechanical presentation are equivalent to the model's equation of motion, with residual forces as momentum sources and the coordinate form on every chart).

Determinant tests. For a $C^1$ map between equal-dimensional phase domains, canonicity is equivalent to the square matrix identity $D{C}^{\mathsf{T}}JDC=J$ in canonical coordinates, and then $\mathrm{\mid }\det DC\mathrm{\mid }=1$ (Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests, A $C^1$ map between phase domains preserves the standard two-form iff $D{C}^{\mathsf{T}}JDC=J$, and a canonical diffeomorphism carries $X_{H\circ C}$ to $X_H$). In dimension $2m\ge 4$ with $m$ the chart half-dimension, that determinant condition is necessary but not sufficient: the companion page records a volume-preserving shear of a four-dimensional phase domain that fails the fundamental bracket $\{Q^2,P_1\}=0$; for a single degree of freedom the symplectic test is exactly $\det DC=1$; absolute determinant $1$ also permits orientation-reversing, antisymplectic maps. Between different dimensions the determinant is not defined; use the rectangular condition $D{C}^{\mathsf{T}}{J}_{m_2}DC=J_{m_1}$ of Canonical maps of a potential presentation: pullback-preserving immersions and the equal-dimensional determinant and field-transport tests.

Projections and caustics. A first-kind generating function exists locally only when the $(q,Q)$-projection of the canonical map is invertible, and the second kind requires the corresponding $(q,P)$ projection (Generating functions of the first and second kind for a local canonical map, A canonical map with invertible $(q,Q)$-projection is locally generated by a first-kind generating function). At points where the required projection degenerates, the whole $J\times O_q\times O_Q$ representation of Generating functions for a potential presentation: first- and second-kind data generate canonical maps locally, the local converse under an invertible projection, and the time-dependent Hamiltonian $K=H+{\mathrm{\partial }}_{t}S$ breaks down even though the map remains canonical; the companion page carries a point transformation whose $(q,Q)$-projection is degenerate identically.

Symplectic map versus globally exact generating function. Canonicity is a local condition on $\omega$; it does not imply the existence of a globally defined real generating function. Under the Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) retained by the cited torus construction, the model-free witness is the cylinder $C=(\mathbb{R}\mathrm{/}\mathbb{Z})\times \mathbb{R}$ and its shift $\mathrm{\Phi }([\theta ],I)=([\theta ],I+a)$ for a fixed real $a\mathrm{\ne }0$. With the descended forms ${\lambda }_C=Id{\theta }_C$ and ${\omega }_C=d{\theta }_C\wedge dI$ defined in lift charts, it preserves ${\omega }_C$ and changes ${\lambda }_C$ by the closed, non-exact form $ad{\theta }_C$, so its local generating functions do not glue to a global one (The symplectic shift of the cylinder changes $\lambda$ by the closed, non-exact form $ad\theta$ and has no global generating function, Scope of the canonical-methods items: locality and topology of generating functions, caustics, and the E7 claims delivered by published root items). The presentation's Cartesian phase domains are open subsets of ${\mathbb{R}}^{6N}$; a generalized-coordinate phase domain $\mathrm{\Omega }\times {\mathbb{R}}^r$ is open in ${\mathbb{R}}^{2r}$, including $r<3N$. These domains need not be simply connected, so no global generating-function existence is claimed here; conversely a momentum shift in Euclidean coordinates has a global second-kind generating function and a degenerate first-kind projection.

Choice assumptions. The explicit phase-chart form calculations, canonical matrix tests and local generating-function calculations use no choice principle. The cylinder obstruction above retains ${\mathrm{A}\mathrm{C}}_{\omega }$ from The symplectic shift of the cylinder changes $\lambda$ by the closed, non-exact form $ad\theta$ and has no global generating function's torus supplier; no choice-free version of that supplier is asserted here. The measure-theoretic volume statement also assumes Countable Choice where the change-of-variables theorem requires it (Liouville's theorem for a potential presentation: the phase-volume form ${\omega }^{3N}\mathrm{/}(3N)!$ and the Lebesgue measure in each canonical chart are preserved by the Hamiltonian evolution on its actual domains, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A separate direct integration argument for an explicit bounded region can avoid that change-of-variables use only when its own mathematical suppliers require no choice assumption.

Numerics. The drift, kick and velocity-Verlet maps of the mechanical splitting are local canonical transformations in exact arithmetic on nonempty admissible domains (The drift, kick and velocity-Verlet maps are canonical: $(DC{)}^{\mathsf{T}}JDC=J$ and $C^{\ast }\omega =\omega$). Restricting the target to each map's open image gives a canonical transformation onto that image; no surjectivity onto the entire declared target phase domain is asserted. The scope of the certified numerical estimates, including roundoff and the absence of all-time energy claims, is recorded in Scope of the certified velocity-Verlet estimates: exact arithmetic, compact tubes, roundoff, and absent all-time claims and is not extended by this page.

Empirical status. Every item of this page is a conditional consequence of the adopted force branch, the declared potential presentation, the declared chart and the cited mathematics. No experimental result, measured constant or statistical claim is used, and none is implied; in particular no statement here identifies the model's phase volume with an observable quantity. Units are those of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework throughout, and the generalized-coordinate charts remain those of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces.