Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Nonlinear stability of the torque-free spin of the rigid-material model about an extreme principal axis, and instability about the intermediate axis, with an explicit escape time and the orientation caveat

Physical scope

Lyapunov stability and instability of the torque-free spin of a nondegenerate rigid-material model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with pairwise distinct principal moments, stated for the model's motions with time window containing $[0, \infty)$ : spins about the largest and smallest principal axes are Lyapunov stable in the body angular momentum, with an explicit conserved deviation bound, and spins about the intermediate axis are unstable with an explicit escape time; the escaping perturbation is realised by an actual torque-free motion. The stability is a statement about the body-frame angular momentum and angular velocity only: two motions with the same constant body angular velocity can have orientations differing by a fixed nontrivial rotation at every time, so no orientation stability is asserted. No measurement, damping, dissipation or empirical value is asserted; the countable-choice assumption is inherited for finite mass moments and inertial integrals, and for the double-cover supplier used in the quaternion constructions. The stability and escape calculations for a given positive diagonal inertia matrix introduce no additional choice principle.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried through Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clause (i) and Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model for finite mass moments, through Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (i) and the inertial-integral suppliers of Euler's equations with torque for the rigid-material model, the rotational power identity, the torque-free reduction and frame covariance and Torque-free motion of the rigid-material model: conservation of the angular momentum about the centre of mass, the energy-momentum invariants, the principal-axis equations and global existence of the spin dynamics, and through the double-cover supplier of The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts for the surjectivity used to choose initial unit quaternions. The stability and escape calculations for a given positive diagonal $I$ introduce no additional choice principle. Adopt a nondegenerate model of the rigid-material branch Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with pairwise distinct principal moments, and label a positively oriented principal basis $(e_{1}, e_{2}, e_{3})$ , $I e_{j} = I_{j} e_{j}$ , so that $I_{1} < I_{2} < I_{3} .$ Use the model in the torque-free branch of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iv). Along any motion write $l : = I ω$ ; the torque-free $l$ -path solves $l^{'} = l \times (I^{- 1} l)$ and is the restriction of a solution defined on all of $R$ (Torque-free motion of the rigid-material model: conservation of the angular momentum about the centre of mass, the energy-momentum invariants, the principal-axis equations and global existence of the spin dynamics clauses 2 and 4). Write $I_{0}$ for the model's time window. All physical realizations below use paths restricted to $I_{0}$ ; clause 3's forward-time realization assumes $[0, \infty) \subseteq I_{0}$ . Let $L \neq 0$ . Then:

Stability of the extreme-axis spins. For $j = 1$ and $j = 3$ the constant path $t \mapsto L e_{j}$ is Lyapunov stable for the $l$ -system (Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system): for every $ε > 0$ there is $δ \in (0, ε]$ such that every maximal solution $l$ with $∥ l (0) - L e_{j} ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies $∥ l (t) - L e_{j} ∥_{2} < ε$ for all $t \geq 0$ . Consequently every torque-free motion of a model of the branch whose time window contains $[0, \infty)$ and whose datum satisfies $∥ l (0) - L e_{j} ∥_{2} < δ$ has, with $Ω : = L / I_{j}$ , $∥ l (t) - L e_{j} ∥_{2} < ε, ∥ ω (t) - Ω e_{j} ∥_{2} < \frac{ε}{I_{\min}} (t \geq 0),$ where $I_{\min} = I_{1}$ .
Explicit deviation bound. For $j = 1$ define on $R^{3}$ $c_{1} : = \frac{1}{2} (\frac{1}{I_{1}} - \frac{1}{I_{2}}) > 0, D_{1} (l) : = \frac{1}{2} (\frac{⟨ l, l ⟩}{I_{1}} - ⟨ l, I^{- 1} l ⟩) \geq 0,$ and for $j = 3$ $c_{3} : = \frac{1}{2} (\frac{1}{I_{2}} - \frac{1}{I_{3}}) > 0, D_{3} (l) : = \frac{1}{2} (⟨ l, I^{- 1} l ⟩ - \frac{⟨ l, l ⟩}{I_{3}}) \geq 0.$ Along every solution of $l^{'} = l \times (I^{- 1} l)$ one has, for all $t$ in its interval, with $l (0)$ evaluated on its unique global extension if the original interval does not contain $0$ , $l_{2} (t)^{2} + l_{3} (t)^{2} \leq \frac{D_{1} (l (0))}{c_{1}}, l_{1} (t)^{2} + l_{2} (t)^{2} \leq \frac{D_{3} (l (0))}{c_{3}} .$
Instability of the intermediate-axis spin. The spin $l_{0} : = L e_{2}$ is not Lyapunov stable. More precisely, with $a : = \sqrt{\frac{1 / I_{1} - 1 / I_{2}}{1 / I_{2} - 1 / I_{3}}} > 0, c_{0} : = a (\frac{1}{I_{2}} - \frac{1}{I_{3}}) \frac{\sqrt{3}}{2} ∣ L ∣ > 0, δ_{0} : = \frac{∣ L ∣}{2 \sqrt{1 + a^{2}}}, ε_{0} : = \frac{∣ L ∣}{2},$ if $[0, \infty) \subseteq I_{0}$ , then for every $δ > 0$ there are a torque-free motion of this model on $I_{0}$ , with body angular velocity $ω$ , and a time $t \geq 0$ such that $∥ l (0) - L e_{2} ∥_{2} < δ, ∥ l (t) - L e_{2} ∥_{2} \geq ε_{0},$ and the escape can be arranged with $t \leq c_{0}^{- 1} \log (δ_{0} / η)$ for any prescribed $η \in (0, \min {δ_{0}, δ})$ whose initial datum $l^{η} : = (η, σ \sqrt{L^{2} - (1 + a^{2}) η^{2}}, - σ a η)$ , $σ : = L / ∣ L ∣$ , satisfies $∥ l^{η} - L e_{2} ∥_{2} < δ$ . Such $η$ exist arbitrarily close to zero, and the datum belongs to the $l$ -system.
Orientation caveat. For $j = 1$ or $j = 3$ and every $Ω \neq 0$ there are two torque-free motions of the model, with $c \equiv 0$ , whose body angular velocities are both constantly equal to $Ω e_{j}$ and whose orientation paths $R_{1}, R_{2}$ satisfy $R_{2} (t) = R_{1} (t) S$ for every $t$ and a fixed rotation $S \neq I_{3}$ . Consequently clause 1 asserts nothing about the closeness of the orientations of two motions.

The statement is conditional on the adopted model and zero-torque balance; it asserts no force, no damping, no measurement and no empirical law.

Facts & Assumptions

Given: A nondegenerate model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with positively oriented principal basis $(e_{j})$ and moments $I_{1} < I_{2} < I_{3}$ , positive; the zero-torque branch; the data $L \neq 0$ and the auxiliary constants $a, c_{0}, δ_{0}, ε_{0}, c_{1}, c_{3}, D_{1}, D_{3}$ of the statement.

[L1]

The torque-free system and its invariants: with $l = I ω$ one has $l^{'} = l \times (I^{- 1} l)$ ; $⟨ l, l ⟩$ and $\frac{1}{2} ⟨ l, I^{- 1} l ⟩$ are constant along solutions; every solution has a unique maximal extension to all of $R$ ; and $l = I ω$ with $I$ diagonal positive in the principal basis, so $∥ ω ∥_{2} \leq ∥ l ∥_{2} / I_{\min}$ and $∥ l ∥_{2} \leq I_{\max} ∥ ω ∥_{2}$ (Torque-free motion of the rigid-material model: conservation of the angular momentum about the centre of mass, the energy-momentum invariants, the principal-axis equations and global existence of the spin dynamics clauses 2-4, Euler's equations with torque for the rigid-material model, the rotational power identity, the torque-free reduction and frame covariance clause 4, Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clauses (a) and (d), Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model).

[L2]

Stability supplier (Lyapunov stability of spin about the largest and smallest principal axes of the torque-free top): for positive reals $I_{1} < I_{2} < I_{3}$ , the diagonal map $I$ and the field $F (l) = l \times (I^{- 1} l)$ on $R^{3}$ , the equilibria $L e_{1}$ and $L e_{3}$ , $L \neq 0$ , are Lyapunov stable in the sense of Equilibria, Lyapunov stability and asymptotic stability of an autonomous first-order system: for $j = 1, 3$ and every $ε > 0$ there is $δ \in (0, ε]$ such that every maximal solution with $∥ l (0) - L e_{j} ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies $∥ l (t) - L e_{j} ∥_{2} < ε$ for all $t \geq 0$ .

[L3]

Instability supplier (Instability of spin about the intermediate principal axis of the torque-free top): its Statement gives the existential instability and time bound. Its Proof, second through sixth numbered steps, gives the stronger prescribed-datum conclusion used here: in the same setting, $L e_{2}$ is an equilibrium which is not Lyapunov stable, and with the constants $a, c_{0}, δ_{0}$ of the statement and $ε_{0} = ∣ L ∣ / 2$ , for every $δ > 0$ there is a solution $l$ of the $l$ -system with $∥ l (0) - L e_{2} ∥_{2} < δ$ and some $t \geq 0$ with $∥ l (t) - L e_{2} ∥_{2} \geq ε_{0}$ ; the escape can be arranged at a time $t \leq c_{0}^{- 1} \log (δ_{0} / η)$ from the datum $l^{η}$ of the statement for every $0 < η < \min {δ_{0}, δ}$ additionally satisfying $∥ l^{η} - L e_{2} ∥_{2} < δ$ ; such data exist arbitrarily close to the spin as $η ↓ 0$ .

[L4]

Realisability: for every continuous $ω : R \to R^{3}$ there is a $C^{1}$ rotation path $R : R \to S O (3)$ whose body angular velocity is $ω$ ; if $ω$ is $C^{1}$ then $R$ is of class $C^{2}$ ; and for $q_{0} \in S^{3}$ with $ρ (q_{0}) = R_{0}$ the path can be chosen with $R (t_{0}) = R_{0}$ , the covering map $ρ$ being surjective with $ρ (q_{1} q_{2}) = ρ (q_{1}) ρ (q_{2})$ on $S^{3}$ (The quaternion lift of a prescribed continuous angular velocity: $q^{'} = \frac{1}{2} q φ (ω (t))$ keeps $S^{3}$ invariant, its rotation path has body angular velocity $ω$ , and the initial value problem has a unique solution on the whole interval clauses 1-2, The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts and its second remark).

[L5]

The balance check ingredients: for a $C^{1}$ curve $R : I \to S O (3)$ and a $C^{1}$ curve $l : I \to R^{3}$ one has $(R l)^{'} = R^{'} l + R l^{'}$ , $R^{'} = R [ω]_{\times}$ , and $[ω]_{\times} z = ω \times z$ ; the cross product is bilinear and alternating with $ω \times l = - (l \times ω)$ (Derivative rules for differentiable matrix-valued curves, and skewness of $R^{T} R^{'}$ for a curve in $S O (3)$ , The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ , The cross product is bilinear, alternating, and orthogonal to both factors, The cross product in $R^{3}$ ); and a $C^{2}$ curve $c$ together with a $C^{2}$ rotation path forms a motion $x = c + R ρ$ of the model (Configuration, configuration space and orientation of a rigid-material model in an inertial frame, Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clause (ii)).

[L6]

The spin motions of Principal-axis uniform rotations of the torque-free rigid-material model: existence, a fixed spatial direction, the invariants, and the converse classification clauses 1-3 realise every constant body angular velocity $Ω e_{j}$ , $j = 1, 2, 3$ , $Ω \neq 0$ , as the body angular velocity of a torque-free motion with $l = Ω I_{j} e_{j}$ constant, and $L_{c} = Ω I_{j} R (t) e_{j}$ is constant along it.

[L7]

The orientation construction uses the imaginary quaternion $φ (e_{j})$ with $N (φ (e_{j})) = 1$ and $φ (e_{j})^{2} = - 1$ , so that $r : = φ (e_{j}) \in S^{3}$ and $r φ (ω) = φ (ω) r$ for $ω = Ω e_{j}$ (The quaternion rotation map $ρ$ : conjugation on the imaginary quaternions, the unit sphere $S^{3}$ , and quaternion lifts, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct; quote the two supplier theorems for the $l$-system, translate to the model's motions through $l=I\omega$, realise the escaping $l$-path as a motion by the quaternion lift and the balance check, and separate the orientation statement from the angular-momentum statement

1.1givenL1L2L5

The positively oriented principal basis has a basis matrix $Q \in S O (3)$ ; the skew/cross correspondence of [L5] gives $Q (u \times v) = (Q u) \times (Q v)$ , so $l \mapsto l \times (I^{- 1} l)$ is the supplier's positive diagonal field in those coordinates. Clause 1 of [L2] gives for $j = 1$ and $j = 3$ and every $ε > 0$ a $δ \in (0, ε]$ such that every maximal solution with $∥ l (0) - L e_{j} ∥_{2} < δ$ is defined for all $t \geq 0$ and satisfies $∥ l (t) - L e_{j} ∥_{2} < ε$ ; by [L1] a maximal solution is a solution defined on $R$ , so this is exactly the Lyapunov stability of $L e_{j}$ for the $l$ -system, and the equilibrium property $F (L e_{j}) = 0$ holds by the alternation of the cross product in [L5].

1.2givenL1algebra

The functions $D_{1}$ and $D_{3}$ of clause 2 are nonnegative, since in the principal basis with $l = \sum_{k} l_{k} e_{k}$ one has $D_{1} (l) = \frac{1}{2} \sum_{k} (1 / I_{1} - 1 / I_{k}) l_{k}^{2}$ and $D_{3} (l) = \frac{1}{2} \sum_{k} (1 / I_{k} - 1 / I_{3}) l_{k}^{2}$ with all coefficients nonnegative by $I_{1} < I_{2} < I_{3}$ ; and $2 D_{1} (l) \geq (1 / I_{1} - 1 / I_{2}) (l_{2}^{2} + l_{3}^{2})$ and $2 D_{3} (l) \geq (1 / I_{2} - 1 / I_{3}) (l_{1}^{2} + l_{2}^{2})$ because the remaining coefficients are at least these. By [L1] the two combinations $⟨ l, l ⟩$ and $⟨ l, I^{- 1} l ⟩$ are constant along every solution, so $D_{1}$ and $D_{3}$ are constant along solutions and the two displayed inequalities of clause 2 follow.

2.1step 1.1givenL1L3algebra

Fix an admissible prescribed $η$ from clause 3. Apply the invariant-branch argument of [L3] with the auxiliary number $η_{0} : = (η + δ_{0}) / 2$ , so $0 < η < η_{0} < δ_{0}$ , as required in that supplier's second proof step. That argument starts with exactly $l (0) = l^{η}$ , with $C = L^{2}$ and $E = L^{2} / (2 I_{2})$ ; until $l_{1}$ first reaches $δ_{0}$ , it obtains $l_{3} = - σ a l_{1}$ , $σ l_{2} \geq \sqrt{3} ∣ L ∣ / 2$ and $l_{1}^{'} \geq c_{0} l_{1}$ . Its fifth and sixth steps therefore give a first hitting time $t \leq c_{0}^{- 1} \log (δ_{0} / η)$ and $∥ l (t) - L e_{2} ∥_{2}^{2} \geq (1 + a^{2}) δ_{0}^{2} = L^{2} / 4$ . The construction applies to every such $η$ , not only to an existentially chosen perturbation. The additional initial-norm condition gives $∥ l (0) - L e_{2} ∥_{2} < δ$ , and $l^{η} \to L e_{2}$ as $η ↓ 0$ ensures admissible data for every $δ > 0$ . This proves the prescribed-datum escape and, since $ε_{0} = ∣ L ∣ / 2 > 0$ is fixed, instability.

2.2step 1.1givenL1

Let a torque-free motion have time window containing $[0, \infty)$ and datum with $∥ l (0) - L e_{j} ∥_{2} < δ$ , $j \in {1, 3}$ , for the $δ$ of step 1.1. Its $l$ -path is a solution of the $l$ -system by [L1], so $∥ l (t) - L e_{j} ∥_{2} < ε$ for all $t \geq 0$ ; and since $l = I ω$ with $I$ diagonal with entries $I_{1}, I_{2}, I_{3}$ , the difference $l (t) - L e_{j} = I (ω (t) - Ω e_{j})$ with $Ω = L / I_{j}$ gives $∥ ω (t) - Ω e_{j} ∥_{2} \leq ∥ l (t) - L e_{j} ∥_{2} / I_{\min} < ε / I_{\min}$ by [L1]. This completes clause 1.

3.1step 2.1givenL1L4L5

Realise the escaping $l$ -path of step 2.1 as a motion. Let $l : R \to R^{3}$ be the global solution of the $l$ -system with datum $l (0) = l^{η}$ (global by [L1]), and put $ω : = I^{- 1} l$ , which is continuous and of class $C^{1}$ because $l$ solves a locally Lipschitz equation; by [L4] there is a $C^{2}$ rotation path $R : R \to S O (3)$ with body angular velocity $ω$ , with any prescribed initial orientation $R (0) = R_{0} \in S O (3)$ by choosing a preimage $q_{0}$ of $R_{0}$ . Assume $[0, \infty) \subseteq I_{0}$ as in clause 3, restrict $R$ and $l$ to $I_{0}$ , and define $x : I_{0} \times R^{3} \to R^{3}$ by $x (t, ρ) : = R (t) ρ$ with $c \equiv 0$ . This is a motion of the same model by [L5], and its domain includes the escape time. Its body angular velocity is $ω$ , so its $l$ -data are $l$ , and its zero-torque balance holds: with [L5], $(R l)^{'} = R^{'} l + R l^{'} = R [ω]_{\times} l + R l^{'} = R (ω \times l + l^{'}) = 0$ because $l^{'} = l \times (I^{- 1} l) = l \times ω = - ω \times l$ . Hence the motion is a torque-free motion of the branch with $l (0) = l^{η}$ , $∥ l (0) - L e_{2} ∥_{2} < δ$ , and $∥ l (t) - L e_{2} ∥_{2} \geq ε_{0}$ at the escape time $t \leq c_{0}^{- 1} \log (δ_{0} / η)$ of step 2.1. This is clause 3.

4.1step 2.2givenL4L6L7algebra∎

For clause 4, put $ω \equiv Ω e_{j}$ and let $r : = φ (e_{j}) \in S^{3}$ , so that $r φ (ω) = φ (ω) r$ by [L7] and $N (r) = 1$ . By [L4] choose $q_{0} \in S^{3}$ with $ρ (q_{0}) = R_{0}$ for a fixed $R_{0} \in S O (3)$ , let $q : R \to S^{3}$ be the lift of $ω$ with $q (0) = q_{0}$ , and put $q_{2} (t) : = q (t) r$ ; then $q_{2}$ is $C^{1}$ , takes values in $S^{3}$ because $N$ is multiplicative, and solves $q_{2}^{'} = q^{'} r = \frac{1}{2} q φ (ω) r = \frac{1}{2} q_{2} φ (ω)$ by the commutation in [L7], with $q_{2} (0) = q_{0} r$ . Let $R_{1} : = ρ \circ q$ and $R_{2} : = ρ \circ q_{2}$ ; by the homomorphism property in [L4], $R_{2} (t) = ρ (q (t)) ρ (r) = R_{1} (t) S$ with $S : = ρ (r) \in S O (3)$ , and $S \neq I_{3}$ because $ρ (r)$ acts as the half-turn about the axis $e_{j}$ : $S e_{j} = e_{j}$ and $S e_{a} = - e_{a}$ for $a \neq j$ by the conjugation formula and $φ (e_{j})^{2} = - 1$ in [L7]. Both $R_{1}$ and $R_{2}$ have body angular velocity $Ω e_{j}$ by [L4] and so, after restriction to $I_{0}$ and with $c \equiv 0$ , are torque-free motions by clause 2 of Principal-axis uniform rotations of the torque-free rigid-material model: existence, a fixed spatial direction, the invariants, and the converse classification; their orientations differ by the fixed rotation $S \neq I_{3}$ at every time, which is clause 4.

Remarks

What is stable. Clauses 1 and 3 are statements about the body-frame angular momentum $l = I ω$ (equivalently, through the fixed invertible $I$ , about the body angular velocity $ω$ ). The orientation of the body in the inertial frame is not controlled, as clause 4 makes explicit; the spin phase about the axis is free.
Why the intermediate axis is different. The two conserved quantities $C = ⟨ l, l ⟩$ and $E = \frac{1}{2} ⟨ l, I^{- 1} l ⟩$ confine the $l$ -path to a compact set, but near $L e_{2}$ that set is a pair of curves through the equilibrium along which $l_{1}$ grows exponentially until the transverse components exchange roles; the explicit datum $l^{η}$ and the escape time of clause 3 make the instability effective, and clause 3's realisation shows that it is a statement about motions, not merely about solutions of a differential equation.
No dissipation. The torque-free branch has no damping; with a nonzero datum $τ$ , the statements of this item do not apply, and a dissipative torque model would require its own analysis. The result also asserts no empirical value and no claim that any concrete body has been observed to spin stably or unstably.

Nonlinear stability of the torque-free spin of the rigid-material model about an extreme principal axis, and instability about the intermediate axis, with an explicit escape time and the orientation caveat

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources