Oscillatory motions in the Restricted 3-Body Problem: A Functional Analytic Approach

After Sitnikov’s work, Moser gave a new proof of the existence of oscillatory motions in the Sitnikov problem [Mos01]. His approach makes use of tools from the geometric theory of dynamical systems, in particular, hyperbolic dynamics. More concretely, Moser considered an invariant periodic orbit “at infinity” (see Section 1.3) which is degenerate (the linearized vector field vanishes) but posseses stable and unstable invariant manifolds. Then, he proved that its stable and unstable manifolds intersect transversally. Close to this intersection, he built a section $\Sigma$ transverse to the flow and established the existence of a (non trivial) locally maximal hyperbolic set $\mathcal{X}$ for the Poincaré map $\Phi_{\Sigma}$ induced on $\Sigma$. The dynamics of $\Phi_{\Sigma}$ restricted to $\mathcal{X}\subset\Sigma$ is moreover conjugated to the shift

acting on the space of infinite sequences. Namely, $\mathcal{X}$ is a horseshoe with ”infinitely many legs” for the map $\Phi_{\Sigma}$. By construction, sequences $\omega=(\dots,\omega_{-n},\omega_{-n+1}\dots\omega_{0},\dots\omega_{n-1}\dots\omega_{n}\dots)\in\mathbb{N}^{\mathbb{Z}}$ for which $\limsup_{n\to\infty}\omega_{n}$ (respectively $\limsup_{n\to-\infty}\omega_{n}$) correspond to complete motions of the Sitnikov problem which are oscillatory in the future (in the past).

Moser’s ideas have been very influential. In [LS80a] Simó and Llibre implemented Moser’s approach in the Restricted Circular $3$-body Problem (RC3BP) in the region of the phase space with large Jacobi constant provided the values of the ratio between the masses of the massive bodies is small enough. Their result was later extended by Xia [Xia94] and closed by Guardia, Martín and Seara in [GMS16] where oscillatory motions for the RC3BP for all mass ratios are constructed in the region of the phase space with large Jacobi constant. The same result is obtained in [CGM⁺21] for low values of the Jacobi constant relying on a computer assisted proof. In [GSMS17] and [SZ20], the Moser approach is applied to the Restricted Elliptic $3$-body Problem and the Restricted 4 Body Problem respectively. For the $3$-body Problem, results in certain symmetric configurations (which reduce the dimension of the phase space) were obtained in [Ale69] and [LS80b]. Another interesting result, which however holds for non generic choices of the 3 masses, is obtained in [Moe07]. In the recent preprint [GMPS22], the first author together with Guardia, Martín and Seara, has proved the existence of oscillatory motions in the planar $3$-body Problem (5 dimensional phase space after symplectic reductions) for all choices of the masses (except all equal) and large total angular momentum.

The first main ingredient in Moser’s strategy is the detection of a transversal intersection between the invariant manifolds of the periodic orbit at infinity. Yet, checking the occurrence of this phenomenon in a physical model is rather problematic, and in general little can be said except for perturbations of integrable systems with a hyperbolic fixed point whose stable and unstable manifolds coincide along a homoclinic manifold. As far as the authors know, all the previous works concerning the existence of oscillatory motions in the $3$-body Problem (restricted or not) adopt a perturbative approach to prove the existence of transversal intersections between the stable and unstable manifolds of infinity. In some cases the perturbative regime is obtained by assuming that certain parameter related to the motion of the massive bodies (in general the ratio between the masses of the massive bodies or the eccentriticy of their orbit) is small, and others by working in a region of the phase space where the massless body is located far away from the primaries. The latter situation falls in what is usually called singular perturbation theory and (in general) needs a much more involved analysis than the former one, usually referred to as regular perturbation theory.

The second key ingredient is the construction of a horseshoe close to the transversal intersections of the invariant manifolds. For the Sitnikov and Isosceles Restricted $3$-body Problem (which is introduced in Section 1.3) are non autonomous Hamiltonian systems with $1+1/2$ degrees of freedom ($3$ dimensional phase space), its dynamics can be reduced to the study of a two dimensional area preserving map in which the periodic orbit at infinity becomes a fixed point which, despite being degenerate, behaves as a hyperbolic fixed point. The same happens in the RC3BP after reducing by rotational symmetry and in certain symmetric configurations of the 3BP. In all of these problems Moser’s ideas for constructing a horseshoe close to the transverse intersections between the invariant manifolds of the parabolic fixed point can be implemented directly. In the planar $3$-body Problem, the dynamics can be reduced to a 4 dimensional symplectic map and the parabolic fixed point becomes a 2 dimensional (degenerate) normally hyperbolic invariant manifold. Due to the existence of central directions the construction of the horseshoe in [GMPS22] becomes much more involved. In [Moe07], the author analyzes orbits which pass close to triple collision. In this setting, the close encounters with triple collision, produce stretching also in the central directions.

An approach different in nature from Moser’s is developed by Galante and Kaloshin in [GK11]. By making use of Aubry-Mather theory and semi-infinite regions of instability, the authors prove the existence of oscillatory orbits for the RC3BP with a realistic value of the mass ratio.

Besides the question of its existence, is the question of their abundance. In the conference in honor of the 70th anniversary of Alexeev, Arnol’d posed the following question (cfr [GK11]).

This question was considered by Arnol’d to be the fundamental issue of Celestial Mechanics. It has been conjectured by Alexeev that the Lebesgue measure is zero. Neverthless, this conjecture remains wide open. The only partial results in this direction are due to Gorodetski and Kaloshin [GK12]. They consider the RC3BP and the Sitnikov problem and prove that for both problems and a Baire generic subset of an open set of parameters (eccentricity in the Sitnikov problem and mass ratio in the RC3BP), the Hausdorff dimension of the set of oscillatory motions is maximal.

In the present work we consider a particular configuration of the Restricted $3$-body Problem known as the Restricted Isosceles $3$-body Problem. In this configuration, the two primaries have equal masses $m_{0}=m_{1}=1/2$ and move periodically on a degenerate ellipse of eccentricity one (a line), according to the Kepler laws for the motion of the $2$-body Problem. The massless particle moves on the plane perpendicular to the line along which the primaries move (see Figure 1.1).

In the plane of motion of the massless body we fix a Cartesian reference frame with origin at the point where the line along which the primaries move intersects the plane. Then, in Cartesian coordinates $(q,p,t)\in\mathbb{R}^{4}\times\mathbb{T}\setminus\{q=0\}$, the motion of the massless body is given by the Hamiltonian system

where $\rho(t):\mathbb{T}\to[0,1/2]$ is a half of the distance between the primaries.

It will be convenient for our analysis to introduce polar coordinates $(r,\alpha,t,y,G)\in\mathbb{R}_{+}\times\mathbb{T}^{2}\times\mathbb{R}^{2}$ where $q=(r\cos\alpha,r\sin\alpha)$ and $(y,G)$ denote the conjugate momenta to $(r,\alpha)$. In polar coordinates, the Hamiltonian of the Restricted Isosceles $3$-body Problem reads

We inmediately notice that $G$ is a conserved quantity for the flow of (1.1). It is therefore natural to consider the one-parameter family of Hamiltonian systems

Since $\lim_{r\to\infty}V(r,t)=0$, for all $G\in\mathbb{R}$ the Hamiltonian (1.2) posses a periodic orbit at infinity

In [GPSV21], the first author together with M. Guardia, T. Seara and C.Vidal, proved the following result.

Theorem 1.3 is proved by exploiting the fact that for $G$ large enough, in a suitable region of the phase space, the Hamiltonian $H_{G}$ can be studied as a perturbation of the (integrable) $2$-body Problem. This allowed the authors to prove that the periodic orbit $\gamma_{\infty}$ posses global stable and unstable invariant manifolds which intersect transversally (see Theorem 4.1). As a corollary of this result, a rather straightforward implementation of Moser’s ideas shows the truth of Theorem 1.3.

The following is the first main result of the present work.

To the best of our knowledge, Theorem 1.4 is the first complete analytic proof of the existence of oscillatory motions relying upon a global analytical approach rather than on perturbative techniques. Some interesting related works, where the existence of oscillatory motions is obtained in a setting which is not close to integrable, are [Moe07] and [CGM⁺21]. While in [Moe07] the author shows the existence of oscillatory motions in the $3$-body Problem close to triple collision (small values of the total angular momentum), in [CGM⁺21] the authors obtain a computer assisted proof of the existence of oscillatory motions in the Restricted Circular $3$-body Problem for small values of the Jacobi constant.

Theorem 1.4 is indeed obtained as a consequence of the following result.

Theorem 1.5 can be read as follows. For almost all $G\in\mathbb{R}$ there exist an orbit $r_{h}$ of (1.2) homoclinic to $\gamma_{\infty}$ such that the following holds. Let $z_{*}=(r,y,t)=(r_{h}(0),\dot{r}_{h}(0),0)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{T}$, let $z_{\infty}=(r,y,t)=(\infty,0,0)=\gamma_{\infty}\cap\{t=0\}\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{T}$ and denote by $\Phi$ the Poincaré map induced on the section $\{t=0\}$ by the flow to the Hamiltonian (1.2). Then, for any $\delta>0$ and any sequence $\{z_{k}\}_{k\in\mathbb{Z}}\subset\{z_{\infty},z_{*}\}^{\mathbb{Z}}$ there exists a point $z\in B_{\delta}(z_{0})$ and a sequence $\{n_{k}\}_{k\in\mathbb{Z}}\in\mathbb{N}^{\mathbb{Z}}$ such that $\Phi^{n_{k}}(z_{0})\in B_{\delta}(z_{k})$ ¹¹1By $B_{\delta}(z_{\infty})$ we mean the set $\{|y|\leq\delta,\ |r|^{-1}\leq\delta\}$.. The statement in Theorem 1.5 is indeed stronger since it also provides control on the orbit in all the intervals $[(n_{k}-n_{k-1})/2,(n_{k}+n_{k+1})/2]$.

The following corollary of Theorem 1.5 can obtained by nowadays well known arguments (see for example [MNT99] and [Koz83]).

As in Moser’s approach, the first main step in our construction is to prove the existence of a homoclinic orbit to $\gamma_{\infty}$. Yet, in the setting of Theorem 1.5, geometric perturbation theory is not available since the Hamiltonian system $H_{G}$ in (1.2) is not nearly integrable. Instead, we will adopt a global approach and deploy the powerful machinery of the theory of calculus of variations. In particular, we rephrase the problem of existence of homoclinic orbits to $\gamma_{\infty}$ as that of the existence of critical points of a certain action functional $\mathcal{A}_{G}$ (cfr 4.2) defined in a suitable Hilbert space $D^{1,2}$ (cfr (4.1)). The existence of critical points of the action functional $\mathcal{A}_{G}$ is obtained by a minmax argument tailored made for the present problem. The use of minmax techniques to study the existence and multiplicity results for homoclinic orbits in Hamiltonian systems has already been widely exploited in the literature (see for example [Sér92, CZES90, CZR91] and [MNT99]). In the variational approach to our problem, we face two main difficulties at this step: the phase space is not compact and the vector field presents singularities (corresponding to possible collision with the massive bodies). In order to overcome the first difficulty we make use of a renormalized action functional (see Remark 4) defined on a appropriately chosen functional space $D^{1,2}$. In order to avoid singularities and gain compactness we then perform a constrained deformation argument. With these techniques, together with a compactness property of the map $\mathrm{d}\mathcal{A}_{G}:D^{1,2}\to D^{1,2}$ (Struwe’s monotonicity trick), we are able to show that, for almost all values of the angular momentum $G$ ²²2See the discussion at the beginning of Section 4.2.1., there exists a Palais-Smale sequence in $D^{1,2}$ which converges to a critical point of the action functional $\mathcal{A}_{G}$. This proves the existence of an orbit $\tilde{r}_{h}$ homoclinic to $\gamma_{\infty}$, which actually correspond to a doubly parabolic motion of our problem. It is worthwhile pointing out that half parabolic and hyperbolic motions for the $n$-body problem have been obtained using variational methods in [MV09, MV20] with a different technique.

The homoclinc orbit $\tilde{r}_{h}$ obtained in this way is associated with an intersection between the stable and unstable manifolds of the periodic orbit $\gamma_{\infty}$. To proceed further, though we can not tell whether this intersection is transversal or not, we may rely on our minmax construction to deduce some topological transversality. This can be achieved by a topological degree argument based on a general result by Hofer ([Hof86]). More precisely, we exploit the mountain pass characterization of $\tilde{r}_{h}$ to show that for almost all values of the angular momentum $G$ (except possibly a finite set of values) there exists a (possibly different) critical point $r_{h}$ of the action functional $\mathcal{A}_{G}$ for which the Leray-Schauder index of the map $\nabla\mathcal{A}_{G}:D^{1,2}\to D^{1,2}$ at $r_{h}$ is well defined and different from zero ³³3In Proposition 5.10 show that the topological degree being non zero implies that the intersection between the invariant manifolds of $\gamma_{\infty}$ at $r_{h}$ is topologically transverse.. This allows us to shadow finite segments of the homoclinic orbit $r_{h}$. The proof of Theorem 1.5 is then obtained by combining a suitable parabolic version of the Lambda lemma close to $\gamma_{\infty}$ with the outer dynamics wich shadows finite segments of $r_{h}$.

In Section 2 we recall some well known facts about the $2$-body Problem. Then, in Section 3 we analyze the dynamics around the periodic orbit $\gamma_{\infty}$. In particular, the existence of stable and unstable manifolds $W^{\pm}(\gamma_{\infty};G)$ and a parabolic version of the lambda lemma close to $\gamma_{\infty}$. In Section 4 we introduce the variational formulation and prove the existence of a homoclinic orbit to $\gamma_{\infty}$ by means of a minmax argument. Then, in Section 5 we obtain a (possibly different) homoclinic orbit associated with a topologically transverse intersection between $W^{\pm}(\gamma_{\infty};G)$. Finally in Section 6 we combine the parabolic Lambda lemma of Section 3 together with the robustness of the topological degree under perturbations to construct “multibump” homoclinics and finish the proof of Theorem 1.5.

Let $\phi_{G}^{s}$ be the time $s$ flow associated with the Hamiltonian $H_{G}$ defined in (1.2). It is a classical result by McGehee [McG73] (see also [BF04]) that $\gamma_{\infty}$ posses local stable and unstable invariant manifolds (by $\pi_{r},\pi_{y}$ we denote the projection on the $r$ and $y$ coordinates of a point $(r,y,t)\in\mathbb{R}_{+}\times\mathbb{R}\times\mathbb{T}$)

It is also a standard fact that $W^{\pm}_{\mathrm{loc},R}(\gamma_{\infty};G)$ are exact Lagrangian submanifolds so they can be parametrized in terms of a generating function. The following result follows directly from the arguments in the proof of Theorem 4.4. in [GPSV21] (see Remark 3).

Define the global stable and unstable invariant manifolds

The analytic dependence of the functions $S^{\pm}(r,t;G)$ on $r$ and $G$ will be key to prove that transversal intersections (whenever they exist) between the global stable and unstable invariant manifolds (3.2) are topologically transverse except for (possibly) a finite subset of values of $G$. This is key for the multibump construction. On the other hand, the estimate $S^{\pm}-S^{0}\sim r^{-3/2}$ as $r\to\infty$ will be needed in the proof of certain technical steps in Lemma 5.5 (see Appendix A).

We now analyze the topology of the flow lines close to the periodic orbit $\gamma_{\infty}$. For that, it is convenient to introduce the McGehee transformation $r=2/x^{2}$ in which the equations of motion associated with the Hamiltonian system $H_{G}$ in (1.2) read

In this variables, the periodic orbit at infinity (1.3) now corresponds to the periodic orbit $\hat{\gamma}_{\infty}=\{x=y=0,\ t\in\mathbb{T}\}$. Following Moser [Mos01], we now straighten the stable and unstable directions associated with this periodic orbit. To that end, we introduce the change of variables

In these coordinates

so it is clear that the local stable and unstable invariant manifolds associated with the periodic orbit $\tilde{\gamma}_{\infty}=\{\tilde{q}=\tilde{p}=0,t\in\mathbb{T}\}$, which, by the work of McGehee [McG73] (see also Proposition 3.1) we already know exist, are close (for small $|\tilde{q}|,|\tilde{p}|$) to $\{\tilde{q}=0\}$ and $\{\tilde{p}=0\}$ respectively. Let now, for sufficiently small $\delta>0$, define the set

and, let $(0,\tilde{p},t)\in Q_{\delta}\to(\tilde{p},\gamma^{s}(\tilde{p},t),t)\subset Q_{\delta}$ and $(\tilde{q},0,t)\in Q_{\delta}\to(\tilde{q},\gamma^{u}(\tilde{q},t),t)\subset Q_{\delta}$ be graph parametrizations of these local invariant manifolds. Introduce new variables on $Q_{\delta}$ given by

From the invariance equation satisfied by $\gamma^{u,s}$ one can deduce their Taylor expansion around $\tilde{q}=\tilde{p}=0$. Then, an easy computation, shows that

so in coordinates $(q,p,t)\subset Q_{\delta}$ the local stable and unstable manifolds are the sets $\{p=0\}\cap Q_{\delta}$ and $\{q=0\}\cap Q_{\delta}$ respectively. Define now, for $a<\delta$ the sections (see Figure 3.2)

and the associated Poincaré map $\Phi_{\mathrm{loc}}:\Sigma^{+}_{a}\to\Sigma^{-}_{a^{\prime}}$, associated with the flow (3.3), whenever is well defined. Lemma 3.2 shows that a parabolic version of the Lambda Lemma holds for the degenerate periodic orbit $\{p=q=0\}$. In order to build orbits whose final motions are hyperbolic, we also introduce the outer sections

The first item in Lemma 3.2 shows that the iterates of curves which are transversal to the local stable manifold accumulate along the unstable manifold (see also Figure 3.2). The second item ensures that orbits with initial conditions on $\Sigma^{+}_{a,\mathrm{hyp}}$ (respectively $\Sigma^{-}_{a,\mathrm{hyp}}$) have forward (respectively backward) hyperbolic final motions. We now translate these results to the original coordinates. To that end we introduce the sections

and the map $\Phi_{\mathrm{loc},R_{1},R_{2}}:\Lambda^{+}_{R_{1},\delta}\to\Lambda^{-}_{R_{2},\delta^{\prime}}$ whenever is well defined. We also define the sections leading to hyperbolic final motions

We introduce the vector space of real valued functions

In the following, we will write $\dot{\varphi}=v_{\varphi}$ (i.e. $v_{\varphi}$ is the weak derivative of $\varphi$). It is easy to chek that

defines an inner product on $D^{1,2}$ for which the functional space $D^{1,2}$ equiped with this inner product is a Hilbert space. We write

for the induced norm. Notice that for all $\varphi\in D^{1,2}$ and all $s\in\mathbb{R}$

After the introduction of the functional space $D^{1,2}$ it is an easy computation to show that the existence of orbits of (1.2) homoclinic to the periodic orbit at infinity $\gamma_{\infty}=\{r=\infty,y=0,t\in\mathbb{T}\}$ is equivalent to the existence of critical points of the action functional $\mathcal{A}_{G}:D^{1/2}\to\mathbb{R}$ given by

where

$V_{G}$ stands for the effective potential

and $V_{0}(r_{0})=\frac{G_{0}^{2}}{2r_{0}^{2}}-\frac{1}{r_{0}}$ with $r_{0}$ being the parabolic orbit of the 2BP with angular momentum $G_{0}\in\mathbb{R}$ (see Remark 5).