Oscillatory motions for the restricted planar circular three body problem

Abstract

The restricted three body problem models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. When they move along circular Keplerian orbits and the third body moves in the same plane, one has the restricted planar circular three body problem (RPC3BP). In suitable coordinates, it is a Hamiltonian system of two degrees of freedom. The conserved energy is usually called the Jacobi constant. Llibre and Simó [Math Ann 248(2):153–184, 1980] proved the existence of oscillatory motions for this system. That is, orbits which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they had to assume the ratio between the masses of the primaries to be small enough. In this paper we prove the existence of such motions for any value of the mass ratio $\mu$ closing the problem of existence of oscillatory motions in the RPC3BP. To obtain such motions, we restrict ourselves to the level sets of the Jacobi constant. We show that, for any value of the mass ratio and for large values of the Jacobi constant, there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets. These transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits. The main achievement is to prove the existence of these orbits without assuming the mass ratio $\mu$ small. When $\mu$ is not small, this transversality can not be checked by means of classical perturbation theory. Since our method is valid for all values of $\mu$, we are able to detect a curve in the parameter space, formed by $\mu$ and the Jacobi constant, where cubic homoclinic tangencies between the invariant manifolds of infinity appear.

Appendix. Computation of the function L: proof of Proposition 3.1

In this appendix, we give the properties of the function $L$ in (28), stated in Proposition 3.1. We first state an auxiliary lemma, which gives some properties of the potential $V$ in (11).

Lemma 7.1

The Fourier coefficients of the function

$$\begin{aligned} \widehat{U}(v,\phi ) = V(\widetilde{r}_{\mathrm {h}}(v),\phi ;\mu ,G_0)=\sum _{\ell \in \mathbb {Z}}\widehat{U}^{[\ell ]}(v)e^{i\ell \phi }, \end{aligned}$$

where $V$ is defined in (11) and $\widetilde{r}_{\mathrm {h}}$ in (10), are

$$\begin{aligned} \widehat{U}^{[\ell ]}(v) = \sum _{j\ge \max \{\delta _0(\ell ),-\ell \}} c_j c_{j+\ell }\frac{\mu (1-\mu )^{2j+\ell }+(-1)^\ell (1-\mu )\mu ^{2j+\ell }}{G_0^{4j+2\ell }\widetilde{r}_{\mathrm {h}}^{2j+\ell +1}(v)}, \end{aligned}$$

where $c_j=\left( \begin{array}{c}-1/2\\ j\end{array}\right)$, $\delta _0(0) = 1$ and $\delta _0(\ell ) = 0$ for $\ell \ne 0$.

Proof

We use the identity

$$\begin{aligned} (1+A\cos \phi )^{-1/2}&= \alpha ^{-1/2}(1+\beta e^{i\phi })^{-1/2}(1+\beta e^{-i\phi })^{-1/2} \\&= \alpha ^{-1/2} \sum _{j\ge 0} \sum _{k \ge 0} c_j c_{k} \beta ^{j+k} e^{i(k-j)\phi }\\&= \sum _{\ell \in \mathbb {Z}} e^{i\ell \phi }\sum _{\begin{array}{c} k-j = \ell \\ j,k \ge 0 \end{array}}c_j c_{k} \beta ^{j+k}\\&= \alpha ^{-1/2} \sum _{\ell \in \mathbb {Z}} e^{i\ell \phi } \sum _{j\ge \max \{0,-\ell \}} c_j c_{j+\ell } \beta ^{2j+\ell }, \end{aligned}$$

where $\alpha = A/(2\beta )$ and $\beta = (1- \sqrt{1-A^2})/A$, and

$$\begin{aligned} \frac{1-\mu }{\sqrt{G_0^4\widetilde{r}_{\mathrm {h}}^2-2\mu G_0^2 \widetilde{r}_{\mathrm {h}}\cos \phi + \mu ^2}}&= \frac{1-\mu }{G_0^2\widetilde{r}_{\mathrm {h}}}\left( 1-\frac{\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}}e^{i\phi }\right) ^{-1/2}\\&\quad \times \left( 1-\frac{\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}}e^{-i\phi }\right) ^{-1/2}\\ \frac{\mu }{\sqrt{G_0^4\widetilde{r}_{\mathrm {h}}^2+2(1-\mu )G_0^2 \widetilde{r}_{\mathrm {h}}\cos \phi + (1-\mu )^2}}&= \frac{\mu }{G_0^2\widetilde{r}_{\mathrm {h}}}\left( 1+\frac{1-\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}}e^{i\phi }\right) ^{-1/2} \\&\quad \times \left( 1+\frac{1-\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}}e^{-i\phi }\right) ^{-1/2} \end{aligned}$$

to obtain

$$\begin{aligned} \frac{1}{G_0^2}V(\widetilde{r}_{\mathrm {h}}(v),\phi )&= \frac{1-\mu }{\sqrt{G_0^4\widetilde{r}_{\mathrm {h}}^2-2\mu G_0^2 \widetilde{r}_{\mathrm {h}}\cos \phi + \mu ^2}}\\&\quad + \frac{\mu }{\sqrt{G_0^4\widetilde{r}_{\mathrm {h}}^2+2(1-\mu )G_0^2 \widetilde{r}_{\mathrm {h}}\cos \phi + (1-\mu )^2}} - \frac{1}{G_0^2 \widetilde{r}_{\mathrm {h}}(v)}\\&= \frac{1-\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}(v)}\sum _{\ell \in \mathbb {Z}} e^{i\ell \phi } \sum _{j\ge \max \{0,-\ell \}} c_j c_{j+\ell } \frac{(-\mu )^{2j+\ell }}{G_0^{4j+2\ell }\widetilde{r}_{\mathrm {h}}^{2j+\ell }(v)} \\&\quad + \frac{\mu }{G_0^2 \widetilde{r}_{\mathrm {h}}(v)}\sum _{\ell \in \mathbb {Z}} e^{i\ell \phi } \sum _{j\ge \max \{0,-\ell \}} c_j c_{j+\ell } \frac{(1-\mu )^{2j+\ell }}{G_0^{4j+2\ell }\widetilde{r}_{\mathrm {h}}^{2j+\ell }(v)} \\&\quad - \frac{1}{G_0^2 \widetilde{r}_{\mathrm {h}}(v)}\\&= \sum _{\ell \in \mathbb {Z}} e^{i\ell \phi } \sum _{j\ge \max \{0,-\ell \}} c_j c_{j+\ell }\\&\quad \times \frac{(-1)^\ell (1-\mu )\mu ^{2j+\ell }+\mu (1-\mu )^{2j+\ell }}{G_0^{4j+2\ell +2}\widetilde{r}_{\mathrm {h}}^{2j+\ell +1}(v)} - \frac{1}{G_0^2 \widetilde{r}_{\mathrm {h}}(v)}. \end{aligned}$$

$\square$

Remark 7.2

A more classical approach to prove this lemma would be to expand the potential in terms of the Legendre polynomials and then extract from these polynomials the Fourier coefficients of $\widehat{U}$. This is the approach followed in [23, 30]. Here we follow the approach in [8] and we provide a direct way to compute an explicit formula for these Fourier coefficients.

Now we prove Proposition 3.1.

Proof Proposition 3.1

First, we observe that the Poincaré function $L$ in (28) can be written as

$$\begin{aligned} L(v,\xi ;\mu ,G_0)=\int _{-\infty }^{+\infty } V(\widetilde{r}_{\mathrm {h}}(t),\xi +G_0^3 v-G_0^3 t+ \widetilde{\alpha }_{\mathrm {h}}(t);\mu ,G_0)dt. \end{aligned}$$

Now, using that

$$\begin{aligned}&V(\widetilde{r}_{\mathrm {h}}(t), \xi +G_0^3 v-G_0^3 t+\widetilde{\alpha }_{\mathrm {h}}(t);\mu ,G_0) \\&\quad = \sum _{\ell \in \mathbb {Z}}\widehat{U}^{[\ell ]}(t) e^{i\ell \widetilde{\alpha }_{\mathrm {h}}(t) } e^{-i\ell G_0^3t} e^{i\ell (\xi + G_0^3 v)} \end{aligned}$$

and

$$\begin{aligned} L(v,\xi ;\mu ,G_0)= \sum L^{[\ell ]} e^{i\ell (\xi + G_0^3 v)} \end{aligned}$$

with

$$\begin{aligned} L^{[\ell ]}= \int _{-\infty }^{+\infty }\widehat{U}^{[\ell ]} (t) e^{i\ell \widetilde{\alpha }_{\mathrm {h}}(t)} e^{-i\ell G_0^3 t} dt \end{aligned}$$

to compute $L^{[\ell ]}$ we use the expansions in Lemma 7.1 obtaining, when $\ell \ne 0$

$$\begin{aligned} L^{[\ell ]} = \sum _{j\ge 0} \left( \begin{array}{c}-\frac{1}{2}\\ j \end{array}\right) \left( \begin{array}{c}-\frac{1}{2}\\ \ell +j \end{array}\right) \frac{\mu (1-\mu )^{2j+\ell }+ (-1)^\ell (1-\mu )\mu ^{2j+\ell }}{G_0^{4j+2\ell }} \mathcal {I}(\ell ,j) \end{aligned}$$

with

$$\begin{aligned} \mathcal {I}(\ell ,j)=\int _{-\infty }^{+\infty }\frac{e^{i\ell \widetilde{\alpha }_{\mathrm {h}}(t)}}{\widetilde{r}_{\mathrm {h}}^{2j+\ell +1}(t)} e^{-i\ell G_0^3 t}\,dt. \end{aligned}$$

To compute $\mathcal {I}(\ell ,j)$ we use the change of variables $v = \frac{1}{2}\left( \frac{1}{3} \tau ^3 + \tau \right)$ given in Lemma 4.1 obtaining

$$\begin{aligned} \mathcal {I}(\ell ,j)= & {} (-1) ^\ell 2^{2j+\ell }\int _{-\infty }^{+\infty }\frac{e^{-i\ell \frac{G_0^3}{2} (\tau +\frac{\tau ^3}{3}) }}{(\tau -i)^{2j} (\tau +i)^{2j+2\ell } } d\tau \\&\quad := (-1) ^\ell 2^{2j+\ell } I(-\ell , j,j+\ell ), \end{aligned}$$

where we have introduced the notation

$$\begin{aligned} I(\ell ,m,n)=\int _{-\infty }^{+\infty }\frac{e^{i\ell \frac{G_0^3}{2} (\tau +\frac{\tau ^3}{3}) }}{ (\tau -i)^{2m} (\tau +i)^{2n}} d\tau \end{aligned}$$

to write the Poincaré function Fourier coefficients as

$$\begin{aligned} L^{[\ell ]}&= \sum _{j\ge 0} \left( \begin{array}{c}-\frac{1}{2}\\ j \end{array}\right) \left( \begin{array}{c}-\frac{1}{2}\\ \ell +j \end{array}\right) \frac{\mu (1-\mu )^{2j+\ell }+ (-1)^\ell (1-\mu )\mu ^{2j+\ell }}{G_0^{4j+2\ell }}\\&\qquad \times (-1) ^\ell 2^{2j+\ell } I(-\ell , j,j+\ell ) . \end{aligned}$$

The first observation is that

$$\begin{aligned} I(-\ell , n,m) = I(\ell ,m,n)=\overline{ I(\ell ,m,n)}. \end{aligned}$$

All the Fourier coefficients are real and we just need to compute them for $\ell >0$:

$$\begin{aligned} L^{[\ell ]}&= \sum _{j\ge 0} \left( \begin{array}{c}-\frac{1}{2}\\ j \end{array}\right) \left( \begin{array}{c}-\frac{1}{2}\\ \ell +j \end{array}\right) \frac{\mu (1-\mu )^{2j+\ell }+ (-1)^\ell (1-\mu )\mu ^{2j+\ell }}{G_0^{4j+2\ell }}\\&\qquad \times (-1) ^\ell 2^{2j+\ell } I(\ell , j+\ell ,j) \end{aligned}$$

and then

$$\begin{aligned} L(\xi , v) =2 \sum _{\ell \in \mathbb {N}} L^{[\ell ]} \cos \ell (\xi + G_0^3 v). \end{aligned}$$

To compute the integrals $I(\ell ,m,n)$ for $\ell >0$, one uses the method in [11] (see also [23, 30]) changing the path of integration to a suitable complex path $\mathrm {Re\, }(\tau + \frac{\tau ^3}{3})= 0$ up to a neighborhood of the singularity $\tau = i$.

Using that $\tau + \frac{\tau ^3}{3}=\frac{2}{3}i +\mathcal {O}((\tau -i)^2)$, to bound the integrals (see [8]) it is enough to reach a neighborhood of the singularity $\tau =i$ of order $\mathcal {O}(G_0^{-3/2})$, obtaining that there exists a constant $K>0$ such that, for any $\ell >0$ and $m,n\ge 1$:

$$\begin{aligned} |I(\ell , m,n)| \le K G_0^{3m-3/2} e^{-\frac{G_0^3}{3}\ell }, \end{aligned}$$

and therefore

$$\begin{aligned} |L^{[\ell ]}| \le (K G_0)^{\ell -3/2} e^{-\frac{G_0^3}{3}\ell }. \end{aligned}$$

To obtain the dominant terms of the function $L$, which correspond to $\ell =1,2$, we can use the results in [8] (see also [30]) to obtain

$$\begin{aligned} I(1,2,1)&= -\frac{1}{6} \sqrt{\frac{\pi }{2}} G_0^{\frac{9}{2}}e^{-\frac{G_0^3}{3}} \left( 1+\mathcal {O}\left( G_0^{-3/2}\right) \right) \\ I(2,2,0)&= \frac{4}{3} \sqrt{\pi } G_0^{\frac{9}{2}}e^{-2\frac{G_0^3}{3}} \left( 1+\mathcal {O}\left( G_0^{-3/2}\right) \right) . \end{aligned}$$

Thus,

$$\begin{aligned} L^{[1]}&= -\frac{\mu (1-\mu )^3-(1-\mu )\mu ^3}{4} \sqrt{\frac{\pi }{2}} G_0^{-\frac{3}{2}}e^{-\frac{G_0^3}{3}} \left( 1+\mathcal {O}\left( G_0^{-1}\right) \right) \\ L^{[2]}&= 2 \mu (1-\mu )\sqrt{\pi } G_0^{\frac{1}{2}}e^{-2\frac{G_0^3}{3}} \left( 1+\mathcal {O}\left( G_0^{-1}\right) \right) . \end{aligned}$$

To obtain the formula for $L^{[1]}$ in Proposition 3.1 it is enough to use that $\mu (1-\mu )^3-(1-\mu )\mu ^3=\mu (1-\mu )(1-2\mu )$. $\square$