Dynamics of an elastic satellite with internal friction.

Front matter

Emanuele Haus Dynamics of an elastic satellite with internal friction Asymptotic stability vs collision or expulsion Ledizioni LediPublishing

Emanuele Haus Department of Mathematics University of Milan Via C. Saldini n.50 20133 Milan - Italy Advisor: Dario Paolo Bambusi Department of Mathematics University of Milan Via C. Saldini 50, Milan, Italy Series Editors: Bert van Geemen, Coordinator of the PhD Program in Mathematics; Giovanni Naldi, Coordinator of the PhD Program in Mathematics and Statistics for Computational Sciences; Luca Barbieri Viale, Director of the Graduate School.

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CHAPTER 2. ASYMPTOTIC BEHA VIOR OF A SATELLITE 55 comes

Let us first consider the outcome (i), i.e. the case of unbounded orbit. The fact that the trajectory of the center of mass of the satellite is unbounded does not necessarily mean that it tends to infinity. In fact, one cannot a priori rule out the possibility that, along with an unbounded trajectory of the center of mass, there existR > 0 and tn→ +∞ such that|X(tn)|<R . Anyway, if one can prove the orbital stability of the synchronous resonance, then a simple application of 2.3.6 proves that an unbounded trajectory of the center of mass must actually tend to infinity. In fact, under the assumption of orbital stability, one proves that, if there wereR> 0 and tn→ +∞ such that|X(tn)| < R, then the orbit should be asymptotic to the synchronous resonance, which cannot be the case if the orbit is unbounded.

The outcome (ii) has the clear meaning of a planet-satellite collision and does not need any further explanation. The outcome (iii) is the one we informally refer to as asymptotically being trapped into the synchronous resonance, since the characterization explained in Section 2.4 shows that the only possible non-dissipating behavior of the satellite is that of synchronous resonance, in the sense that the satellite always shows the same face to the planet, revolving about it at a fixed distance. Actually, in the least degenerate case, where the non-dissipating invariant manifold NDinv is made up of only one orbit, the outcome (iii) clearly implies

CHAPTER 2. ASYMPTOTIC BEHA VIOR OF A SATELLITE 56 asymptoticity to such an orbit; however, in presence of very degenerate non-dissipating invariant manifolds, the outcome (iii) could a priori not result in asymptoticity to a synchronous resonant orbit. In order to perform some more accurate analysis of the manifold NDinv, one should add some assumptions about the structure of the satellite. However, in the next chapter we analyze, in the planar approximation (which we will specify in the next chapter), the two relevant cases of a triaxial satellite and of a satellite with spherical symmetry. In the triaxial case, the non-dissipating invariant manifold turns out to be completely non-degenerate; in the spherically symmetric case, the non-dissipating invariant manifold turns out to be a slightly degenerate one, in a sense that we will discuss later. In such a slightly degenerate situation, the outcome (iii) can still be referred to as asymptotically approaching the synchronous resonance. A remarkable fact is that, anyway, the discussion of the present chapter excludes the possibility that other periodic orbits exist, different from the synchronous resonant ones. This may seem quite surprising, since, for instance, some celestial bodies are known to be trapped in spin-orbit resonances different from the synchronous one (think, for example, of the 3:2 of Mercury, seen as a satellite of the Sun). At the same time, from our point of view, it is natural to think that any periodic orbit different from the synchronous resonant one would cause some periodic deformation of the satellite. Such a periodic deformation would, in turn, produce some periodic dissipation.

CHAPTER 2. ASYMPTOTIC BEHA VIOR OF A SATELLITE 57 But one cannot continue dissipating periodically the same amount of energy for an infinite time, since this would lead the energy to −∞, implying that a collision is going to occur. Our interpretation is that situations like the 3:2 resonance of Mercury are very likely to be metastable, and it would be very interesting to investigate further in this direction. For instance, one could try to give an estimate of the time of such a metastability: it would not be so surprising if such a time were of the same order of magnitude as the estimated lifespan of the Solar system.

Chapter 3 Orbital stability In this chapter, as we did in the previous one, we will deal with a dynamical system made up of a a pointlike massM (the “planet”), whose space coordinates are fixed at the origin of the Euclidean space R3, and an elastic body (the “satellite”) with internal friction, subject to the force of gravity exerted byM.

Our aim here is to prove the orbital stability of the synchronous resonance. We will give the proof of the stability in two cases: (i) triaxial satellite; (ii) satellite with spherical symmetry.

The proof of the orbital stability will be done under some approximations. Namely, we will make the planar approximation (which we

CHAPTER 3. ORBITAL STABILITY 60 will explain later) and we will truncate the expansion of the gravitational potential at the quadrupole terms, neglecting higher order effects.

In both cases, we make the convenient assumption that the elasticity moduli of the satellite are very large. Before making some kinematic considerations, let us say what we mean by “triaxial satellite”. To this end, we have to recall some basic facts about moments and axes of inertia.

We first need to recall what the matrix of inertia is. In principle, one may evaluate the matrix of inertia of an extended body with respect to any point in space; however, we will always refer to the matrix of inertia evaluated with respect to the center of mass. Let B be an extended body which occupies a volumeV in the three-dimensional Euclidean space. Fix a Cartesian frame of reference (x,y,z ), with origin in the center of mass ofB and letρ(x,y,z ) be the density function ofB. Then the matrix of inertia ofB is a symmetric matrix ) ]) I11 I12 I13 I12 I22 I23 I13 I23 I33 ( [ where I11 = ∫ ∫ ∫ V ρ(x,y,z )(y2 +z2)dxdydz (3.0.1) I22 = ∫ ∫ ∫ V ρ(x,y,z )(x2 +z2)dxdydz (3.0.2)

CHAPTER 3. ORBITAL STABILITY 61 I33 = ∫ ∫ ∫ V ρ(x,y,z )(x2 +y2)dxdydz (3.0.3) I12 =− ∫ ∫ ∫ V xyρ(x,y,z )dxdydz (3.0.4) I13 =− ∫ ∫ ∫ V xzρ(x,y,z )dxdydz (3.0.5) I23 =− ∫ ∫ ∫ V yzρ(x,y,z )dxdydz . (3.0.6) Since the matrix of inertia is a real symmetric matrix, it has real eigenvalues. The eigenvalues of the matrix of inertia are theprincipal moments of inertiaof B. If the three eigenvalues are distinct, then the directions of the three associated eigenvectors are well defined and individuate theprincipal axes of inertiaof B. As we will show later, the principal moments and axes of inertia are very relevant objects in the study of our problem. On the one side, the kinematic study of extended bodies rotating in space involves the concept of moment of inertia. On the other side, which is more specific to our problem, if one expands in multipoles the gravitational interaction of an extended body with a pointlike mass and neglects terms beyond quadrupole, the only relevant parameters of the extended body (apart from its mass and the position of its center of mass) are found to be the principal moments of inertia and the directions of the principal axes of inertia. A rigid body is called triaxial if its three moments of inertia are distinct. In our framework, we deal with a body (the satellite) which

CHAPTER 3. ORBITAL STABILITY 62 is not rigid. When we talk about triaxiality of the satellite, we mean thatthesatelliteistriaxialwhenitreachesitsconfigurationofequilibrium between the elastic and self-gravitating forces. In other words, the satellite is triaxial if the three eigenvalues of the matrix of inertia are distinct when the satellite is in such an equilibrium configuration. In the triaxial case, the orbital stability of the synchronous resonance is a consequence of the orbital stability for a triaxialrigid body, the deformable case being a small perturbation of the rigid one.

In the spherically symmetric case, instead, there is no orbital stability of the synchronous resonance for a rigid body, and such a stability appears as a consequence of the tidal deformation of the satellite.

For this reason, the proof of orbital stability will be quite straightforward in the triaxial case, while it will require a careful and rather technical kinematic analysis in the spherically symmetric case. 3.1 General setting and global rotations Let us now recall some notation we have already used in the previous chapter, only to fix ideas. Let Ω be the material space and let ζ : Ω → R3 be the body. In this chapter we assume that, in the reference configurationζ0(x) = x, elastic forces and forces of self-gravitation are in equilibrium in the satellite. As in the previous chapter, we denote byρ0 : Ω → R the density function in the

CHAPTER 3. ORBITAL STABILITY 63 reference configuration and byρ : Ω→ R the density function in a generic configurationζ, which is related toρ0 by (2.1.8). Without loss of generality, we assume that the center of mass in the reference configuration is at the origin, i.e. ∫ Ω xρ0(x)d3x = 0 . (3.1.1) The map ζ describes both the deformation of the body and its position and rotation in space, so it is natural to try to decomposeζ into a rigid translation, a rigid rotation and an internal deformation. As in the previous chapter, we define the center of mass of the body by X = 1m ∫ Ω ζ(x)ρ(x)d3x , (3.1.2) and decompose the configuration vector fieldζ as ζ(x) =X +v(x) , (3.1.3) where v is such that ∫ Ω v(x)ρ(x)d3x = 0 . (3.1.4) Denote byC the space of thev’s such that (3.1.4) holds. Remark 3.1.1. The spaceC is an infinite dimensional function space, so in order to discuss dynamics one should introduce a suitable norm in it, prove an existence and uniqueness theorem for the solutions to the Cauchy problem and, in order to use energy conservation

CHAPTER 3. ORBITAL STABILITY 64 (or dissipation) to prove dynamical properties, one should also prove that dynamics is well posed in the energy space, which is in general unknown [26].

Here, we do not want to enter such a kind of mathematical problems, so we will simply assume existence and uniqueness for the solutions to the Cauchy problem and well-posedness of the dynamics in the energy space.

Now the body configuration is uniquely parameterized through the position X of the center of mass and the functionv∈C . The reference configuration corresponds toX0 = 0,v 0(x) =x. Then we would like to factor out rotations in a way similar to translations, however this requires a careful discussion. The point is that it is clear what it means to rotate a body, but it is not clear how to say that a deformation does not rotate the body: as we will see, this is not a well defined concept.

To understand this point, we recall the standard analysis of the local deformation in linear elasticity theory, as explained in the previous chapter. Fix a pointx0∈ Ω. Then, the displacement vector field is defined byu(x) :=ζ(x)−x. The gradient∇u(x0) is decomposed as the sum of its symmetric partε and its skew-symmetric partω. Then, at the pointx0, the local deformation is described by the strain tensor ε, while the skew-symmetric part of the gradientω describes the local rotation. Thus, at a local level, the separation between deformation and rotation is well-defined and completely standard.

CHAPTER 3. ORBITAL STABILITY 65 On the other hand, when considering the global configuration, the situation is more complicated, because it is not trivial at all to answer the following question. Let the configurationζ be assigned. Is the corresponding displacement vector fieldu a “pure deformation”, in the sense that it does not “globally rotate the body”, or is it given by the composition of a rotation of the body with some “pure deformation”? The answer to this question is easy if one considers only affine deformations, i.e. if one allows only displacements whose gradient is spatially constant. In fact, in this case, the rotation is simply described by the skew-symmetric part of the gradient of the displacement (evaluated at any point, since it is constant). Nevertheless, if onewantstodealwithgeneral displacementfields, no obvious answer can be given to the previous question. One could try to give some reasonable definitions of what a “pure deformation” is. For instance, we could fixx0∈ Ω and say thatu is a “pure deformation” if the displacement,locally at x0, does not contain any rotation, i.e. ∇u(x0) is symmetric. Anyway, this seems quite arbitrary; moreover, different choices of x0 would result in different definitions of “pure deformation”.

In order to make it independent of the particular choice of a point, one could try to give the definition of “pure deformation” through an integral condition. This is done, for instance, in [31,35], where such a definition (in the case of an incompressible homogeneous body) is

CHAPTER 3. ORBITAL STABILITY 66 given by imposing that the integral of the curl vanishes, i.e. ∫ Ω curl(u(x))d3x = 0 . (3.1.5) But the choice of this definition is again arbitrary. The real point is that there is no natural way of defining what it means that a displacement does not globally rotate the body. However, it is even trivial to explain what it means to rotate a body or to say that a configuration is obtained by rotating another configuration. Mathematically, this corresponds to the fact that there exists a group actionA1 of the rotation groupSO(3) on the configuration spaceC, defined by A1 : SO(3)×C →C (Γ,v ) ↦→ Γv . (3.1.6) Thisgroupactionallowsonetointroduceastructureofprincipalfibre bundle inC, the base manifold being the quotientM :=C/SO(3). The elements of such a quotient manifold are what one could call “pure deformations”.

The fact that the quotient of a manifold under a group action is still a manifold is not always true, so we have to recall some basic facts about group actions in order to justify our assertion. Definition 3.1.2. Let G be a group acting on a setX. The action is said to befree if the following condition is verified: if there exist x∈X and g∈G such thatgx =x, theng is the identity.

CHAPTER 3. ORBITAL STABILITY 67 Definition 3.1.3. The action of a topological groupG on a topological spaceX is said to beproperif the mapping G×X→X×X (g,x )↦→ (gx,x ) is proper, i.e., inverses of compact sets are compact. Theorem 3.1.4.LetG be a topological group acting on a topological spaceM. IfG is compact, then the action is proper. Theorem 3.1.5. If a Lie group G acts freely and properly on a smooth manifold, then the quotientM/G is a smooth manifold. In our case, the fact thatSO(3) is a compact group and that the actionA1 is obviously free implies thatM is a manifold. As usual in this geometric context, it is useful to introduce coordinates in which a point ofC is represented by an element ofSO(3) and an element of the base manifold. However a concrete representation of the elements of the quotient manifold can be obtained only locally, by introducing a local section of the bundle, namely by choosing a submanifold S ofC, transversal to the group orbit. Now it is clear that there are infinitely many possible choices of the section of the bundle, which correspond to infinitely many admissible definitions of “pure deformations”. Nonetheless, the physics is independent of the choice of the sectionS: a change in the choice of the section simply results in a change of coordinates. The fact that the section is only local is not a problem, as long as only small deformations are allowed.

CHAPTER 3. ORBITAL STABILITY 68 Now, letA1(v0) be the orbit of the reference configuration under the action of the groupSO(3). Once the section has been chosen, it naturally induces a smooth one-to-one correspondence between a neighborhood ofA1(v0) (obtained as the union of the orbits of the points of S under the group actionA1) andSO(3)× S. Such a correspondence allows one to parameterize the spaceC of configurations through an element of the groupSO(3) and a point of the section S. In the context of classical mechanics, this is useful, since, due to the isotropy of space, the Lagrangian of the body is going to be independent from the element ofSO(3) and will depend only on the variable belonging toS.

Weremarkthatthemachinerywehavejustintroduceddescribesa general fact, which is independent of all the assumptions we will make later in the paper. We can summarize the result of our discussion in the following theorem: Theorem 3.1.6. There exist a neighborhoodU⊂C ofA1(v0) and a one-to-one smooth functionf :SO(3)×S→U with the property that f(Γ,w ) = A1(Γ)w. Therefore, Γ and w can be used as coordinates on the configuration spaceC.

CHAPTER 3. ORBITAL STABILITY 69 3.2 The matrix of inertia as a function of the configuration Since the satellite is deformable, in our setting the matrix of inertia of the satellite is not a fixed object, but it is a function of the body configuration. In a compact notation, the elements{Iij}3i,j=1 of the inertia matrixI are given by the following formula: Iij =Iij(v) :=ei· ∫ Ω v(x)∧ (ej∧v(x))ρ(x)d3x . (3.2.1) In the rest of the chapter we will always denote with the double subscript the elementsIij of the matrix of inertia, while the principal moments of inertia (i.e., the eigenvalues of the matrix I) will be denoted byI1, I2, I3. Again, we remark that the principal moments of inertia are a function of the body configuration. The principal axes of inertia are also functions of the body configuration and we will denote them byu1, u2, u3. 3.3 Planar restriction We are going to study the dynamics of the satellite in the special case when the motion of the center of mass of the satellite is planar and the spin axis of the body is orthogonal to the plane of the orbit and coincides with one of the principal axes of inertia of the body. For this reason, we will restrict to “planar motion of the center of mass”, “planar deformations” and “planar rotations”. Precisely, we

CHAPTER 3. ORBITAL STABILITY 70 make the following assumptions. (Here and below we denote bye1, e2, e3 the vectors of the canonical base ofR3.) Assumption 6 (Planar motion of the center of mass). We assume that X lies in the plane generated bye1 and e2. Assumption 7 (Planar deformations). The configuration is such that e3 is an eigenvector ofI. We label the principal axes of inertia so that u3 = e3.

In other words, we are assuming the matrix of inertia of the satellite to be of the form I(v) = ) ]) I11(v) I12(v) 0 I12(v) I22(v) 0 0 0 I33(v) ( [ , (3.3.1) so thatI3(v)≡I33(v).

Assumption 8(Planar rotations). Γ is a rotation about thee3-axis, i.e. it has the form Γ = Γ(α) := ) ]) cosα − sinα 0 sinα cosα 0 0 0 1 ( [ . (3.3.2) 3.4 The gravitational interaction In this section we prove that, in the quadrupole approximation, the gravitational potential of the satellite, in the Newtonian field generated byM, depends only on:

CHAPTER 3. ORBITAL STABILITY 71 (i) the mass of the satellite; (ii) the position of the center of mass of the satellite; (iii) the principal moments of inertia of the satellite; (iv) the orientation of the principal axes of inertia of the satellite. To start with, we fix some notation:(R,ψ ) are the polar coordinates of the center of mass of the satellite in the plane of the orbit. We denote withγ the angle between the principal axisu1 and the line joining the planet to the center of mass of the satellite. Such a line is usually referred to as theline of centers. Then, we have the following result.

Proposition 3.4.1. In the quadrupole approximation the gravitational potential energy of the body in the field generated by the mass M is given by Vg(X,I 1,I 2,I 3,γ ) :=−GMmR +GMR3 [−I1+2I2−I3+3(I1−I2) cos2γ]. (3.4.1) Proof. Let (x1,x 2,x 3) be the Cartesian coordinates referred to the system with originX and axesu1u2u3. Then we introduce the spherical coordinates(r,ϑ,φ ) of the generic pointPin the satellite, defined

CHAPTER 3. ORBITAL STABILITY 73 The gravitational potential energyVg is: Vg =− ∫ ζ(Ω) GMρ(ξ) |ξ| d3ξ =− ∫ ζ(Ω) GMρ(ξ)√ R2 +r2− 2Rr cosηd3ξ , (3.4.14) whereη is the angle between the line of centers andXP. Notice that the relation cosη = cosφ cos (ϑ +γ) (3.4.15) holds. Let us recall now how the multipole expansion arises. We have |ξ| = 1√R2 +r2− 2Rr cosη = 1R 1∑1 +(rR{2− 2(rR{ cosη . (3.4.16) In terms of the Legendre polynomialsPn(z), one has 1√ 1 +x2− 2xz =  n≥0 xnPn(z) . (3.4.17) In particular, we recall that P0(z) = 1 P1(z) =z P2(z) = 3z2− 12 .

Taking the quadrupole approximation means to cut the sum atn = 2. We get |ξ| = 1R n≥0 (r R {n Pn(cosη)≃ 1R  1 + rR cosη +(rR{2 3 cos2η− 12  , (3.4.18)

CHAPTER 3. ORBITAL STABILITY 75 The proposition we have proved in this section is the reason why it is convenient to use the principal moments of inertia of the satellite as Lagrangian coordinates in the study of our dynamical system. 3.5 Triaxial satellite As we said above, triaxiality means that the three principal moments of inertia of the satellite in its reference configurationv0 are all distinct. Without loss of generality, we may assume thate1, e2 and e3 are the principal axes of inertia of the satellite in the reference configuration.

Therefore, in the present section, we will assume that the following assumption is satisfied.

Assumption 9(Triaxiality). The reference configurationv0 is such that I12(v0) = 0 and I1(v0) =I11(v0)<I 2(v0) =I22(v0)<I 3(v0) =I33(v0) . (3.5.1) Remark 3.5.1. The assumption onI3(v0) is actually useless, once we have done the planar restriction. However, this is the correct assumption that one should make in order to look for stability of the complete three-dimensional model, without the assumption of planarity. We leave the assumption in this form, in order to make physically relevant the situation that we are studying. Anyway, in the planar model, we could only assumeI1(v0)⁄= I2(v0), the sign of the inequality betweenI1 andI2 being determined by a choice of notation.

CHAPTER 3. ORBITAL STABILITY 76 We are now ready to prove the orbital stability of the synchronous resonance for a triaxial satellite. The scheme of the proof is the following: we will first prove orbital stability for the conservative Lagrangian system that one obtains neglecting internal friction. Such a proof involves conservation of energy. Then we observe that, if one adds some extra dissipative force which makes energy a nonincreasing function of time, then one still has orbital stability (in the future). 3.5.1 Lagrangian coordinates By virtue of Assumption 8 and Theorem 3.1.6, we can use the following set of Lagrangian coordinates: • the polar coordinates(R,ψ ) of the center of mass of the satellite; • the angleχ :=α−ψ, describing the rigid rotation of the satellite, measured with respect to the line of centers; • a set of Lagrangian coordinates forw∈ S.

We start with the idea of using the principal moments of inertia and an angle describing the orientation of the principal axes of inertia as Lagrangian coordinates; however, there is the problem of singularity in the definition of the axes of inertia when two principal moments of inertia have the same value. In the triaxial case, since we need not

CHAPTER 3. ORBITAL STABILITY 77 exploit any further symmetry (as, instead, we will have to do in the spherically symmetric case), we can get rid of the problem of such a singularity by simply using the matrix elementsIij(w) as coordinates instead of the eigenvalues and eigenvectors of the matrix of inertia. In order to do this, we need the following assumption. Assumption 10. The functions Iij : S→ R , (i,j ) = (1, 1), (2, 2), (3, 3), (1, 2) are independent in a neighborhood ofv0.

Remark 3.5.2. The previous assumption is satisfied, for instance, if for any ij there exists a deformation which modifiesIij, leaving unaltered the other elements of the matrixI. The same assumption would not be satisfied if, for example, one added some additional constraint, like the incompressibility constraint. In that case, one would have to drop one degree of freedom.

If the assumption of independence is satisfied, then one can complete the set ofIij’s to a system of coordinates on the local section S, which is expressed by the following proposition. Proposition 3.5.3. There exist functions(z1,z 2,... ), with zj : S→ R (j = 1, 2,... ) such that (I11,I 22,I 33,I 12,z 1,z 2,... ) is a set of smooth coordinates on S.

CHAPTER 3. ORBITAL STABILITY 78 Without loss of generality (just by applying a translation), we may also assumezj(v0) = 0.

Now that we have a set of Lagrangian coordinates, we seek for the expression of the Lagrangian function in these coordinates, which gives the equations of motion for the conservative system. 3.5.2 Potential energy The potential energy is the sum of three terms: (1) the potential energy of the satellite in the gravitational field generated byM, (2) the elastic potential energy, (3) the energy of self-gravitation of the satellite.

Gravitational potential energy Since we are using as coordinates theIij’s instead of theIj’s and the angle γ, we have to adjust, in terms of the new coordinates, the expression of the gravitational potential that was calculated in Proposition 3.4.1.

We obtain the following proposition.

Proposition 3.5.4. In the quadrupole approximation the gravitational potential energy of the body in the field generated by the mass M is given by Vg(R,χ,I 11,I 22,I 33,I 12) =−GMmR + (3.5.2) +GMR3 [−I11 + 2I22−I33 + 3(I11−I22) cos2χ− 3I12 sin(2χ)] .

CHAPTER 3. ORBITAL STABILITY 79 Proof. We follow the proof of Proposition 3.4.1. Everything works in the same way, except for the fact that we replaceγ with χ and that, correspondingly, in the frame co-rotating with the satellite, the product of inertiaI12 does not vanish. Then we have the thesis. Elastic and self-gravitational energy In our model, the elastic and self-gravitational forces play, in some sense, the same role, since the combined action of both forces tends to restore the reference configuration of the body and since they are both independent of the rotation angleχ. Therefore, we will not distinguish between the two corresponding potential energies. (With an abuse of language, we will refer to the sum of the elastic and self-gravitational potential energies simply as “elastic potential energy”.) We will also assume that the satellite has very large moduli of elasticity, which will correspond to having a very small parameter ϵ. Using the notation I = (I11,I 22,I 33,I 12), z = (z1,z 2,... ), we summarize these facts in the following assumption. Assumption 11. The elastic potential energy has the form Ve(I,z ) = 1ϵV0(I,z ) , (3.5.3) whereϵ is a small parameterQ andV0 has a nondegenerate minimum at (I,z ) = (I0, 0), having setI0 := (I11(v0),I 22(v0),I 33(v0),I 12(v0)).

CHAPTER 3. ORBITAL STABILITY 80 3.5.3 Kinetic energy When evaluating the expression for the kinetic energy, it is convenient to express it in terms of the angleα = χ +ψ, which describes the actual speed of rotation of the satellite with respect to an inertial frame of reference. By König’s second Theorem, the kinetic energy T can be written as the sum of two terms: the former is the kinetic energy of the center of mass Tcm = m2 ( ˙R2 +R2 ˙ψ2) (3.5.4) and the latter is the kinetic energy of the satellite with respect to its center of mass Tr :=Tr( ˙α, ˙I, ˙z;I,z ) . (3.5.5) Remark 3.5.5.Tr is independent ofα and due to theA1-invariance. Since the kinetic energy with respect to the center of mass is a quadratic form in the corresponding velocities, we will use the notation Tr := 12 +∞ i,k=1 aik(I,z ) ˙qi ˙qk , (3.5.6) where q = (α,I 11,I 22,I 33,I 12,z 1,z 2,... ). Observe that the coefficients aik(I,z ) are such that the quadratic form is positive definite onC.

Lemma 3.5.6. The coefficienta11(I,z ) satisfies a11(I,z ) =I33 . (3.5.7)

As the vector fieldw(x) is independent ofα, we observe thatTr is the sum of three integrals, the first of which gives the term in˙α2, while the third one is a quadratic form in( ˙I, ˙z) and the second one gives mixed terms in˙α and in the other velocities. Therefore, one gets a11(I,z ) = ∫ Ω {u3× [x +u(x)]}2ρ(x)d3x , (3.5.11)

CHAPTER 3. ORBITAL STABILITY 82 and it can easily be seen that this expression equals the moment of inertia related to the vertical axis, which concludes the proof of the lemma. 3.5.4 The reduced Lagrangian The Lagrangian of the system is given by L =T−V =Tcm +Tr−Vg−Ve , (3.5.12) where each of the terms is given by the expressions calculated in the previous sections. As a result, one gets L = m2 ( ˙R2 +R2 ˙ψ2{ +Tr( ˙χ + ˙ψ, ˙I, ˙z;I,z ) +GMmR + + GMR3 I11− 2I22 +I33 + 3(I22−I11) cos2χ + 3I12 sin(2χ) + −Ve(I,z ;ϵ) . (3.5.13) Now, we observe that the Lagrangian does not depend on the cyclic coordinateψ, so the total angular momentum p := ∂L ∂ ˙ψ =mR2 ˙ψ +I33( ˙χ + ˙ψ) + 2 +∞ k=2 a1k(I,z ) ˙qk , (3.5.14) is a constant of motion. We can invert relation (3.5.14), to get the expression of ˙ψ as a function of the other variables: ˙ψ = p−I33 ˙χ− 2 ∑+∞ k=2a1k(I,z ) ˙qk mR2 +I33 . (3.5.15)

CHAPTER 3. ORBITAL STABILITY 83 Then, we can drop one degree of freedom and study the reduced Lagrangian L∗ =L− ˙ψ∂L ∂ ˙ψ , (3.5.16) where ˙ψ must be thought of as a function of the other Lagrangian coordinates and velocities. After some calculations, we get L∗ =T2 +T1− ˜V , (3.5.17) where T2 = m2 ˙R2 +Tr( ˙χ, ˙I, ˙z;I,z )−  I33 ˙χ + 2 ∑+∞ k=2a1k(I,z ) ˙qk 2 2(mR2 +I33) T1 = p  I33 ˙χ + 2 ∑+∞ k=2a1k(I,z ) ˙qk  mR2 +I33 ˜V = p2 2(mR2 +I33)− GMmR + − GMR3 I11− 2I22 +I33 + 3(I22−I11) cos2χ + 3I12 sin(2χ) + + Ve(J,z ;ε) .