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A free label: a nonempty regular common level set of a completely integrable datum that is noncompact and has no period lattice

Physical scope

A counterexample in the autonomous zero-force one-label presentation on R6: the smooth three-integral datum F=(H,p1,p2) on p3>0 has connected regular fibres diffeomorphic to R3, hence noncompact. Its commuting R3 action has trivial stabilizer, so no full period lattice exists, and the physical Hamiltonian flow has no nonzero period. This is declared model data, with no measured value or apparatus claim.

Statement refuted

In a completely integrable datum of a presentation of the framework's force branch, every nonempty regular common level set of the integral map is compact, and along it the datum's Hamiltonian flow is periodic with a full lattice of periods.

Facts & Assumptions

Given: The one-label model of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance with the zero force map, presented by the potential $U \equiv 0$ and $d \equiv 0$ (Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, The action functional of a force model with a potential presentation: the mechanical Lagrangian, its chart instance and its dimensions): configuration domain $Q = R^{3}$ , time window $R$ , mass $m > 0$ and no interactions. Its presentation is hyperregular with the Cartesian phase domain $W = R^{6}$ and branch Hamiltonian $H (x, p) = ∣ p ∣^{2} / (2 m)$ (Every potential presentation is hyperregular in its charts, with inverse branch $v = M^{- 1} p$ (or $G (q)^{- 1} p$ ) and the explicit kinetic-plus-potential Hamiltonian, The Hamiltonian of a hyperregular potential presentation: the Legendre transform of the mechanical Lagrangian, its domain and its dimensions, The momentum of a potential presentation: the fibre derivative of the mechanical Lagrangian, the Cartesian momentum Mv and generalized momenta, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L1]

Datum and fields (Completely integrable data of a declared presentation: $n$ smooth phase functions with $F_{1} = H$ , pairwise in involution and independent on the regular locus, First integrals of the model's Hamiltonian and families in involution: the bracket criterion and differential independence, The Hamiltonian field of a phase function for a potential presentation: the unique field $X_{f}$ with $ι_{X_{f}} ω = d f$ , and $X_{H} = (H_{p}, - H_{x})$ for the branch Hamiltonian): on the open phase domain $W_{0} = {(x, p) \in R^{6} : p_{3} > 0}$ take $F = (H, p_{1}, p_{2})$ . All three functions are smooth and depend only on $p$ , so their pairwise brackets vanish. Their momentum derivatives are the rows $p / m$ , $(1, 0, 0)$ and $(0, 1, 0)$ , with determinant $p_{3} / m > 0$ . Thus the regular locus is all of $W_{0}$ . The fields are $X_{H} = (p / m, 0)$ , $X_{p_{1}} = (e_{1}, 0)$ and $X_{p_{2}} = (e_{2}, 0)$ .

[L2]

Regular fibres: for $c_{1}, c_{2} \in R$ and $E > (c_{1}^{2} + c_{2}^{2}) / (2 m)$ set $ρ = \sqrt{2 m E - c_{1}^{2} - c_{2}^{2}} > 0$ and $p_{*} = (c_{1}, c_{2}, ρ)$ . The common fibre of $F$ at $(E, c_{1}, c_{2})$ is exactly $N = R^{3} \times {p_{*}}$ . It is nonempty, connected and regular at every point, and the projection to $x$ is a smooth diffeomorphism to $R^{3}$ .

[L3]

Compactness and continuity (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Continuity of $f : A \to R$ at a point of $A$ and on $A$ : the $ε$ - $δ$ condition, its agreement with $\lim_{x \to c} f (x) = f (c)$ at a limit point, and continuity at an isolated point, The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement): a continuous real function on a compact metric space attains its maximum; the coordinate function $(x, p) \mapsto x_{1}$ on $W$ is continuous.

[L4]

The torus and lattice conclusion (Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice clauses 2 and 3): compactness of the fibre is a hypothesis there for the completeness of the restricted fields, for the full period lattice and for the torus identification; the item asserts no conclusion for a noncompact level set.

Counterexample

1.1L1L2given

By [L1], $F = (H, p_{1}, p_{2})$ is a completely integrable datum with $n = 3$ on the six-dimensional domain $W_{0}$ . By [L2], $N = F^{- 1} (E, c_{1}, c_{2})$ is a nonempty connected regular common fibre.

1.2L1L2algebra

Use fixed SI numerical coordinates for the physical quantities, as in Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. Give the flow parameters the dimensions $[a] = τ$ and $[b] = [c] = ℓ$ : $a$ is the Hamiltonian time parameter, while $b, c$ are position-translation parameters for the momentum integrals. With these units each summand in the displacement has length dimension. On $N$ the commuting fields have the globally defined joint action $Ψ^{(a, b, c)} (x, p_{*}) = (x + a p_{*} / m + b e_{1} + c e_{2}, p_{*})$ . In these fixed numerical units the translation matrix has determinant $ρ / m > 0$ , so it is transitive and its stabilizer is ${0}$ at every point. There is therefore no full rank-three period lattice. In particular the physical Hamiltonian flow $Φ^{t} (x, p_{*}) = (x + t p_{*} / m, p_{*})$ has no nonzero period since $p_{*} \neq 0$ .

2.1L3step 1.1

The continuous coordinate $x_{1}$ takes every real value on $N$ , at points $((c, 0, 0), p_{*})$ . If $N$ were compact, [L3] would make this coordinate bounded. Thus $N$ is noncompact.

3.1L4step 1.1step 2.1step 1.2∎

The compactness hypothesis in [L4] fails, while connectedness and regularity hold. Thus the qualified torus and action-angle conclusions remain consistent, and the refuted conclusion without compactness fails inside the declared free model.

Remarks

What is refuted. Only the version of the fibre claims without the compactness hypothesis. The qualified theorem Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice keeps that hypothesis, and its conclusion that compact connected regular fibres are tori remains untouched.
The model reading. The free presentation is a legitimate model of the force branch with zero force; the counterexample is a statement about that declared model and carries no measured value and no claim about any apparatus.

A free label: a nonempty regular common level set of a completely integrable datum that is noncompact and has no period lattice

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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