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Linearization of the torque-free spin of the rigid-material model at a principal-axis spin: the oscillatory/exponential dichotomy and linear stability

Physical scope

The first-order linearization of the torque-free body-angular-velocity dynamics of a nondegenerate rigid-material model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope at a principal-axis spin $Ω e_{j}$ , for pairwise distinct principal moments: the total derivative of the spin field, the explicit transverse equations, the sign criterion $λ_{j} < 0$ for an extreme axis and $λ_{j} > 0$ for the intermediate axis, the bounded oscillatory solutions in the extreme case and the exponentially growing solutions in the intermediate case, and the corresponding linear-stability verdicts in the sense of Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium. The statement concerns the linearized system only; no nonlinear Lyapunov conclusion is drawn here, and no measurement, damping, friction or empirical value is asserted.

Statement

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 in a positively oriented principal basis $(e_{1}, e_{2}, e_{3})$ , $I e_{j} = I_{j} e_{j}$ with $I_{j} > 0$ (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clauses (a) and (d)), and use it 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). In the body-angular-velocity variable $ω$ the torque-free dynamics is the autonomous field $F (ω) : = - I^{- 1} (ω \times I ω), ω \in R^{3},$ on $R^{3}$ (Euler's equations with torque for the rigid-material model, the rotational power identity, the torque-free reduction and frame covariance clause 4), and for every axis $j$ and every $Ω \neq 0$ the constant path $\overset{ˉ}{ω} \equiv Ω e_{j}$ is a solution (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). Fix such $j$ and $Ω$ , and choose $(a, b, j)$ cyclically: $(a, b, j) = (2, 3, 1)$ , $(3, 1, 2)$ or $(1, 2, 3)$ . Thus $(e_{a}, e_{b}, e_{j})$ is positively oriented. Put $λ_{j} : = \frac{Ω^{2} (I_{a} - I_{j}) (I_{j} - I_{b})}{I_{a} I_{b}} \neq 0, κ_{j} : = \sqrt{∣ λ_{j} ∣} .$ Then:

The linearized field. $F$ is totally differentiable at $Ω e_{j}$ in the sense of The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, with total derivative $A_{j}$ given on the principal basis by $A_{j} e_{a} = \frac{Ω (I_{j} - I_{a})}{I_{b}} e_{b}, A_{j} e_{b} = \frac{Ω (I_{b} - I_{j})}{I_{a}} e_{a}, A_{j} e_{j} = 0,$ and there is a constant $C \geq 0$ with $∥ F (Ω e_{j} + δ) - A_{j} δ ∥_{2} \leq C ∥ δ ∥_{2}^{2}$ for all $δ \in R^{3}$ .
The linearized system and its dichotomy. Write $δ = δ_{a} e_{a} + δ_{b} e_{b} + δ_{j} e_{j}$ $δ = δ_{a} e_{a} + δ_{b} e_{b} + δ_{j} e_{j}$ .
- If $I_{j}$ is a largest or smallest principal moment, then $λ_{j} < 0$ , every solution of the linearized system $δ^{'} = A_{j} δ$ has a unique maximal extension defined on all of $R$ and bounded on $[0, \infty)$ , with $δ_{a} (t) = δ_{a} (0) \cos (κ_{j} t) + \frac{Ω (I_{b} - I_{j})}{I_{a} κ_{j}} δ_{b} (0) \sin (κ_{j} t),$ $δ_{b} (t) = δ_{b} (0) \cos (κ_{j} t) + \frac{Ω (I_{j} - I_{a})}{I_{b} κ_{j}} δ_{a} (0) \sin (κ_{j} t),$ and $δ_{j}$ constant; consequently the spin is linearly stable in the sense of Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium.
- If $I_{j}$ lies strictly between $I_{a}$ and $I_{b}$ , then $λ_{j} > 0$ , the same two formulas hold with $\cosh$ and $\sinh$ in place of $\cos$ and $\sin$ and $κ_{j} = \sqrt{λ_{j}}$ , the linearized system has unbounded solutions (for instance with $δ_{b} (0) = 0$ and $δ_{a} (0) \neq 0$ satisfying $δ_{a} (0) + \frac{Ω (I_{b} - I_{j})}{I_{a} κ_{j}} δ_{b} (0) \neq 0$ ), and the spin is not linearly stable.

The statement is a statement about the linearized system at the spin; the corresponding nonlinear Lyapunov statements are separate items of this page, and no conclusion about the nonlinear stability of the spin is drawn here.

Facts & Assumptions

Given: A nondegenerate model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with pairwise distinct principal moments $I_{j} > 0$ in the positively oriented principal basis $(e_{j})$ , the zero-torque branch, and the data $j$ , $Ω \neq 0$ of the statement.

[L1]

The torque-free spin equation: for the zero-torque branch the body angular velocity satisfies $ω^{'} = F (ω)$ with $F (ω) = - I^{- 1} (ω \times I ω)$ (Euler's equations with torque for the rigid-material model, the rotational power identity, the torque-free reduction and frame covariance clause 4, Body and spatial angular velocity of a rigid-material motion, and the material velocity field).

[L2]

The mathematical supplier (Linearization of the torque-free Euler system at a principal-axis spin, and the oscillatory/exponential dichotomy): let $I_{1}, I_{2}, I_{3} > 0$ be pairwise distinct, let $I$ be the diagonal map with $I e_{k} = I_{k} e_{k}$ on the standard orthonormal basis and $F (ω) = - I^{- 1} (ω \times I ω)$ ; for $Ω \neq 0$ the field $F$ is totally differentiable at $Ω e_{3}$ with total derivative $A$ satisfying $(A δ)_{1} = Ω (I_{2} - I_{3}) δ_{2} / I_{1}$ , $(A δ)_{2} = Ω (I_{3} - I_{1}) δ_{1} / I_{2}$ , $(A δ)_{3} = 0$ and $∥ F (Ω e_{3} + δ) - A δ ∥_{2} \leq C ∥ δ ∥_{2}^{2}$ for the explicit constant $C = ∥ I^{- 1} ∥_{F} ∥ I ∥_{F}$ ; with $λ = Ω^{2} (I_{2} - I_{3}) (I_{3} - I_{1}) / (I_{1} I_{2})$ the linearized system $δ^{'} = A δ$ has constant $δ_{3}$ and solves the transverse pair by $\cos, \sin$ when $λ < 0$ (with $κ = \sqrt{- λ}$ , bounded solutions) and by $\cosh, \sinh$ when $λ > 0$ (with $κ = \sqrt{λ}$ , unbounded solutions and the growth criterion of the supplier), and $λ < 0$ exactly when $I_{3}$ is a largest or smallest moment, $λ > 0$ exactly when it lies strictly between $I_{1}$ and $I_{2}$ (The total (Fréchet) derivative $D f (a)$ as the linear first-order approximation with $o (∥ h ∥_{2})$ remainder, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

[L3]

Linear stability: for an equilibrium of an autonomous system with total derivative $L$ , the linearized system is $y^{'} = L y$ , and the equilibrium is linearly stable when every maximal solution of the linearized system is defined for all $t \geq 0$ and bounded on $[0, \infty)$ (Invariant sets, orbital stability of an invariant set, and linear stability of an equilibrium).

[L4]

The spin is a solution: for every axis $j$ and rate $Ω$ the constant body angular velocity $Ω e_{j}$ belongs to a torque-free motion with $l = I ω = Ω I_{j} e_{j}$ constant (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).

[L5]

Principal data: $(e_{1}, e_{2}, e_{3})$ is a positively oriented orthonormal basis and $I$ is diagonal with positive entries in it, $∥ \cdot ∥_{2}$ denoting the Euclidean norm (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clauses (a)-(b) and (d), The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product in $R^{3}$ ).

[L6]

For $Q \in S O (3)$ the cross product is equivariant, $Q (u \times v) = (Q u) \times (Q v)$ (The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ clause 3 and proof step 1.3). An orthonormal principal basis can be made positively oriented by replacing one eigenvector by its negative; the associated eigenvalues are unchanged.

Proof

technique · direct; identify the principal basis with the supplier's standard basis so that the spin axis is third, apply its clauses, and translate the sign criterion and the boundedness conclusion

1.1givenL1L4L5

The spin $Ω e_{j}$ is a solution of $ω^{'} = F (ω)$ on every interval with at least two points, because it is the constant body angular velocity of the torque-free motion of [L4] and [L1] is the same equation; with $l = I ω = Ω I_{j} e_{j}$ and $I^{- 1} l = Ω e_{j}$ the right-hand side $F (Ω e_{j}) = - I^{- 1} (Ω e_{j} \times Ω I_{j} e_{j}) = 0$ vanishes.

1.2givenL2L5L6algebra

Identify $R^{3}$ with the supplier's $R^{3}$ of [L2] by the map $Q$ carrying its standard basis to the cyclic basis $(e_{a}, e_{b}, e_{j})$ . By the positive orientation stipulated in [L5], $Q \in S O (3)$ , so $Q (u \times v) = (Q u) \times (Q v)$ ([L6]); then the supplier's diagonal map with entries $(I_{a}, I_{b}, I_{j})$ is the model's inertia operator $I$ expressed in the principal basis [L5], and the fields $F (ω) = - I^{- 1} (ω \times I ω)$ correspond. Clause 1 of [L2] applied at the spin about the third basis vector gives the total differentiability of $F$ at $Ω e_{j}$ and the derivative $A_{j}$ of clause 1: the components are $(A_{j} δ)_{a} = Ω (I_{b} - I_{j}) δ_{b} / I_{a}$ , $(A_{j} δ)_{b} = Ω (I_{j} - I_{a}) δ_{a} / I_{b}$ , $(A_{j} δ)_{j} = 0$ , so evaluating at $e_{a}$ and $e_{b}$ gives exactly the stated basis images. The supplier also gives the remainder bound and explicit constant, and clauses 2-3 of [L2] give the transverse formulas, the criterion for the sign of $λ_{j} = Ω^{2} (I_{a} - I_{j}) (I_{j} - I_{b}) / (I_{a} I_{b})$ and the bounded/unbounded alternatives. The displayed sine/cosine and hyperbolic formulas are defined on all of $R$ , satisfy the linear system, and have the prescribed initial data; the uniqueness assertion in the supplier proof makes every local solution their restriction. This is clause 1 and the case analysis of clause 2 up to the sign criterion.

2.1step 1.2givenL2algebra

The product $(I_{a} - I_{j}) (I_{j} - I_{b})$ is positive exactly when $I_{j}$ lies strictly between $I_{a}$ and $I_{b}$ and negative exactly when $I_{j}$ is larger than both or smaller than both; since $Ω^{2} / (I_{a} I_{b}) > 0$ , the criterion of clause 3 of [L2] reads: $λ_{j} < 0$ (oscillatory, bounded, the displayed $\cos, \sin$ formulas with $κ_{j} = \sqrt{- λ_{j}}$ ) exactly for the extreme axes, and $λ_{j} > 0$ (the $\cosh, \sinh$ formulas with $κ_{j} = \sqrt{λ_{j}}$ , unbounded solutions, the stated growth criterion) exactly for the intermediate axis. This completes clause 2.

3.1step 1.2step 2.1givenL3∎

By [L3] the boundedness of every global maximal solution of the linearized system on $[0, \infty)$ in the extreme case is exactly linear stability of the spin, and the existence of an unbounded solution in the intermediate case is the failure of linear stability, since a solution that is unbounded on $[0, \infty)$ is not bounded there.

Remarks

The linearization variable. The linearization is taken in the body-angular-velocity variable $ω$ ; because $l = I ω$ with $I$ a fixed invertible linear map, stability and boundedness statements transfer between the $ω$ - and $l$ -variables up to the fixed constants $I_{\min}$ and $I_{\max}$ , and the two formulations are equivalent for the linearized system.
Linear versus nonlinear. Clause 2 is a statement about the linearized system. Exponential growth of a linearized solution does not by itself prove nonlinear instability, and boundedness of the linearized system does not by itself prove nonlinear Lyapunov stability; the nonlinear statements are proved separately in 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, which uses the conserved quantities rather than the linearization.
Distinctness is a real hypothesis. The supplier requires pairwise distinct principal moments and the case analysis uses it; repeated moments change the equilibrium set and are not covered by this statement.

Linearization of the torque-free spin of the rigid-material model at a principal-axis spin: the oscillatory/exponential dichotomy and linear stability

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources