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The two-rotor model: a positive-measure family of invariant Lagrangian tori with Diophantine frequencies for sufficiently small coupling, with the explicit units and the stated scope

Physical scope

The conditional KAM statement for the adopted two-rotor model: two ideal fixed-axis rotors of equal positive moment of inertia with the smooth interaction potential E0 epsilon cos theta1 cos theta2. For every Diophantine exponent tau>1 there is a constant c>0 determined by the declared dimensionless data such that for 0<alpha<c and 0<epsilon<=c alpha^2 the set of frequencies satisfying the Diophantine bound has positive Lebesgue measure in the declared frequency cube, and for every such frequency the model has a C1 invariant Lagrangian embedding whose image carries quasi-periodic motion with angular frequencies omega0 nu_i. The item asserts nothing about the complement of that frequency set (resonant or non-Diophantine labels), nothing about all initial data, no chaos or diffusion statement, no measured value of any parameter and no empirical realization; epsilon<=c alpha^2 is a model-data smallness condition.

Statement

Assume the Axiom of Countable Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ )), inherited from the analytic KAM suppliers. Adopt the two-rotor model: two ideal rotors with fixed axes, equal moments of inertia $I > 0$ , angular coordinates $θ_{1}, θ_{2}$ in $R / 2 π Z$ (radians, dimensionless), conjugate angular momenta $Π_{1}, Π_{2}$ , the physical Hamiltonian $H_{p h y s} (θ, Π) = \frac{Π_{1}^{2}}{2 I} + \frac{Π_{2}^{2}}{2 I} + E_{0} ε \cos θ_{1} \cos θ_{2}$ on the phase domain $(R / 2 π Z)^{2} \times R^{2}$ , with the canonical two-form $d θ_{1} \land d Π_{1} + d θ_{2} \land d Π_{2}$ and Hamilton equations $θ^{'} = \partial_{Π} H_{p h y s}$ , $Π^{'} = - \partial_{θ} H_{p h y s}$ ; here $E_{0} > 0$ is an energy scale and $ε \geq 0$ a dimensionless coupling. All quantities are model data; $[H_{p h y s}] = μ ℓ^{2} τ^{- 2}$ , $[Π_{i}] = μ ℓ^{2} τ^{- 1}$ , $[I] = μ ℓ^{2}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Put $J_{0} : = \sqrt{I E_{0}}, t_{0} : = \frac{J_{0}}{E_{0}} = \sqrt{\frac{I}{E_{0}}}, ω_{0} : = \frac{1}{t_{0}} = \sqrt{\frac{E_{0}}{I}},$ and read the dimensionless model through $I_{i} : = Π_{i} / J_{0}$ and the dimensionless time $s : = t / t_{0}$ , $H (I, ϑ) = \frac{1}{2} (I_{1}^{2} + I_{2}^{2}) + ε \cos ϑ_{1} \cos ϑ_{2} .$ Let $C^{*} : = {ν \in R^{2} : ∣ ν ∣_{\infty} \leq \frac{1}{16}}$ be the declared frequency cube and, for $τ > 1$ and $α > 0$ , let $Ω_{α, τ} : = {ν \in C^{*} : ∣ k \cdot ν ∣ \geq α ∣ k ∣_{1}^{- τ} for all k \in Z^{2} ∖ {0}}$ be its $(α, τ)$ -Diophantine subset (Analytic KAM data: angular tori, action-angle neighbourhoods, Fourier cutoffs and Diophantine frequency labels). Then for every $τ > 1$ there is a constant $c > 0$ , depending only on the fixed dimensionless analytic data (the action cube, the strip width $s_{0}$ , the exponent $τ$ , $M_{2} = 2$ , and the normalized perturbation profile $g (ϑ) = \cos ϑ_{1} \cos ϑ_{2}$ with strip bound $∣ g ∣ \leq \cosh^{2} (s_{0})$ ), and uniform in $α$ and the amplitude $ε$ , such that for every $0 < α < c$ and every $ε > 0$ with $ε \leq c α^{2}$ :

(Positive measure of frequencies.) The Diophantine set has positive Lebesgue measure, with $∣ Ω_{α, τ} ∣ \geq ∣ C^{*} ∣ - C_{2, τ} (diam C^{*}) α = \frac{1}{64} - \frac{C_{2, τ}}{8} α,$ where $C_{2, τ}$ is the lattice constant of Small-divisor slabs and the positive measure of the Diophantine set: the resonance covering has measure $O (α)$ .
(Invariant Lagrangian tori.) For every $ν \in Ω_{α, τ}$ there is a $C^{1}$ embedding ${\hat{Ψ}}_{ν} : T^{2} \to (R^{2}) \times T^{2}$ (a $C^{1}$ injective immersion that is a homeomorphism onto its image), and it is Lagrangian ( ${\hat{Ψ}}_{ν}^{*} ω = 0$ for the two-form $ω$ of the dimensionless model), and satisfies $X_{H} ({\hat{Ψ}}_{ν} (ϑ)) = \sum_{j = 1}^{2} ν_{j} \partial_{ϑ_{j}} {\hat{Ψ}}_{ν} (ϑ);$ consequently a torus parameter phase follows $φ (s) = φ_{0} + ν s$ , and its dimensionless phase-space trajectory is ${\hat{Ψ}}_{ν} (φ_{0} + ν s)$ . At physical time $t$ it is the scaled image of ${\hat{Ψ}}_{ν} (φ_{0} + ω_{0} ν t)$ : the physical angle is the angular component of this embedding, not generally the linear parameter phase. Thus the torus carries quasi-periodic motion with fundamental angular frequencies $ω_{0} ν_{1}, ω_{0} ν_{2}$ . The parametrisation is close to the unperturbed one and Lipschitz in $ν$ with the bounds of Positive-measure families of invariant Lagrangian tori for nondegenerate real analytic Hamiltonians: persistence for $ε \leq c α^{2}$ clause 3.
(Scope.) Nothing is asserted about the complement of $Ω_{α, τ}$ (resonant or non-Diophantine labels), about tori surviving outside the smallness threshold, about all initial data, about diffusive or chaotic behaviour of the complement, or about any laboratory rotor: the smallness condition $ε \leq c α^{2}$ is a model-data condition, the constant $c$ is not computed numerically here, and no measured value of $I$ , $E_{0}$ or $ε$ is used or implied.

Facts & Assumptions

Given: the two-rotor model data $I, E_{0}, ε$ , the scaling constants $J_{0}, t_{0}, ω_{0}$ , the dimensionless Hamiltonian $H (I, ϑ) = \frac{1}{2} (I_{1}^{2} + I_{2}^{2}) + ε \cos ϑ_{1} \cos ϑ_{2}$ and the frequency cube $C^{*}$ of the Statement.

[L1]

The analytic KAM theorem (Positive-measure families of invariant Lagrangian tori for nondegenerate real analytic Hamiltonians: persistence for $ε \leq c α^{2}$ , whose notation and hypotheses are those of Analytic KAM data: angular tori, action-angle neighbourhoods, Fourier cutoffs and Diophantine frequency labels): for $n \geq 2$ , $τ > n - 1$ , data $h, U, K, r_{0}, Ω, b, M_{2}, F, ε, s_{0}, σ_{0}$ and a compact cube $C^{*}$ satisfying the listed analyticity, nondegeneracy ( $D h (b (ν)) = ν$ ), boundedness ( $M_{2}$ ) and smallness conditions, there is $c > 0$ such that for $0 < α < c$ and $0 < ε \leq c α^{2}$ the set $Ω_{α, τ}$ has the stated positive measure with the lattice constant of Small-divisor slabs and the positive measure of the Diophantine set: the resonance covering has measure $O (α)$ , and for every $ν \in Ω_{α, τ}$ there is an invariant Lagrangian torus with $X_{H} ({\hat{Ψ}}_{ν} (ϑ)) = \sum_{j} ν_{j} \partial_{ϑ_{j}} {\hat{Ψ}}_{ν} (ϑ)$ and the closeness and Lipschitz bounds of clause 3.

[L2]

Diophantine resonance slabs (Small-divisor slabs and the positive measure of the Diophantine set: the resonance covering has measure $O (α)$ ): the resonance slabs inside a bounded frequency cube have total measure at most $C_{n, τ} (diam C^{*})^{n - 1} α$ for the stated lattice constant, so the Diophantine set has the complementary lower bound.

[L3]

Angular regularity of the limiting embedding (Convergence of the composed KAM transformations, with derivative and frequency-Lipschitz bounds): the composed KAM maps converge uniformly together with their first angular derivatives; for each fixed frequency label, the limit is holomorphic in the angular variables on a complex strip. Its real torus map is therefore $C^{1}$ (indeed real analytic). The derivative bounds and smallness $2 A E_{0} \leq 1 / 2$ in the two-dimensional application of Positive-measure families of invariant Lagrangian tori for nondegenerate real analytic Hamiltonians: persistence for $ε \leq c α^{2}$ give $∥ D_{ϑ} w ∥ < 1$ for the periodic angular correction in ${\hat{Ψ}}_{ν} (ϑ) = (b (ϑ, ν), ϑ + w (ϑ, ν))$ ; hence its angular component has invertible derivative and the full map is an immersion. The pullback of the canonical two-form is defined.

Proof

technique · direct

1.1givenL1algebra

Verification of the analytic data for [L1], part 1: take $n : = 2$ , the open set $U : = {J \in C^{2} : {dist}_{\infty} (J, K) < \frac{1}{4}}$ with $K : = {J \in R^{2} : ∣ J ∣_{\infty} \leq \frac{3}{4}}$ and $r_{0} : = \frac{1}{8}$ ; then ${J : {dist}_{\infty} (J, K) < 2 r_{0}} \subseteq U$ and $h (J) : = \frac{1}{2} (J_{1}^{2} + J_{2}^{2})$ is holomorphic and conjugation-symmetric on $U$ with $h (K) \subseteq R$ . Take $Ω : = {ν \in R^{2} : ∣ ν ∣_{\infty} < \frac{1}{8}}$ , $Ω^{'} : = {ν \in C^{2} : ∣ ν ∣_{\infty} < \frac{1}{4}}$ and $b : = {i d}_{Ω^{'}}$ ; then $D h (b (ν)) = ν$ on $Ω^{'}$ , and $b (Ω) + (B_{r_{0}} \cap R^{2}) \subseteq K$ because $∣ ν ∣_{\infty} + ∣ ξ ∣_{\infty} \leq \frac{1}{8} + \frac{1}{8} = \frac{1}{4} \leq \frac{3}{4}$ . The Hessian bound holds with $M_{2} : = 2$ , since $∣ I^{T} D^{2} h (J) I ∣ = ∣ I_{1}^{2} + I_{2}^{2} ∣ \leq ∣ I_{1} ∣^{2} + ∣ I_{2} ∣^{2} \leq 2 ∣ I ∣_{\infty}^{2}$ for every $I \in C^{2}$ and $J \in U$ .

1.2givenL1L3

Angular regularity and embedding: for each fixed $ν \in Ω_{α, τ}$ , the angular parametrization of the torus in [L1] is the quotient of the limiting KAM map in [L3]. Uniform convergence of the maps and their first angular derivatives makes that limit $C^{1}$ on the real torus; holomorphic convergence gives real-analytic regularity. The angular derivative block is $I + D_{ϑ} w$ , which is invertible because $∥ D_{ϑ} w ∥ < 1$ by [L3]; thus ${\hat{Ψ}}_{ν}$ is an immersion. Injectivity and the homeomorphism onto its image are supplied by [L1], so it is a $C^{1}$ embedding and ${\hat{Ψ}}_{ν}^{*} ω$ is defined.

2.1givenL1step 1.1algebra

Verification of the analytic data for [L1], part 2: fix $s_{0} : = 1$ , $σ_{0} : = \frac{1}{24} = \min (1, s_{0} / 24)$ and $F (J, ϑ) : = ε \cos ϑ_{1} \cos ϑ_{2}$ , which is holomorphic, conjugation-symmetric and $2 π$ -periodic in $ϑ$ ; on the strip $∣ Im ϑ ∣_{\infty} \leq s_{0}$ one has $∣ \cos ϑ_{j} ∣ \leq \cosh s_{0}$ , hence $∣ F ∣ \leq ε \cosh^{2} (s_{0}) = : ε_{1}$ on the declared domain. Take the compact cube $C^{*} = {∣ ν ∣_{\infty} \leq \frac{1}{16}}$ , whose closure satisfies $\overline{C^{*}} \subseteq Ω$ with ${dist}_{\infty} (\overline{C^{*}}, \partial Ω) = \frac{1}{16} > 0$ . All the hypotheses of [L1] are therefore satisfied with the perturbation size $ε_{1} = ε \cosh^{2} s_{0}$ in place of $ε$ .

3.1step 1.1step 1.2step 2.1L1L2L3algebra

Application of [L1]: its threshold $c_{1} > 0$ depends on the fixed action and analytic domains, $M_{2}$ , the exponent $τ$ and the normalized profile $g$ , but is uniform in the perturbation-size bound and $α$ . By step 2.1 the actual perturbation has bound $ε_{1} = ε \cosh^{2} (s_{0})$ , so [L1] applies whenever $0 < α < c_{1}$ and $0 < ε_{1} \leq c_{1} α^{2}$ . Put $c : = c_{1} / \cosh^{2} (s_{0})$ ; then these conditions follow from $0 < α < c$ and $0 < ε \leq c α^{2}$ . For such $α, ε$ , clause 1 of [L1] and [L2] give $∣ Ω_{α, τ} ∣ \geq ∣ C^{*} ∣ - C_{2, τ} (diam C^{*}) α$ , which is the displayed bound because $∣ C^{*} ∣ = (1 / 8)^{2} = 1 / 64$ and $diam C^{*} = \frac{1}{8}$ in the supremum metric; clause 2 of [L1] gives, for every $ν \in Ω_{α, τ}$ , the injective Lagrangian torus parametrisation with $X_{H} ({\hat{Ψ}}_{ν} (ϑ)) = \sum_{j} ν_{j} \partial_{ϑ_{j}} {\hat{Ψ}}_{ν} (ϑ)$ ; step 1.2 supplies its $C^{1}$ regularity, and clause 3 of [L1] supplies its closeness and Lipschitz bounds.

4.1step 3.1givenalgebra

Physical reading: define $C (I, ϑ) : = (ϑ, J_{0} I)$ from dimensionless to physical phase space. It is a smooth diffeomorphism with $C^{*} ω_{p h y s} = J_{0} ω$ , where $ω = \sum_{i} d ϑ_{i} \land d I_{i}$ ; hence $(C \circ {\hat{Ψ}}_{ν})^{*} ω_{p h y s} = J_{0} {\hat{Ψ}}_{ν}^{*} ω = 0$ , so the scaled embedding is Lagrangian. Together with $t = t_{0} s$ , the scaling preserves the form of Hamilton's equations: from $θ_{i}^{'} = I_{i}$ and $I_{i}^{'} = ε \sin ϑ_{i} \cos ϑ_{3 - i}$ (the dimensionless equations) one obtains ${\dot{θ}}_{i} = Π_{i} / I$ and ${\dot{Π}}_{i} = E_{0} ε \sin θ_{i} \cos θ_{3 - i} = - \partial_{θ_{i}} U_{p h y s}$ by the identities $J_{0}^{2} = I E_{0}$ and $J_{0} t_{0} = I$ ; hence the dimensionless and physical systems correspond under this scaling of solutions, and an invariant torus of the dimensionless model is an invariant torus of the physical model whose physical trajectory is the scaled embedding evaluated at parameter phase $φ_{0} + ω_{0} ν t$ . The identity $J_{0} t_{0} = I$ follows from $t_{0} = J_{0} / E_{0}$ and $J_{0}^{2} = I E_{0}$ .

5.1step 3.1step 4.1L1L2∎

Clause 3 (scope): no clause of [L1] or [L2] contains a statement about the complement of $Ω_{α, τ}$ , about other frequencies, about tori outside the smallness threshold or about empirical realizations, and no step above produced such a statement; the constant $c = c_{1} / \cosh^{2} (s_{0})$ is not numerically evaluated in this item.

Remarks

What is and is not proved. The theorem is a conditional persistence statement: for the adopted potential and sufficiently small coupling, the labels of the surviving Diophantine tori form a positive-measure subset of the declared frequency cube. It does not assert that non-Diophantine or resonant labels fail, that the complementary motion is chaotic, or that every trajectory is quasi-periodic. The mathematical core is entirely the supplier [L1]; this item performs the rotor-specific verification of its hypotheses and the unit restoration.
Interaction with the horseshoe result. A smooth periodically forced one-label Cartesian model with a transverse homoclinic horseshoe: the physical stroboscopic map has a compact invariant set conjugate to the shift, chaotic in the stated sense establishes a chaotic invariant set for a different, explicitly driven full three-dimensional force model whose pulsed one-coordinate dynamics lies on an invariant axis and whose two transverse coordinates have adopted inverted-oscillator forces. The two results are about different model classes and are not contradictory: KAM persistence concerns invariant tori of a nearly integrable two-frequency autonomous two-rotor model, while the horseshoe concerns the pulsed model and its invariant set.
Units. $ω_{0} = \sqrt{E_{0} / I}$ has dimension $τ^{- 1}$ ; the frequencies $ν$ are dimensionless labels in the frequency cube $C^{*}$ , so the physical angular frequencies $ω_{0} ν_{i}$ have the right dimension (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The measure bound concerns the dimensionless frequency cube, and no probability, sampling or frequency-of-occurrence interpretation is attached to it.

The two-rotor model: a positive-measure family of invariant Lagrangian tori with Diophantine frequencies for sufficiently small coupling, with the explicit units and the stated scope

Statement

Facts & Assumptions

Proof

Remarks

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