Nonlinear Stability Analysis of a Spinning Top with an Interior Liquid-Filled Cavity

Consider the motion of the the coupled system, $\mathscr S$, constituted by a (non-necessarily symmetric) top, $\mathscr B$, with an interior cavity, $\mathscr C$, completely filled up with a Navier-Stokes liquid, $\mathscr L$. A particular steady-state motion $\bar{\sf s}$ (say) of $\mathscr S$, is when $\mathscr L$ is at rest with respect to $\mathscr B$, and $\mathscr S$, as a whole rigid body, spins with a constant angular velocity $\bar{\V\omega}$ around a vertical axis passing through its center of mass $G$ in its highest position ({\em upright spinning top}). We then provide a completely characterization of the nonlinear stability of $\bar{\sf s}$ by showing, roughly speaking, that $\bar{\sf s}$ is stable if and only if $|\bar{\V\omega}|$ is sufficiently large, all other physical parameters being fixed. Moreover we show that, unlike the case when $\mathscr C$ is empty, under the above stability conditions, the top will eventually return to the unperturbed upright configuration.


Introduction
The stability of an upright spinning top is a renowned, classical problem in rigid body dynamics. More precisely, the top is a rigid body, B, that moves while keeping one of its points, O, fixed at all times. Among the many motions that B can execute, particularly interesting is the one where B spins with constant angular velocity,ω, around the vertical axis, a, passing through O and its center of mass G at its highest position, and coinciding with one of the principal axes of inertia e B (say). The abovementioned problem consists exactly in studying the stability properties of such rotational motion. It is then well known that, if |ω| is "sufficiently large" and the principal moment of inertia around e B is a maximum, then the motion is stable and the perturbed motion will be a combined precession-nutation around a [25, Example 9.7C].
In this paper we investigate the analogous question when the top has an interior cavity, C, entirely filled with a viscous liquid, L. In this case the basic motion is the same as above, with L at rest with respect to B and the coupled system S := B ∪ L rotating as a whole rigid body around the direction a coinciding with e S . We denote bys this particular unperturbed steady-state. Mainly due to its important technological applications ( [2] and the references therein), this problem has attracted the attention of a number of applied mathematicians, especially from the Russian School. As a matter of fact, Sobolev [31] was the first to give sufficient stability conditions for the stability ofs, in the case when the liquid is inviscid (Euler), within the linearized approximation, namely, when all nonlinear terms in the relevant equations are entirely disregarded. Precisely, let A, B and C be the principal moments of inertia of S with respect to O with C moment of inertia about e S . In [31] it is then shown that a sufficient condition for linear stability is that where β 2 is the material constant defined in (1.2). Successively, by a clever but formal 1 application of the classical Liapunov method, Rumyantsev [28] proved that the requests in (0.1) in fact ensure

Formulation of the Problem and Preliminary Considerations
Let S := B ∪ L the coupled system constituted by a rigid body B possessing an interior cavity entirely filled with a Navier-Stokes liquid L. More precisely, let Ω and C be two bounded domains of R 3 with C ⊂ Ω. Then Ω\C represents the spatial region occupied by B, while C is the cavity, which we shall suppose throughout of class C 2 . We assume that S moves under the action of the gravity, g, while keeping one of its points, O, fixed at all times with respect to an inertial frame. Denote by I the inertia tensor of S with respect to O, e 1 , e 2 , and e 3 its (ortho-normalized) eigenvectors and A, B, and C corresponding eigenvalues (principal moments of inertia). Moreover, let F := {O, e 1 , e 2 , e 3 } denote the principal frame of inertia. We shall suppose that the center of mass, G, of S belongs to the axis O e 3 , and G = O. Also, we orient F in a way that the only non-zero coordinate, ℓ, of G in F is positive. Thus, the equations governing the motion of S in the body-fixed frame F are given by [21,24] v (1.1) Here v, ρ, µ ≡ ρν and ω denote, respectively, relative velocity, density, shear viscosity coefficient of L and angular velocity of B. Moreover, where p is the pressure field of L, while M is the total mass of S, and g = |g|. Finally, γ := g/g stands for the direction of the gravity, which, in the frame F , is time-dependent. Notice that As it is immediately checked, (1.1) admits a class of time independent (steady-state) solutions (v(x),ω,γ), characterized by the conditions v ≡ 0 ,ω = λγ ,γ = −e 3 , λ ∈ R − {0} . (1.4) From the physical viewpoint, such solutions describe those motions of S, where the liquid is "frozen" in the cavity and S rotates as a whole rigid body around e 3 with constant angular velocity ω, and its center of mass at its highest position ("upright spinning top").
The main objective of this paper is to characterize the stability properties of these motions. To this end, let be a generic "perturbed motion" around (1.4). Then, in view of (1.1) the "perturbation" (v, ω, z) satisfies the following set of equations (1.6) Because of (1.3). the perturbation field z must satisfy the constraint However, dot-multiplying (1.6) 4 a first time by z, a second time by −e 3 and summing side by side the resulting equations, we find d dt As a consequence, (1.7) is equivalent just to require that the initial data z(0) satisfies: Our next objective is to rewrite (1.6) as an evolution equation in an appropriate Hilbert space. To this end, let L 2 σ (C) := {v ∈ L 2 (C) : div v = 0 in C, v · n = 0 on ∂C} and define the Hilbert space endowed with the inner product and associated norm u := u, u .
We then introduce the operators: (1.10) with P Helmholtz projection from L 2 (C) onto L 2 σ (C). From [16, § 6.2.3], we deduce the following. In view of this lemma, we may set where D(L) = D(N) ≡ D(A), and deduce that the system of equations (1.6) can be formally written as an evolution equation in the Hilbert space X: The stability of the steady-state solution (1.4) is then reduced to the stability of the solution u = 0 to (1.12). The study of the latter will be performed as a result of the general stability theory -presented in in next section that is founded upon suitable functional and spectral properties of the linear operator L.

Stability Properties for an Abstract Evolution Problem
Objective of this section is to study the stability of the zero solution to a suitable evolution problem in a Banach space. The peculiarity of this problem is that 0 is an eigenvalue of the relevant linear operator, so that the classical "linearization principle" (e.g. [13, § 5.1]) does not apply. We need, instead, a "generalized linearization principle" in the spirit of [27, Theorem 2.1]. Here, we shall follow the approach of [9], based on an operator fractional powers method, that appears to be more specific and direct for the type of fluid-structure interaction problems considered in this paper; see, however, also [22,23]. In fact, the stability result stated in the following Theorem 2.2 is similar to its counterpart in [9, Theorem 1.1], but obtained under slightly more general assumptions on the nonlinear operator and with a simpler argument. For completeness and reader's sake, in Theorem 2.3 we also provide a short proof of the complementary instability result.
In a (real) Banach space X, we consider the following evolution problem with L and N to be defined next. Let A : X → X be a linear, sectorial operator with compact inverse and Re σ(A) > 0. For α ∈ [0, 1], set It is well known that, for α > 0, X α is a Banach space compactly embedded in X, e.g. [13,Theorem 1.4.8]. Let B : X → X be a linear operator with D(B) ⊃ D(A), and such that We then assume and that, by the properties of A, it follows [14, Theorem 3.17 at p. 214] that L has a compact resolvent and, therefore, a discrete spectrum.
On the operator L, we further assume: and We then have the following.
Lemma 2.1. The space X admits the decomposition Moreover, denoting by Q and P the spectral projections according to the spectral sets

7)
and holds, so that we may take S = N[L], which proves (2.5). The remaining properties stated in the lemma are then a consequence of (2.5) and classical results on spectral theory (e.g., [ We now turn to the operator N. We begin to assume Furthermore, we observe that, by (2.5), every u ∈ X can be written as Thus, setting we suppose there is a non-negative, continuous function ǫ = ǫ(ρ) with ǫ(0) = 0 such that We are now in a position to prove the following stability result; see also [9].
there is a unique corresponding solution u = u(t) to (2.1) for all t > 0, satisfying, for any T > 0, Moreover, the solution u = 0 to (2.1) is exponentially stable in X α , namely: (a) For any ε > 0 there is δ > 0 such that Proof. Under the stated assumptions on A, B and (H4) 1 , the existence of a unique solution u to (2.1) in some time interval (0, t ⋆ ) satisfying (2.8) for each T ∈ (0, t ⋆ ) is guaranteed by classical results on semilinear evolution equations (e.g., [26, p. 196-198]). Moreover, (0, t * ) is maximal, in the sense that either t * = ∞ or else lim t→t * u(t) α = ∞. We shall next show that, in fact, only the former situation occurs for sufficiently "small" initial data. Applying Q and P on both sides of (2.1) and taking into account (2.7) we get Since the operator L, being sectorial, is the generator of an analytic semigroup in X, so is Also, by assumption and the spectral property of L, there is γ > 0 such that which implies that the fractional powers L α 1 , α ∈ (0, 1), are well defined in X (1) . Thus, setting from (2.10) we get From the local existence theory considered earlier on, we know that for any given ρ > 0 there exists an provided u(0) α < η, for some η > 0. Let us show that η and ρ can be chosen sufficiently small so that (2.13) holds also with τ = t ⋆ , thus implying, in particular, that the solution u = u(t) to (2.1) exists for all times t > 0. In fact, suppose, by contradiction, that there is τ 0 < t ⋆ such that (2.14) In view of the stated properties of L 1 it results (e.g. [13, Theorem 1.4.6]), Moreover, we recall that in X (1) it is from (2.12), (2.14)-(2.17), and (H5) we show On the other hand, (2.9) 2 , with the help of (2.14), (H5) and (2.18), furnishes Combining (2.18) and (2.19), and choosing c 3 η/ρ + ǫ(ρ) < 1/4 we conclude in particular contradicting (2.14). As a result, by what we observed early on, we may take t * = ∞ in (2.13) and conclude as well sup proving, as a byproduct, the desired global existence property. Now, employing in (2.12), the inequalities (2.20), (2.15) along with the assumption (H5), we easily deduce, for ρ small enough, namely, Also, using (2.21) into (2.19), we infer Therefore, from (2.22) and (2.23) we recover the stability property stated in (a). Moreover, integrating (2.9) 2 between arbitrary t 1 , t 2 > 0 using (2.22) and reasoning in a way similar to what we did to obtain (2.19) we get Employing this information into (2.24) in the limit t 2 → ∞, and with t 1 = t we show which, once combined with (2.22), proves the exponential rate of decay stated in (b). The proof of the theorem is thus concluded.
We next show the following instability result. 2 Theorem 2.3. Let L be the operator defined in (2.3), and suppose that N satisfies hypotheses (H1)-(H4) Then, if Re σ(L) ∩ (−∞, 0) = ∅ the solution u = 0 to (2.1) is unstable in X α . Precisely, there is ε 0 > 0 with the following property: for any given δ ∈ (0, 1] there is a solution u(t) to (2.1) in the class where (0, t * ) is the maximal interval of existence.
Proof. We follow the ideas of [3, Theorem 2.3 in Chapter VII]. From (2.1) we deduce Re λ .

Preliminary Properties of the Operator L
The main purpose of this section is to find conditions under which the operator L defined in (1.11), (1.10) satisfies the assumptions (H1)-(H3) stated in the previous sections. In this regard we begin to observe that, by the well known properties of the Stokes operator with domain D(A 0 ) := W 2,2 (C) ∩ D 1,2 0 (C) and range L 2 σ (C), it follows that A has a compact inverse and, therefore, a discrete spectrum which, in addition, lies on the positive real axis. Since I −1 is symmetric (and bounded), the operator A enjoys the same properties as A. Furthermore, B is bounded and therefore B satisfies (2.2) with α = 0. We shall now check the validity of the other assumptions.
We start with the following result which, in particular, gives sufficient conditions in order that L satisfies (H1).

(3.4)
By dot-multiplying both sides of (3.4) 1 by v, integrating by parts over C and using (3.4) 2,3 , we deduce v ≡ 0. As a result, the elements of N[L] must have v ≡ 0, while ω and z solve the following equations −λ e 3 × I · ω + e 3 × λ C ω − β 2 z = 0 , Replacing (3.5) 2 into (3.5) 1 and recalling that I = diag (A, B, C) we thus get which, in turn, under the given assumptions furnishes z = ze 3 , z ∈ R. The desired property is then a consequence of the latter and of (3.5) 2 .
We now pass to the investigation of the validity of the assumption (H2).
In order to investigate further properties of the operator L, we need the following.
where ω * := ω − a (with a given in (1.2)) and along the solutions to the linear problem (3.10) Proof. From (1.12) we deduce that the evolution equation in (3.10) is equivalent to the following set of equations (3.11) Due to the analyticity of the semigroup generated by L, solutions to (3.11) with initial data in X are smooth. We dot-multiply both sides of (3.11) 1 by v and integrate by parts over C to get We next dot-multiply both sides of (3.11) 3 by z and both sides of (3.11) 4 by I·ω * , and sum the resulting equations side-by-side. We thus infer By dot-multiplying both sides of (3.11) 3 by ω * + a we show Finally, dot-multiplying both sides of (3.11) 4 by z we obtain If we form the linear combination (3.12) + (3.14) + δ (3.15) − λ (3.13) and use (3.8), we deduce which completes the proof.
A first important consequence of this lemma is provided by the following result that furnishes sufficient conditions for the validity of (H3).
Proof. Contradicting the stated property means that the system (3.11) has at least one time-periodic solution (v, ω * , z) of period T > 0 (say) such that Integrating (3.9) from 0 to T and using the periodicity we deduce T 0 ∇v 2 2 dt = 0 which, in turn, by (3.11) 5 furnishes v ≡ 0. Then, by (3.11) 1 it isω * = −∇p and by operating with curl on both sides we get ω * = const, implying, by (3.16), ω * = 0. The latter in conjunction with (3.11) 3 delivers z = z 3 e 3 that once combined with (3.11) 4 allows us to deduce z 3 = const. Thus, by (3.16) we find also that z = 0, and the proof of the proposition is completed.
We conclude this section by collecting in the following theorem some relevant consequences of Proposition 3.1, Proposition 3.2 and Proposition 3.4. In the next section we shall give detailed information on the spectrum of L.

On the Location of the Spectrum of the Operator L
We now turn to the study of the location of the eigenvalues of the operator L. As a matter of fact, according to the decomposition proved in Lemma 2.1, this amounts to investigate the same property for the restriction, L 1 , of L to the subspace R[L].
To this end, we propose two preparatory results collected in the form of as many lemmas.  for some κ = κ(u(0), A, B, C) > 0.
The dynamics on the ω-limit set is then characterized by v ≡ 0, and ω ≡ ω * , and z satisfyinġ By taking the curl of both sides of the first equation, it immediately followsω = 0, which, in turn, by the second equation, furnishesż i = 0, i = 1, 2. Using this information in the third equation dot-multiplied by e 3 we get alsoż 3 = 0. As a consequence, the previous system reduces to which coincides with (3.5). As a result, proceeding as in the proof of Proposition 3.1, we show Ω(u 0 ) ⊆ N[L].
We are now in a position to prove the main result of this section.
(iii) By contradiction, assume, instead, Reσ(L 1 ) ⊂ (0, ∞). From (3.11) 4 it follows that z 3 = const., so that (3.9) can also be equivalently written as where Thus, integrating (4.12) from 0 to t and arguing exactly like in (ii), we arrive again at (4.11). Now, if C < A, B, we get an immediate contradiction. If A < C < B we take z 01 = 0, thus getting a contradiction again. The case B < C < A is treated analogously by taking z 02 = 0 instead, and this completes the proof of the theorem.

Properties of the Operator N
We will now prove that the nonlinear operator N defined in (1.11), (1.10) satisfies the assumptions in (H4), (H5). The latter, together with the results proved in the preceding two sections, will thus allow us to use Theorem 2.2 and Theorem 2.3, and provide a complete characterization of the stability and asymptotic stability of the steady motions for (1.1); see Section 6.
Let us use the canonical decomposition and the operator A 0 is the Stokes operator introduced in (3.1). Then, where [·, ·] α denotes the complex interpolation. The nonlinear operator N defined in (1.11), (1.10) has the bilinear structure N(u) = B(u, u) with B : we easily deduce, for every v, v 1 and v 2 ∈ Y α , and α ∈ (3/4, 1) that This shows that the remaining conditions in (H4) are satisfied with p = 2 and α as above. We next pass to the proof of (H5). To this end, let Then, from (1.10) we deduce (5.4) Thus, if K A , K B = 1, from Proposition 3.1 we infer

Nonlinear Stability Properties
In view of the results obtained in the previous three sections, we are now able to employ the general theory developed in Section 2, and use Theorem 2.2 and Theorem 2.3 to provide a complete characterization of the stability and asymptotic stability properties of the steady-state motion (1.4).
We begin to show the following nonlinear stability result. Then, the steady-state (1.4) is exponentially stable in X α , namely, for any ε > 0 there is δ > 0 such that (v(t), ω(t), z(t)) α < ε . all the above limits occurring at an exponential rate.
Proof. By Theorem 4.3, we know that, under the stated assumptions (6.1), we have σ(L 1 ) ⊂ (0, ∞). Therefore, in view of what we have shown in Proposition 3.1-Proposition 3.4, and Section 5, 3 we may employ Theorem 2.2 and deduce (6.2) and (6.3) 1,2 . Furthermore, there exists z ∈ R such that lim t→∞ z(t) = ze 3 , at exponential rate. We now observe that the perturbation field z(t) must satisfy (1.7) (or, equivalently, the data satisfy (1.8)). This implies that either z = 2 or else z = 0. In the first case, taking into account (1.5), we deduce that the terminal state of the coupled system will be of the type (v ≡ 0, ω = rγ, γ = e 3 ) which, as it is easily checked, is a steady-state solution to (1.1) corresponding to a constant rigid rotation of S around e 3 with its center of mass G in its lowest position. However, by the stability property (6.1), G must be at all times in a neighborhood of its highest position. Therefore, we can only have z = 0, which completes the proof of the theorem. Remark 6.2. As an illustration of the results obtained in the previous theorem, consider a "classical" symmetric top, T, spinning at sufficiently fast rate around its axis a in the vertical direction, d, passing through the fixed point O and center of mass G (in its highest position). It is then well known (e.g. [25,Example 9.7C]) that a small disorientation of a from d will produce a stable precession of T around d with a performing small oscillations (nutation). If, however, T possess an interior cavity filled up with a viscous liquid, Theorem 6.1 tells us that under the same above circumstances, the axis a will eventually reposition itself in the vertical direction d, at an exponential rate. This fact provides a further example of the stabilizing influence of an interior liquid-filled cavity on the motion of a rigid body [4,9,11,20,21,22,23].
We also have the following instability result, which is an immediate consequence of Proposition 3.1, Proposition 3.2, Theorem 4.3, Section 5, and Theorem 2.3. Theorem 6.3 (Nonlinear instability). With the same notation as in Theorem 6.1, the steady state (1.4) is unstable in X α if either C > M , and λ 2 < β 2 C − M .
or C < M.