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The zero-energy level of the harmonic model: a compact connected common level set of a completely integrable datum that is a point, not a torus

Physical scope

A counterexample inside the declared harmonic model of the framework's force branch: the one-degree-of-freedom harmonic datum $(H)$ on the generalized-coordinate phase plane of the explicit straight-line immersion is a completely integrable datum, and its zero common level set is compact, connected and nonempty but consists of the single rest point, so it is not diffeomorphic to a torus; equivalently, the level is a critical level at which the regular-fibre hypotheses of the torus theorem fail. The item asserts no measured value and no claim about any apparatus; it shows that regularity cannot be dropped from the hypotheses.

Statement refuted

In a completely integrable datum of a presentation of the framework's force branch, every nonempty compact connected common level set of the integral map is diffeomorphic to a torus $T^{n} = R^{n} / Z^{n}$ .

Facts & Assumptions

Given: The scalar harmonic model of The harmonic (linear restoring) model: a declared linear force law with parameter $k > 0$ with $m > 0$ , $k > 0$ , $[m] = μ$ , $[k] = μ τ^{- 2}$ and length-valued $x$ . To place its phase plane in the presentation class, declare the ambient one-label force model on $Q_{a m b} = R^{3}$ , time window $R$ , mass operator $m I_{3}$ , smooth potential $U_{a m b} (y) = k y_{1}^{2} / 2$ and $d = 0$ , with force $F_{a m b} (y, v) = (- k y_{1}, 0, 0)$ (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets). Choose the smooth immersion $X : R \to Q_{a m b}$ , $X (x) = (x, 0, 0)$ , of Generalized-coordinate maps of the model, the substituted Lagrangian, the kinetic metric and generalized forces. Its derivative $(1, 0, 0)^{T}$ is injective, its kinetic metric is $G = m$ , and its substituted potential is $k x^{2} / 2$ ; thus clause 2 of 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 gives the generalized-coordinate phase domain $W = R \times R$ and chart Hamiltonian (write it as $H$ ) $H (x, p) = p^{2} / (2 m) + k x^{2} / 2$ , with $p = m x^{'}$ and $[p] = μ ℓ τ^{- 1}$ , $[H] = μ ℓ^{2} τ^{- 2}$ (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, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). This identifies the scalar model with the substituted equations exactly: $x$ solves $m x^{''} = - k x$ if and only if $X \circ x$ solves $m y^{''} = (- k y_{1}, 0, 0)$ along this line; the transverse equations are $0 = 0$ .

[L1]

Datum and integral map (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): with $ω = \sqrt{k / m} > 0$ so that $H (x, p) = (p^{2} + m^{2} ω^{2} x^{2}) / (2 m)$ , the single function $F_{1} : = H$ on $W_{0} : = W$ satisfies ${H, H} = 0$ by skew-symmetry of the bracket, is $C^{\infty}$ , and has differential $d H = (m ω^{2} x) d x + (p / m) d p$ , which vanishes exactly at $(0, 0)$ ; hence $F = (H)$ is a completely integrable datum of the presentation on $W_{0}$ , whose integral map is $H$ itself and whose only critical value is $0$ . The complement of $(0, 0)$ is open and dense, so the independence requirement holds on a dense open subset.

[L2]

Fibre and diffeomorphism vocabulary (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, $C^{k}$ Euclidean maps and diffeomorphisms, Injection, surjection, bijection, Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent): a diffeomorphism is in particular a bijection, so it preserves cardinality; the one-dimensional torus $T^{1} = R / Z$ is an infinite set; and the common level set of the datum at a value $b$ is $H^{- 1} (b) \subseteq W_{0}$ .

[L3]

The torus theorem's hypotheses (Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice): the conclusion that a common level set is a torus is asserted there only for a compact connected regular fibre of a regular value, among further hypotheses; the item makes no claim about critical levels.

Counterexample

1.1L1given

By [L1] the datum $F = (H)$ is completely integrable on $W_{0}$ and $H^{- 1} (0) = {(0, 0)}$ : indeed $H (x, p) = 0$ forces $p = 0$ and $x = 0$ because $H$ is a sum of two squares divided by the positive number $2 m$ . The set $H^{- 1} (0)$ is therefore nonempty, compact (a single point) and connected.

1.2L1L3

The point $(0, 0)$ is a critical point of $H$ by [L1], so $0$ is a critical value and the fibre $H^{- 1} (0)$ is a critical (not regular) fibre; the hypothesis of [L3] fails, and the torus conclusion is not available. This already shows that the regular-value hypothesis is needed, but the point of the counterexample is stronger.

2.1L2step 1.1algebra

The fibre is not diffeomorphic to $T^{1}$ : it has exactly one point, while $T^{1}$ is infinite, and a diffeomorphism is a bijection by [L2]; no bijection exists between a one-point set and an infinite set. Hence the refuted statement fails at the declared model datum.

2.2L1step 1.1algebra

The failure is confined to the critical level: for every $E > 0$ the level $H^{- 1} (E)$ is the ellipse ${(x, p) : p^{2} / (2 m) + k x^{2} / 2 = E}$ , a one-dimensional compact connected manifold, the phase curve of a nonconstant harmonic motion; the smooth parametrization $u \mapsto (\sqrt{2 E / k} \sin (2 π u), \sqrt{2 m E} \cos (2 π u))$ is a diffeomorphism from $R / Z$ to that ellipse. Indeed the Pythagorean identity gives its image, the bijection of $t \mapsto (\cos t, \sin t)$ is a bijection from $[0, 2 π)$ onto the real unit circle and the $2 π$ period fix $u$ modulo one and cover the whole ellipse. At each point either the derivative of its $x$ coordinate or of its $p$ coordinate is nonzero, since sine and cosine cannot vanish together. Applying The Euclidean inverse function theorem to that scalar coordinate in a real representative interval gives a smooth local inverse, proving the claimed diffeomorphism. These trigonometric properties are The zero sets of sine and cosine and the least positive common period 2 pi, The derivatives of sine and cosine are cosine and minus sine and Parity and the Pythagorean identity for sine and cosine. Thus the regular fibres have the torus structure, whereas the critical fibre is a point. The refuted claim therefore fails exactly at the critical value.

3.1step 1.1step 1.2step 2.1step 2.2∎

Collecting: the common level set $H^{- 1} (0)$ of the completely integrable datum $(H)$ is nonempty, compact and connected but not diffeomorphic to a torus, so the refuted statement is false; the regular values of the same datum do satisfy the torus conclusion, so no claim of Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice is affected.

Remarks

What is refuted. Only the version of the torus claim without the regularity hypothesis. The qualified theorem Compact connected regular fibres of a completely integrable datum of the presentation are Lagrangian tori with a full period lattice is untouched: it assumes a regular value with a compact connected regular fibre, exactly the hypothesis that fails here.
No empirical content. The model data $m, k$ are declared model parameters and the level computation is an identity about the model's Hamiltonian; the counterexample asserts no measured value and no claim about a real oscillator.

The zero-energy level of the harmonic model: a compact connected common level set of a completely integrable datum that is a point, not a torus

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

Dependency tree · 2 levels

Sources