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On the Co-orbital Motion in the Three-Body Problem: Existence of Quasi-periodic Horseshoe-Shaped Orbits

Abstract

Janus and Epimetheus are two moons of Saturn with very peculiar motions. As they orbit around Saturn on quasi-coplanar and quasi-circular trajectories whose radii are only 50 km apart (less than their respective diameters), every four (terrestrial) years the bodies approach each other and their mutual gravitational influence lead to a swapping of the orbits: the outer moon becomes the inner one and vice-versa. This behavior generates horseshoe-shaped trajectories depicted in an appropriate rotating frame. In spite of analytical theories and numerical investigations developed to describe their long-term dynamics, so far very few rigorous long-time stability results on the “horseshoe motion” have been obtained even in the restricted three-body problem. Adapting the idea of Arnol’d (Russ Math Surv 18:85–191, 1963) to a resonant case (the co-orbital motion is associated with trajectories in 1:1 mean motion resonance), we provide a rigorous proof of existence of 2-dimensional elliptic invariant tori on which the trajectories are similar to those followed by Janus and Epimetheus. For this purpose, we apply KAM theory to the planar three-body problem.

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Notes

  1. 1.

    For two of these configurations the three bodies are located at the vertices of an equilateral triangle. These equilibria correspond to the fixed points \(L_4\) and \(L_5\) in the restricted three-body problem (RTBP). The other three are the Euler collinear configurations (\(L_1\), \(L_2\), and \(L_3\) in the RTBP).

  2. 2.

    The horseshoe trajectories are depicted in the frame that rotates with the moons’ average mean motion.

  3. 3.

    Which is the gravitational sphere of influence where the primary acts as a perturbator.

  4. 4.

    In this approximation, it is assumed that the massless one does not affect the motion of the other two, which is consequently Keplerian.

  5. 5.

    Indeed, Janus is only 3 times more massive than Epimetheus. This is a particular case since for all the co-orbital pairs of celestial objects observed up to now, one is very small with respect to the other hence the RTBP is a good model except for Janus-Epimetheus.

  6. 6.

    Which is the averaged perturbation along the Keplerian flows.

  7. 7.

    This result was initially stated by Melnikov [27] and independently proved by Eliasson [15] and Kuksin [21].

  8. 8.

    According to Gascheau [19], when the planetary orbits are circular, the equilateral configurations are linearly stable if the mass of the three bodies satisfy the relation \(27(m_0{\varepsilon }m_1+m_0{\varepsilon }m_2 + {\varepsilon }m_1{\varepsilon }m_2) < (m_0 + {\varepsilon }m_1 +{\varepsilon }m_2)^2 \).

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Acknowledgements

The authors are indebted to Jacques Féjoz for key discussions concerning KAM theory. A.P. acknowledges the support of the H2020-ERC project 677793 StableChaoticPlanetM and this research is part of this project. L.N. acknowledges the support of the ANR project BEKAM (ANR-15-CE40-0001) and the NSF-Grant No. DMS-1440140 as well as the MSRI-Berkeley where he was in residence.

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Correspondence to Laurent Niederman.

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Communicated by C. Liverani

Appendix A. Proofs

Appendix A. Proofs

Theorem 4.1: Estimates on \(H_K\), \(H_P\)

By the real analyticity of the transformation in Poincaré resonant complex variable \(\tilde{\Upsilon }\circ \Upsilon \), there exists \(\rho _0>0\) and \(\sigma _0>0\) such that the differential of its complex extension,

$$\begin{aligned} \tilde{\Upsilon }\circ \Upsilon : \quad \Bigg \{ \begin{array}{ccc} {\hat{{{\mathcal {K}} }}}_{\rho _0, \sigma _0} &{}\longrightarrow &{} {{\mathbb {C}}}^8 \\ ({\mathbf{Z} }, \varvec{\zeta }, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) &{}\longmapsto &{}(\tilde{\mathbf{r}} _1, \mathbf{r} _1, \tilde{\mathbf{r}} _2, \mathbf{r} _2 )\, , \end{array} \end{aligned}$$

admits a norm uniformly bounded on the collisionless domain \({\hat{{{\mathcal {K}} }}}_{\rho _0, \sigma _0} \) (defined in Sect. 4.1) by a constant \(C>0\) independent of \({\varepsilon }\).

In the following, we will denote \(D_{\rho _0,\sigma _0}\) the image of \({\hat{{{\mathcal {K}} }}}_{\rho _0, \sigma _0}\) by the transformation \(\tilde{\Upsilon }\circ \Upsilon \).

Hence, as \(\left\| ({\mathbf{Z} }, \varvec{\zeta }, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) \right\| _{{\hat{{{\mathcal {K}} }}}_{\rho _0,\sigma _0}} \leqslant \rho _0 + \sigma _0 + 2\sqrt{\rho _0\sigma _0}\) then

$$\begin{aligned} \left\| \mathbf{r} _j - {{\,\mathrm{Re}\,}}(\mathbf{r} _j) \right\| _{D_{\rho _0,\sigma _0}} \leqslant C(\rho _0 + \sigma _0 + 2\sqrt{\rho _0\sigma _0}) . \end{aligned}$$

Thus, one has

$$\begin{aligned} \begin{aligned} \left\| \mathbf{r} _1 - \mathbf{r} _2 \right\| _{D_{\rho _0,\sigma _0}}&\geqslant \left\| {{\,\mathrm{Re}\,}}(\mathbf{r} _1) - {{\,\mathrm{Re}\,}}(\mathbf{r} _2) \right\| _{D_{\rho _0,\sigma _0}} - \sum _{j\in \{1,2\}}\left\| \mathbf{r} _j - {{\,\mathrm{Re}\,}}(\mathbf{r} _j) \right\| _{D_{\rho _0,\sigma _0}} \\&\geqslant \Delta - 8C\sigma _0 \geqslant \frac{\Delta }{2} \end{aligned} \end{aligned}$$

since \(\rho _0<\sigma _0\) and where \(\hat{\Delta }\) is an arbitrary fixed value on \({{\mathbb {T}}}\) such that the minimum distance \(\Delta \) between two planets in circular motion is reached (see Sect. 4.1 for more details).

Consequently, and

as \(\Delta \) (resp. \(\hat{\Delta }\)) does not depend on the small parameter \({\varepsilon }\).

Finally, since then there exists a constant \(c>0\) such that

$$\begin{aligned} c\leqslant \left\| \Lambda _{1,0} + Z_1 \right\| _{{{\mathcal {K}} }_{\rho _0, \sigma _0}} \quad \text{ and } \quad c\leqslant \left\| \Lambda _{2,0} + Z_2-Z_1 \right\| _{{{\mathcal {K}} }_{\rho _0, \sigma _0}} \end{aligned}$$

which implies that

Theorem 4.1: First Averaging Theorem

First of all, we define an iterative lemma of averaging. Let us introduce some notations: \((\xi _k)_{k\in \{1,2,3\}}\) are given positive numbers such that

$$\begin{aligned} 0<\xi _1< \rho , \quad 0< \xi _2<\sigma , \quad 0<\xi _{3}<\sqrt{\rho \sigma } \end{aligned}$$

and, for \(0\leqslant r\leqslant 1\), we denote \({\hat{{{\mathfrak {K}} }} }_r\) the domain such as

$$\begin{aligned} {\hat{{{\mathfrak {K}} }} }_r={{\mathcal {B}}}^2_{\rho -r\xi _1}\times {{\mathcal {V}}} _{\sigma -r\xi _2}{\hat{{{\mathcal {I}} }} }\times {{\mathcal {B}}}^4_{\sqrt{\rho \sigma } - r\xi _{3}} . \end{aligned}$$

Hence, we set out the following

Lemma A.1

(First iterative lemma). Let \(\rho ^-\), \(\sigma ^-\), \(\xi _1\), \(\xi _2\) be fixed positive real numbers that depend on the small parameter \({\varepsilon }\) and

$$\begin{aligned} \begin{aligned} \rho ^+&= \rho ^- - \xi _1>0,\quad \sigma ^+ = \sigma ^- - \xi _2>0, \\ \xi _{3}&= \sqrt{\rho ^-\sigma ^-} - \sqrt{\rho ^+\sigma ^+} . \end{aligned} \end{aligned}$$
(A.1)

Let \(H^-\) be a Hamiltonian of the form

$$\begin{aligned} \begin{aligned} H^-({\mathbf{Z} }, \varvec{\zeta }, {\mathbf{x} }, {\widetilde{\mathbf{x}} })&= H_K({\mathbf{Z} }) + \overline{H} _P({\mathbf{Z} }, \zeta _1, {\mathbf{x} }, {\widetilde{\mathbf{x}} })\\&\quad + {H_{*}^{0,-}}({\mathbf{Z} }, \zeta _1, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) + {H_{*}^{1,-}}({\mathbf{Z} }, \varvec{\zeta }, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) \end{aligned} \end{aligned}$$

which is analytic on the domain \({\hat{{{\mathfrak {K}} }} }_0^-={\hat{{{\mathcal {K}} }}}_{\rho ^-, \sigma ^-}\) and such that

$$\begin{aligned} {\overline{H} _{*}^{1,-}}({\mathbf{Z} }, \zeta _1, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) = \frac{1}{2\pi }\int _0^{2\pi } {H_{*}^{1,-}}({\mathbf{Z} }, \zeta _1, \zeta _2, {\mathbf{x} }, {\widetilde{\mathbf{x}} }){\mathrm {d}}\zeta _2 =0 . \end{aligned}$$

Let \(\eta ^-\), \((\mu _l^-)_{l\in \{0,1,2,3\}}\) be fixed positive real numbers, which depend on \({\varepsilon }\), such that

$$\begin{aligned} \begin{aligned} \left\| {H_{*}^{1,-}} \right\| _{{\hat{{{\mathfrak {K}} }} }_0^-}\leqslant \eta ^-, \quad \left\| {H_{*}^{0,-}} \right\| _{ {\hat{{{\mathfrak {K}} }} }_0^-}\leqslant \mu _0^- \end{aligned} \end{aligned}$$
(A.2)

and

$$\begin{aligned} \begin{aligned}&\left\| \partial _{{\mathbf{Z} }}\big (\overline{H} _P + {H_{*}^{0,-}}\big ) \right\| _{ {\hat{{{\mathfrak {K}} }} }_0^-}\leqslant \mu _1^-,&\quad&\left\| \partial _{\varvec{\zeta }}\big (\overline{H} _P +{H_{*}^{0,-}}\big ) \right\| _{{\hat{{{\mathfrak {K}} }} }_0^-}\leqslant \mu _2^-,&\\&\left\| \partial _{({\mathbf{x} },{\widetilde{\mathbf{x}} })}\big (\overline{H} + {H_{*}^{0,-}}\big ) \right\| _{ {\hat{{{\mathfrak {K}} }} }_0^-}\leqslant \mu _3^- .&\end{aligned} \end{aligned}$$

If we assume that

(A.3)

then there exists a canonical transformation

(A.4)

and such that, in the new variables, the Hamiltonian \(H^+= H^- \circ \overline{\Upsilon }^+\) can be written

$$\begin{aligned} \begin{aligned} H^+&= H_K + \overline{H} _P + {H_{*}^{0,-}} + H_*^+ \\&= H_K + \overline{H} _P + {H_{*}^{0,+}} + {H_{*}^{1,+}} \end{aligned} \quad \text{ with } \quad \left\{ \begin{array}{l} {H_{*}^{0,+}} = {H_{*}^{0,-}} + \overline{H} _{*}^+ \\ {H_{*}^{1,+}} = H_*^+ - \overline{H} _*^+ \end{array} \right. \end{aligned}$$

and

Furthermore, we have the thresholds

$$\begin{aligned} \begin{aligned} \left\| {H_{*}^{1,+}} \right\| _{{\hat{{{\mathfrak {K}} }} }_1^-}\leqslant \eta ^+ , \quad \left\| {H_{*}^{0,+}} \right\| _{ {\hat{{{\mathfrak {K}} }} }_1^-}\leqslant \mu _0^+ , \end{aligned} \end{aligned}$$
(A.5)

and

$$\begin{aligned} \begin{aligned}&\left\| \partial _{{\mathbf{Z} }}\big (\overline{H} _P + {H_{*}^{0,+}}\big ) \right\| _{ {\hat{{{\mathfrak {K}} }} }_1^-}\leqslant \mu _1^+ ,&\quad&\left\| \partial _{\varvec{\zeta }}\big (\overline{H} _P + {H_{*}^{0,+}}\big ) \right\| _{{\hat{{{\mathfrak {K}} }} }_1^-}\leqslant \mu _2^+ ,&\\&\left\| \partial _{({\mathbf{x} }, {\widetilde{\mathbf{x}} })}\big (\overline{H} _P + {H_{*}^{0,+}}\big ) \right\| _{ {\hat{{{\mathfrak {K}} }} }_1^-}\leqslant \mu _3^+,&\end{aligned} \end{aligned}$$
(A.6)

with the following quantities:

(A.7)

Proof

We define \(\overline{\Upsilon }^+:{\hat{{{\mathfrak {K}} }} }_1^-\longrightarrow {\hat{{{\mathfrak {K}} }} }_0^- \) that is the time-one map of the Hamiltonian flow generated by the auxiliary function \(\chi ^+\), i.e. \(\overline{\Upsilon }^+=\Phi _1^{\chi ^+}\) with

$$\begin{aligned} {\chi ^+}({\mathbf{Z} },\varvec{\zeta }, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) = \frac{2\pi }{\upsilon _0}\int _0^1 s {H_{*}^{1,-}}({\mathbf{Z} },\zeta _1, \zeta _2 + 2\pi s, {\mathbf{x} }, {\widetilde{\mathbf{x}} }){\mathrm {d}}s \end{aligned}$$

such that

(A.8)

Thus, in the new variables, the Hamiltonian reads

with the remainder

that is given by the Eqs. (3.2) and (3.3) while \((*)\) is equal to zero by (A.8).

We have to estimate the size of \(H_*^+\) to prove the thresholds (A.5) and (A.6). Firstly, by the conditions (A.2), we have as \(\upsilon _0={{\mathcal {O}}} (1)\). One then applies the Cauchy inequalities to obtain the partial derivatives

and deduces the estimates on the Poisson brackets

(by the threshold given by (4.1) and the mean value theorem),

(as (A.1) implies that \((\xi _{3})^2 \geqslant \xi _1 \xi _2\)), and

As a consequence, the remainder of the transformation \(\overline{\Upsilon }^+\) is bounded such that

where \(\theta ^+\) is given by (A.7). Moreover, taking into account that \(\overline{\chi }^+=0\) (given by (A.8)), we have

$$\begin{aligned} \overline{H} _{*}^+({\mathbf{Z} }, \zeta _1, {\mathbf{x} }, {\widetilde{\mathbf{x}} }) = \frac{1}{2\pi } \int _0^{2\pi } \int _0^1 s \left\{ {\chi ^+},{H_{*}^{1,-}}\right\} \circ \Phi _s^{\chi ^+}({\mathbf{Z} }, \zeta _1, \tau , {\mathbf{x} }, {\widetilde{\mathbf{x}} }) {\mathrm {d}}s {\mathrm {d}}\tau \end{aligned}$$

and therefore

Hence, if we denote \({H_{*}^{0,+}} = {H_{*}^{0,-}} + \overline{H} _{*}^+\) and \({H_{*}^{1,+}} = H_*^+ - \overline{H} _*^+\) then the triangle inequality gives the estimates (A.5) and (A.6) (together with the Cauchy inequalities for the last).

Finally, by the Eq. (3.2) and the Cauchy inequalities, we can estimate the size of the transformation \(\overline{\Upsilon }^+\). Hence, the condition (A.3) provides the following estimates

which yields (A.4). \(\quad \square \)

Now, in order to prove Theorem 4.1, one applies a first time Lemma A.1. Thus, we define the following

$$\begin{aligned} (\xi _1, \xi _2, \xi _3) = \frac{\sigma _0}{3}({\varepsilon }^\beta , 1, {\varepsilon }^{\beta /2}) \quad \text{ for } \quad 1/7<\beta <1/2 \end{aligned}$$

such that \({\hat{{{\mathfrak {K}} }} }_r = {\hat{{{\mathcal {K}} }}}_{1-\frac{r}{3}}\) for \(0\leqslant r\leqslant 1\). By Theorem 4.1 and the notations of Lemma A.1, the Hamiltonian H is analytical on \({\hat{{{\mathcal {K}} }}}_1\) and of the form

$$\begin{aligned} H({\mathbf{Z} },\varvec{\zeta },{\mathbf{x} },{\widetilde{\mathbf{x}} }) = H_K({\mathbf{Z} }) + \overline{H} _P({\mathbf{Z} },\zeta _1,{\mathbf{x} },{\widetilde{\mathbf{x}} }) + \big [H_P - \overline{H} _P\big ]({\mathbf{Z} },\varvec{\zeta },{\mathbf{x} },{\widetilde{\mathbf{x}} }) \end{aligned}$$

with

Hence, the condition (A.3) is fulfilled and Lemma A.1 provides the existence of the transformation \(\overline{\Upsilon }^{0}: {\hat{{{\mathcal {K}} }}}_{2/3} \longrightarrow {\hat{{{\mathcal {K}} }}}_{1}\) such that

$$\begin{aligned} H^{0}=H\circ \overline{\Upsilon }^{0}= H_K + \overline{H} _P + H_{*}^{0,0} + H_{*}^{1,0} \end{aligned}$$

with the following thresholds:

(A.9)

and

Moreover, by the Eq. (3.2) and the Cauchy inequalities, one has:

Then, we apply iteratively Lemma A.1 to reduce the fast component of the Hamiltonian until an exponentially small size with respect to \({\varepsilon }\). To do so, let s be a non-zero integer such that \(s = {\mathrm E }({\varepsilon }^{-\alpha }) + 1\) where

$$\begin{aligned} \alpha = \frac{1-2\beta }{5} \quad \text{ for } \quad 1/7<\beta <1/2 . \end{aligned}$$

We define

$$\begin{aligned} (\xi _1, \xi _2, \xi _3) = \frac{\sigma _0}{3s}({\varepsilon }^\beta , 1, {\varepsilon }^{\beta /2}) \end{aligned}$$

as well as the sequences \((\rho ^{j})_{j\in \{0,1,\ldots ,s\}}\), \((\sigma ^{j})_{j\in \{0,1,\ldots ,s\}}\) with

$$\begin{aligned} (\rho ^{j}, \sigma ^{j}) = \frac{2s-j}{3s} (\rho ,\sigma ) \quad \text{ for } \quad j\in \{1,\ldots ,s\} \end{aligned}$$

such that \({\hat{{{\mathfrak {K}} }} }^j_r = {\hat{{{\mathcal {K}} }}}_{\frac{2}{3}-\frac{j + r}{3s}}\) for \(0\leqslant r\leqslant 1\).

Replacing the notation \( ^-\) and \( ^+\) of Lemma A.1 by \(^{j-1}\) and \(^{j}\) and assuming that for all \(0< j\leqslant s\) the following condition (associated with (A.3)) is fulfilled:

(A.10)

an iterative application of Lemma A.1 to the Hamiltonian \(H^{0}\) provides a sequence of canonical transformations \((\overline{\Upsilon }^{j})_{j\in \{1,\ldots ,s\}}\) such that \(H^0\circ \overline{\Upsilon }^{1}\circ \cdots \circ \overline{\Upsilon }^{s}\) is equal to the Hamiltonian of the formula (4.3) with

In order to complete the proof, let us consider \(n\in \{1,\ldots ,s\}\) such that the sequences \((\eta ^{j})_{j\in \{1,\ldots ,n\}}\) and \((\mu _l^{j})_{j\in \{1,\ldots , n\}}\) satisfy the following induction hypothesis:

$$\begin{aligned} \eta ^{j}\leqslant \eta ^{j-1}\exp (-1) \quad \text{ and } \quad \mu _l^{j} - \mu _l^{j-1} \leqslant \frac{\mu _l^{0}}{s} \quad (l\in \{1,2,3\}) . \end{aligned}$$

For \(n=1\), as \(0<\alpha<1/7<\beta <1/2\), (A.10) is satisfied and implies that

for and

for .

For a fixed integer n, \((\eta ^j)_{j\in \{0,\ldots ,n\}}\) is decreasing while

$$\begin{aligned} \mu _l^{n} \leqslant \mu _l^{n-1} + \frac{\mu _l^{0}}{s} \leqslant \cdots \leqslant \mu _l^{0} + n\frac{\mu _l^{0}}{s} \leqslant 2 \mu _l^{0}\quad (l\in \{1,2,3\}), \end{aligned}$$

then the induction is immediate. Indeed, (A.10) is satisfied and implies that

As a consequence, we have

which prove (4.5). Likewise,

and then proves (4.4).

At last, and in the same way as for the first application of Lemma A.1, for each transformation \(\overline{\Upsilon }^j\) with \(j\in \{1, \ldots , s\}\), the Eq. (3.2) and Cauchy inequalities lead to

Consequently, the size of the transformation \(\overline{\Upsilon }\) is dominated by that of the transformation \(\overline{\Upsilon }^{0}\) which provides the estimates (4.6) and yields (4.2).

Lemma 4.2: D’Alembert rule in the Averaged Problem

The D’Alembert rule, given by (2.4), derives from the preservation of the angular momentum denoted \({\tilde{{{\mathcal {C}}}}}= \sum _{j\in \{1,2\}} \tilde{\mathbf{r}} _j \times \mathbf{r} _j\). By the transformation in the resonant Poincaré complex variables \(\tilde{\Upsilon }\circ \Upsilon \), we have \( {{\mathcal {C}}}({\mathbf{Z} },\varvec{\zeta },{\mathbf{x} },{\widetilde{\mathbf{x}} }) = {\tilde{{{\mathcal {C}}}}}\circ \tilde{\Upsilon }\circ \Upsilon ({\mathbf{Z} },\varvec{\zeta },{\mathbf{x} },{\widetilde{\mathbf{x}} }) = Z_2 + ix_1{\widetilde{x} }_1 + ix_2{\widetilde{x} }_2\). \({\tilde{{{\mathcal {C}}}}}\) being an integral of the motion, it turns out that

$$\begin{aligned} 0 = \{{\tilde{{{\mathcal {C}}}}}, {{\mathcal {H}} }\} = \{ {{\mathcal {C}}}, {{\mathcal {H}} }\circ \tilde{\Upsilon }\circ \Upsilon \} = \{ {{\mathcal {C}}}, H \} . \end{aligned}$$
(A.11)

Injecting the expansion (2.3) in (A.11) we get

$$\begin{aligned} \begin{aligned} 0&= \{ {{\mathcal {C}}}, \sum _{(k,{\mathbf{p} },{\widetilde{{\mathbf{p} }} }) \in {{\mathscr {D}}}}\quad f_{k,{\mathbf{p} },{\widetilde{{\mathbf{p} }} }}({\mathbf{Z} },\zeta _1)x_1^{p_1}x_2^{p_2}{\widetilde{x} }_1^{{\widetilde{p} }_1}{\widetilde{x} }_2^{{\widetilde{p} }_2}\exp (i k\zeta _2) \} \\&= - i \quad \sum _{(k,{\mathbf{p} },{\widetilde{{\mathbf{p} }} }) \in {{\mathscr {D}}}} \quad ( k + p_1 -{\widetilde{p} }_1 + p_2 - {\widetilde{p} }_2 ) f_{k,{\mathbf{p} },{\widetilde{{\mathbf{p} }} }}({\mathbf{Z} },\zeta _1)x_1^{p_1}x_2^{p_2}{\widetilde{x} }_1^{{\widetilde{p} }_1}{\widetilde{x} }_2^{{\widetilde{p} }_2}\exp (i k\zeta _2) . \end{aligned} \end{aligned}$$

As a consequence, one has

$$\begin{aligned} k + p_1 -{\widetilde{p} }_1 + p_2 - {\widetilde{p} }_2 = 0. \end{aligned}$$

In order to prove Lemma 1, it only needs to be shown that the expression of \({\overline{{{\mathcal {C}}}}}= {{\mathcal {C}}}\circ \overline{\Upsilon }\) is equal to . As the averaging transformation \( \overline{\Upsilon }\) is generated by the composition of the transformations \((\Phi _1^{\chi ^{j}})_{j\in \{0,\ldots ,s\}}\) (see Sect. A.2), the result holds if \(\{ {\mathcal \chi ^{j}} , {\overline{{{\mathcal {C}}}}}\} = 0\).

At first iteration, the generating function \(\chi ^{0}\) reads

$$\begin{aligned} \chi ^{0}({\mathbf{Z} },\varvec{\zeta },{\mathbf{x} },{\widetilde{\mathbf{x}} }) = \frac{2\pi }{\upsilon _0}\int _0^1 s \left[ H_P - \overline{H} _P\right] _{({\mathbf{Z} },\zeta _1,\zeta _2 + 2\pi s,{\mathbf{x} },{\widetilde{\mathbf{x}} })} {\mathrm {d}}s . \end{aligned}$$

As \(H_P\) satisfies the D’Alembert rule, one has

$$\begin{aligned} \{ \chi ^{0} , {{\mathcal {C}}}\} = \frac{2\pi }{\upsilon _0}\int _0^1 s\{ H_P - \overline{H} _P, {{\mathcal {C}}}\}{\mathrm {d}}s = 0 , \end{aligned}$$

which leads to: \( {{\mathcal {C}}}\circ \Phi _1^{\chi ^{0}} = {{\mathcal {C}}}\). The same holds true for the other iterations.

Finally, let a real function f that satisfies the D’Alembert rule and does not depend on the fast angle . Hence, the total degree in , , , in the monomials appearing in the Taylor expansion of f in neighborhood of is even. As a consequence f can be decomposed such as \(f= f_0 +f_2\) with the properties (4.8).

Theorem 4.2: Reduction

The Hamiltonian of Theorem 4.2 is obtained by a suitable expansion of the averaged Hamiltonian \(\overline{H} \) in the neighborhood of the quasi-circular manifold \({\mathrm C}_0\).

First of all, by Lemma 4.2, \(\overline{H} \) and \(\overline{H} _*\) can be decomposed respectively such as

Regarding the eccentricities, a polynomial expansion of \(\overline{H} \) of the degree two with respect to provides

The size of the remainder involved in this approximation is estimated thanks to the mean value theorem applied on the function \(g_{(j,k)}\) for together with the bound (4.1) of Theorem 4.1. Hence, this yields

Now, we consider the expansion of \(\overline{H} \) with respect to the exact resonant action . The Keplerian part can be written:

where the quadratic form \(\tilde{ Q }\) reads

and its approximation

The application of the Taylor formula on the function \(g(t)= H_K(t{\mathbf{Z} })\) for \((t,{\mathbf{Z} })\in [0,1]\times {{\mathcal {B}}}^2_{\rho /3}\) leads to

and, together with the bound (4.1), provides the estimates

Regarding the estimate of \(R^3_K\), as

$$\begin{aligned} {\widehat{m}}_j = m_j + {{\mathcal {O}}} ({\varepsilon }), \quad \mu _j = m_0 + {{\mathcal {O}}} ({\varepsilon }), \end{aligned}$$
(A.12)

then \({\tilde{A}}- A = {{\mathcal {O}}} ({\varepsilon })\) and \({\tilde{\kappa }}- \kappa = {{\mathcal {O}}} ({\varepsilon })\) provide the following bound:

In the case of the the perturbation part, one can split \(\overline{H} _{P,0}\) and \((\overline{H} _{P,(j,k)})_{1\leqslant j,k,\leqslant 2}\) in the sum of three terms as follows:

and

where

and

$$\begin{aligned} \left( G_{(j,k)}\right) _{1\leqslant j,k\leqslant 2} = i{\varepsilon }\frac{m_1m_2}{\sqrt{m_0}} \begin{pmatrix} \displaystyle \frac{A_0}{m_1a_{1,0}^{1/2}} &{} \displaystyle \frac{B_0}{\sqrt{m_1m_2}(a_{1,0}a_{2,0})^{1/4}} \\ \displaystyle \frac{{{\,\mathrm{conj}\,}}(B_0)}{\sqrt{m_1m_2}(a_{1,0}a_{2,0})^{1/4}} &{} \displaystyle \frac{A_0}{m_1a_{1,0}^{1/2}}\end{pmatrix} \end{aligned}$$

with

With similar reasonings as for the eccentricities, we use the mean value theorem to evaluate the remainder in the truncation at order 0 of \(\overline{H} _{P,0}\) and \((\overline{H} _{P,(j,k)})_{1\leqslant j,k,\leqslant 2}\). Hence, this yields

Moreover the following estimates:

are obtained with the approximation of the formula (A.12).

Finally, in order to get a more tractable expression, one can shift the perturbation parts to

with \(\Lambda _{j,\star } = {\widehat{m}}_j \mu _j^{1/2}m_0^{1/6}\upsilon _0^{-1/3}\) where the two associated semi-major axes are both equal to the same value given by \(a_\star = m_0^{1/3} \upsilon _0^{-2/3}\). This yields

$$\begin{aligned} G_0= {\varepsilon }\upsilon _0B {{\mathcal {F}} }+ R^4_{P,0} \quad \text{ and } \quad G_{(j,k)} = \tilde{{\mathscr {Q}} }_{(j,k)} + R^4_{P,(j,k)} \end{aligned}$$

with the following thresholds:

that are estimated thanks to the bound .

As a consequence,

Lemma A.2

The averaged Hamiltonian can be written

with such that

$$\begin{aligned} \begin{aligned} R_0&= R_K^2 + R_{P,0}^2 + R_K^3 + R_{P,0}^3 + R_{P,0}^4 + \overline{H} _{*,0}\\ R_{(j,k)}&= R_{P,(j,k)}^1 + R_{P,(j,k)}^2 + R_{P,(j,k)}^3 + R_{P,(j,k)}^4 + \overline{H} _{*,(j,k)} \end{aligned} \end{aligned}$$

and

Moreover, if we assume \(\beta >1/3\), we can ensure that

Remark that this last bound comes from the threshold

that is obtained by application of the Cauchy inequalities.

In order to uncouple the fast and semi-fast degrees of freedom, we perform the symplectic linear transformation which diagonalizes the quadratic form Q. This leads to the Hamiltonian \(\tilde{{\mathscr {H}} }\) and its remainder \(\tilde{{\mathscr {R}} }= R\circ \tilde{\Psi }\). The inclusions (4.9) are ensured since \(\kappa \leqslant 1/2\).

Lemma 4.3: Semi-fast Frequency

Let us first prove the expression (4.13) which gives the lower bound of \(\varphi _1\) along a \(h_\delta \)-level curve.

A straightforward calculation shows that \(\varphi _{1,\delta }^{\min }\) is given by the smallest positive root of the polynomial equation \(4X^3 -(5+3\delta )X +1 =0\), where \(X = \sin (\varphi _{1,\delta }^{\min }/2)\). It follows that \(\varphi _{1,\delta }^{\min }\) is an analytic function of \(\delta \) in a neighborhood of 0, which satisfies

$$\begin{aligned} \varphi _{1,\delta }^{\min }= 2 \arcsin \left( \frac{\sqrt{2} -1}{2}\right) - \frac{3(\sqrt{2} -1) }{(3\sqrt{2} -2)\sqrt{1+2\sqrt{2}}}\delta + {{\mathcal {O}}} (\delta ^2). \end{aligned}$$

In order to prove the relations (4.14), let us begin to derive an asymptotic expansion of the integral \({{\mathcal {I}} }_\delta = {\int }_{\varphi _{1,\delta }^{\min }}^\pi \displaystyle \frac{{\mathrm {d}}\varphi }{\sqrt{U_\delta (\varphi )}}\) involved in the expression (4.4). \({{\mathcal {I}} }_\delta \) can be splitted in three different terms:

$$\begin{aligned} \begin{aligned}&{{\mathcal {I}} }_\delta = {{\mathcal {I}} }_\delta ^{1} + {{\mathcal {I}} }_\delta ^{2} + {{\mathcal {I}} }_\delta ^{3} \quad \text{ with } \quad \\&{{\mathcal {I}} }_\delta ^{1} = \int _{\varphi _{1,\delta }^{\min }}^\frac{\pi }{3} \frac{{\mathrm {d}}\varphi }{\sqrt{U_\delta (\varphi )}}, \quad {{\mathcal {I}} }_\delta ^{(2)} = \int _{\frac{\pi }{3}}^{\pi }\frac{{\mathrm {d}}\varphi }{\sqrt{U_\delta ^0(\varphi )}} \quad \text{ and } \quad \\&{{\mathcal {I}} }_\delta ^{3} = \int _{\frac{\pi }{3}}^{\pi } \left( \frac{1}{\sqrt{U_\delta (\varphi )}} - \frac{1}{\sqrt{U_\delta ^0(\varphi )}} \right) {\mathrm {d}}\varphi \\&\text{ where } \quad U_\delta ^0(\varphi ) = \delta + \frac{7}{24}(\varphi -\pi )^2 . \end{aligned} \end{aligned}$$

As \(U_\delta (\varphi _{1,\delta }^{\min }) =0\), Taylor formula leads to

$$\begin{aligned} \begin{aligned} {{\mathcal {I}} }_\delta ^{1}&= \left( \frac{\pi }{3} - \varphi _{1,\delta }^{\min }\right) \int _0^1 \frac{{\mathrm {d}}u}{\sqrt{u}\sqrt{G_\delta (u)}}\quad \text{ where } \quad \\ G_\delta (u)&= \int _0^1 {{\mathcal {F}} }'\left( \varphi _{1,\delta }^{\min }+ \left( \frac{\pi }{3} - \varphi _{1,\delta }^{\min }\right) uv \right) {\mathrm {d}}v . \end{aligned} \end{aligned}$$

As \(G_\delta (u) > G_\delta (1)\) and \(G_0(1) >1\) and if \(\delta >0\) is small enough, one has

As a consequence, \({{\mathcal {I}} }_\delta ^{1}\) is analytic with respect to \(\delta \).

The integral expression \({{\mathcal {I}} }_\delta ^{2}\) can be calculated explicitly as

$$\begin{aligned} {{\mathcal {I}} }_\delta ^{2} = \sqrt{\frac{24}{7}} \mathrm{arcsinh}\left( \sqrt{\frac{7}{54}}\frac{\pi }{\sqrt{\delta }} \right) = \sqrt{\frac{6}{7}} \vert \ln \delta \vert + {\mathscr {I}}_\delta ^{2} \end{aligned}$$

where \({\mathscr {I}}_\delta ^{2}\) is analytic in \(\delta \).

All that remains is to estimate the size of \({{\mathcal {I}} }_\delta ^{3}\) and of its first derivative. First of all, \(U_\delta \) being an infinitely differentiable function of \(\varphi \in [\pi /3, \pi ]\) satisfying the additional relations:

$$\begin{aligned} U_\delta (\pi ) = 1 + \delta + {{\mathcal {F}} }(\pi ) = \delta , \quad \frac{{\mathrm {d}}U_\delta }{{\mathrm {d}}\varphi }(\pi ) = \frac{{\mathrm {d}}^3 U_\delta }{{\mathrm {d}}\varphi ^3}(\pi )=0 \quad \text{ and } \quad \frac{{\mathrm {d}}^2 U_\delta }{{\mathrm {d}}\varphi ^2}(\pi )=\frac{7}{12} , \end{aligned}$$

Taylor formula leads to From the inequalities

that hold for \((\delta ,\varphi ) \in [\delta ^*,2\delta ^*]\times [\pi /3,\pi ]\), one can derive the following relations:

It follows that \({{\mathcal {I}} }_\delta ^{3}\) is analytic on \([\delta ^*,2\delta ^*]\) and that its first derivative is bounded by

As a consequence

$$\begin{aligned} T_\delta = \frac{2\pi }{\upsilon _0 \sqrt{{\varepsilon }}K} \left| \ln \delta \right| \left[ 1 + g(\delta ) \right] \end{aligned}$$

with and . As

$$\begin{aligned} \nu _\delta = \frac{\upsilon _0 \sqrt{{\varepsilon }}K}{\vert \ln \delta \vert } \left[ 1 - g(\delta ) + \frac{ g(\delta )^2}{1+ g(\delta )}\right] , \end{aligned}$$

we get the expressions (4.14).

Theorem 4.3: Semi-fast Holomorphic Extension

We consider the mechanical system

$$\begin{aligned} \tilde{{\mathscr {H}} }_1(I_1, \varphi _1) = \upsilon _0\left( -AI_1^2 + {\varepsilon }B {{\mathcal {F}} }(\varphi _1)\right) \end{aligned}$$

where A, B are two positive constants and the real function \({{\mathcal {F}} }\) is defined on \(]0,2\pi [\) by (4.11).

On the domain \({{\mathfrak {D}}} _*\), defined as

$$\begin{aligned} {{\mathfrak {D}}} _* = \left\{ \begin{array}{c}(I_1,\varphi _1) \in {\mathbb {R}}\times ]0,2\pi [ \quad \text{ such } \text{ that } \quad \tilde{{\mathscr {H}} }_1(I_1,\varphi _1) = h_\delta \\ \text{ with }\quad \delta ^* \leqslant \delta \leqslant 2\delta ^* \end{array}\right\} \end{aligned}$$

for some \(\delta ^*>0\), we can build a system of action-angle variables denoted \((J_1,\phi _1)\) such that

$$\begin{aligned} {{\mathscr {H}} }_1(J_1)= \tilde{{\mathscr {H}} }_1\circ {{\mathfrak {F}}}(J_1, \phi _1) =h_\delta \quad \text{ and } \quad {{\mathscr {H}} }_1'(J_1) = \nu _\delta . \end{aligned}$$

The transformation in action-angle variables, which will be denoted \({{\mathfrak {G}}}\), satisfies

$$\begin{aligned} {{\mathfrak {G}}}: \quad \Bigg \{ \begin{array}{ccc} {{\mathfrak {D}}} _{*} &{}\longrightarrow &{} {{\mathcal {S}} }_*\times {{\mathbb {T}}}\\ (I_1,\varphi _1) &{}\longmapsto &{} (J_1, \phi _1) \end{array} \end{aligned}$$

with \( {{\mathcal {S}} }_* = \left[ a, b\right] \) for some \(a< 0 <b\). We also denote \({{\mathfrak {F}}}={{{\mathfrak {G}}}}^{-1}\) the inverse of the action-angle transformation as in (4.16).

We rewrite the Hamiltonian in a suitable form for the complex extension,

$$\begin{aligned} \begin{aligned} \tilde{{\mathscr {H}} }_1(I_1, \varphi _1)&=-{\varepsilon }\upsilon _0 B(1+{\mathrm h }(I_1, \varphi _1))\\ \text{ with } \quad {\mathrm h }(I_1, \varphi _1)&= \frac{A}{{\varepsilon }B}I_1^2-1-{{\mathcal {F}} }(\varphi _1 ), \end{aligned} \end{aligned}$$

and the transformation \({{{\mathfrak {G}}}}\) can be defined explicitly by a classical integral formulation. The action is given by

$$\begin{aligned} \begin{aligned} J_1&= 4 \sqrt{{\varepsilon }\frac{B}{A}}\left( \pi -\varphi _{(I_1,\varphi _1)}^{\min }\right) \int _{0}^{1}\sqrt{{{\mathcal {U}} }^\pi _{(I_1, \varphi _1) }(x)} {\mathrm {d}}x\\ \text{ with } \quad {{\mathcal {U}} }^{\theta }_{(I_1,\varphi _1)}(x)&= 1 + {{\mathcal {F}} }\left( (1-x)\varphi _{(I_1,\varphi _1)}^{\min } + x\theta \right) + {\mathrm h }(I_1, \varphi _1) \end{aligned} \end{aligned}$$

where \(\varphi _{(I_1,\varphi _1)}^{\min }={{\mathcal {F}} }^{-1}(-1-{\mathrm h }(I_1,\varphi _1))\) and \(J_{1}\) is the action linked to an energy curve corresponding to an arbitrary shift of energy \(\delta \in ]\delta ^* ,2\delta ^* [\). The lower angle \(\varphi _{(I_1,\varphi _1)}^{\min }\) is well defined since \({{\mathcal {F}} }'(\varphi _0^{\min })\ne 0\) where \(\varphi _0^{\min }=2\arcsin \left( \frac{\sqrt{2} -1}{2}\right) \) is the minimal value of the angle \(\varphi _1\) along the separatrix hence \({{\mathcal {F}} }(\varphi _0^{\min })=-1\), consequently \({{\mathcal {F}} }^{-1}\) is analytic around \(-1\). Concerning the angle \(\phi _1\), we have to consider the time of transit from the point \(( 0,\varphi _{(I_1,\varphi _1)}^{\min })\) to \((I_1,\varphi _1)\) which is given by

$$\begin{aligned} \begin{aligned} \tau (I_1,\varphi _1)&=\frac{2}{\sqrt{{\varepsilon }AB}}\left( \varphi _1 -\varphi _{(I_1,\varphi _1)}^{\min }\right) \int _{0}^{1} \frac{{\mathrm {d}}x}{\sqrt{{{\mathcal {U}} }^{\varphi _1}_{(I_1,\varphi _1)}(x)}}\\&\text{ and }\quad \phi _1 =\displaystyle \frac{\pi }{2}\frac{\tau (I_1,\varphi _1)}{\tau (I_1,\pi )} . \end{aligned} \end{aligned}$$

Now, we look for the complex domain of holomorphy of the integrable Hamiltonian \({{\mathscr {H}} }_1\). We first consider the complex domain

$$\begin{aligned} D_{*,\hat{\rho }} = \left\{ (I_1,\varphi _1 )\in {{\mathbb {C}}}^2 / \exists (I_1^*,\varphi _1^*)\in {{\mathfrak {D}}} _{*} \text{ with: } \begin{array}{l} \left| I_1 - I_1^* \right| \leqslant \sqrt{{\varepsilon }} \hat{\rho }\\ \left| \varphi _1 - \varphi _1^* \right| \leqslant \hat{\rho }\end{array}\right\} \end{aligned}$$

for \(\hat{\rho }>0\) and \({\varepsilon }\) small enough (, ).

In order to disentangle the dependance of the complex domain \(D_{*,\hat{\rho }}\) with respect to \(\delta ^*\) and \({\varepsilon }\), we perform the following scalings:

$$\begin{aligned} I_1 =\sqrt{{\varepsilon }}{\hat{I}} _1 \quad \text{ and } \quad J_1 =\sqrt{{\varepsilon }} \hat{J } _1 \quad \text{ for } \quad ({\hat{I}} _1,\varphi _1)\in {\hat{{{\mathfrak {D}}} }}_{*} \end{aligned}$$

with

$$\begin{aligned} {\hat{{{\mathfrak {D}}} }}_{*} = \left\{ \begin{array}{c}({\hat{I}} _1,\varphi _1) \in {\mathbb {R}}\times ]0,2\pi [ \quad \text{ such } \text{ that } \quad {\hat{{\mathrm h }} }({\hat{I}} _1,\varphi _1) = \delta \\ \text{ with }\quad \delta ^* \leqslant \delta \leqslant 2\delta ^* \end{array}\right\} \end{aligned}$$

for the real analytic function

$$\begin{aligned} {\hat{{\mathrm h }} }({\hat{I}} _1, \varphi _1)=\frac{A}{B}{\hat{I}} _1^2-1-{{\mathcal {F}} }(\varphi _1), \end{aligned}$$

and we consider the complex extension \({\hat{D}}_{*,\hat{\rho }} = {{\mathcal {B}}}_{\hat{\rho }}{\hat{{{\mathfrak {D}}} }}_{*}\) with

$$\begin{aligned} {\hat{D}}_{*,\hat{\rho }} = \left\{ ({\hat{I}} _1,\varphi _1 )\in {{\mathbb {C}}}^2 / \exists ({\hat{I}} _1^*,\varphi _1^*)\in {\hat{{{\mathfrak {D}}} }}_{*} \text{ with: } \begin{array}{l} \left| {\hat{I}} _1 - {\hat{I}} _1^* \right| \leqslant \hat{\rho }\\ \left| \varphi _1 - \varphi _1^* \right| \leqslant \hat{\rho }\end{array}\right\} \end{aligned}$$

where \(\hat{\rho }>0\) is small enough ().

Likewise, we have the following real analytic functions:

$$\begin{aligned} \hat{J } _1({\hat{I}} _1, \varphi _1) = 4 \sqrt{\frac{B}{A}}\left( \pi -\hat{\varphi {}} _{({\hat{I}} _1,\varphi _1)}^{\min }\right) \displaystyle \int _{0}^{1}\sqrt{{\hat{{{\mathcal {U}} }} }^\pi _{({\hat{I}} _1, \varphi _1)}(x)} {\mathrm {d}}x \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \begin{aligned} {\hat{{{\mathcal {U}} }} }^{\theta }_{({\hat{I}} _1,\varphi _1)}(x)\!&=\! 1\! +\! {{\mathcal {F}} }\left( (1-x)\hat{\varphi {}} _{({\hat{I}} _1,\varphi _1)}^{\min }\! + x\theta \right) \! + {\hat{{\mathrm h }} }({\hat{I}} _1, \varphi _1),\\ \hat{\varphi {}} _{({\hat{I}} _1,\varphi _1)}^{\min }\!\!&=\! {{\mathcal {F}} }^{-1}(-1 - {\hat{{\mathrm h }} }({\hat{I}} _1, \varphi _1)), \end{aligned} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \phi _1 ({\hat{I}} _1,\varphi _1)= \displaystyle \frac{\pi }{2} \frac{\varphi _1 -\hat{\varphi {}} _{({\hat{I}} _1,\varphi _1)}^{\min }}{\pi -\hat{\varphi {}} _{({\hat{I}} _1,\varphi _1)}^{\min }} \frac{\displaystyle \int _{0}^{1}\frac{{\mathrm {d}}x}{\sqrt{{\hat{{{\mathcal {U}} }} }^{\varphi _1}_{({\hat{I}} _1, \varphi _1)}(x)}}}{\displaystyle \int _{0}^{1}\frac{{\mathrm {d}}x}{\sqrt{{\hat{{{\mathcal {U}} }} }^{\pi }_{({\hat{I}} _1, \varphi _1)}(x)}}}. \end{aligned}$$

Hence we consider the transformation

$$\begin{aligned} {\hat{{{\mathfrak {G}}}}}: \quad \Bigg \{ \begin{array}{ccc} {\hat{{{\mathfrak {D}}} }}_{*} &{}\longrightarrow &{} {{\hat{{{\mathcal {S}} }}} }_* \times {{\mathbb {T}}}\\ ({\hat{I}} _1,\varphi _1) &{}\longmapsto &{} ( \hat{J } _1, \phi _1) \end{array} \end{aligned}$$

with

$$\begin{aligned} {{\hat{{{\mathcal {S}} }}} }_* = \left[ \frac{a}{\sqrt{{\varepsilon }}}, \frac{b}{\sqrt{{\varepsilon }}}\right] \quad \text{ for } \text{ some } \quad a<0 <b \end{aligned}$$

which corresponds to the action-angle variables for the mechanical system

$$\begin{aligned} \hat{{\mathscr {H}} }_1({\hat{I}} _1, \varphi _1) = \upsilon _0\left( -A {\hat{I}} _1^2 + B {{\mathcal {F}} }(\varphi _1)\right) \end{aligned}$$

and its inverse mapping will be denoted \({\hat{{{\mathfrak {F}}}}}={\hat{{{\mathfrak {G}}}}}^{-1}\). Moreover, these transformations are independent of \(\varepsilon \).

By classical theorem of complex analysis, \({\hat{{{\mathfrak {F}}}}}\) (resp. \({\hat{{{\mathfrak {G}}}}}\)) can be extended in a unique way to a map F (resp. G) holomorphic on a complex set

$$\begin{aligned} \begin{aligned} {{\mathcal {B}}}_r{{\hat{{{\mathcal {S}} }}} }_*\times {{\mathcal {V}}} _{s}{{\mathbb {T}}}= \left\{ \begin{array}{c} ( \hat{J } _1,\phi _1 )\in {{\mathbb {C}}}^2\, / \, \exists \hat{J } _1^*\in {{\hat{{{\mathcal {S}} }}} }_* \quad \text{ such } \text{ that } \quad \\ \left| \hat{J } _1- \hat{J } _1^* \right| \leqslant r ,\quad {{\,\mathrm{Re}\,}}(\phi _1)\in {{\mathbb {T}}}\\ \text{ and }\quad \left| {{\,\mathrm{Im}\,}}(\phi _1) \right| \leqslant s \end{array}\right\} \end{aligned} \end{aligned}$$

and G holomorphic over the set \({\hat{D}}_{*,\hat{\rho }}\) for some \(r>0\), \(s>0\) and \(\hat{\rho }>0\) small enough. We want to compute a lower bound on the analyticity widths r, s. For \(\hat{\rho }>0\), we denote

$$\begin{aligned} \left\| \hat{J } _1 \right\| _{\hat{\rho }} =\sup \limits _{{\hat{D}}_{*,\hat{\rho }}} \left| \hat{J } _1( {\hat{I}} _1,\varphi _1) \right| , \quad \left\| \phi _1 \right\| _{\hat{\rho }} = \sup \limits _{{\hat{D}}_{*,\hat{\rho }}}\left| \phi _1({\hat{I}} _1,\varphi _1) \right| , \end{aligned}$$

moreover, we consider

$$\begin{aligned} {\tilde{M}} =\left\| \hat{J } _1 \right\| _{\hat{\rho }} +\left\| \phi _1 \right\| _{\hat{\rho }} . \end{aligned}$$

Finally, since the real mapping \({\hat{{{\mathfrak {G}}}}}\) is symplectic, it is non-degenerate at each point of the domain \({\hat{{{\mathfrak {D}}} }}_{*}\) and we denote

$$\begin{aligned} {\tilde{L}} =\sup \limits _{{\hat{{{\mathfrak {D}}} }}_{*}} \left| {\mathrm {d}}{\hat{{{\mathfrak {G}}}}}^{-1}_{({\hat{I}} _1,\varphi _1)} \right| . \end{aligned}$$

By a standard application of the Lipschitz inverse function theorem (see [18]), we obtain the main estimate of this section.

Theorem A.1

Suppose that U is an open subset of a Banach space \((E,\vert \vert .\vert \vert )\) and that \(g : U\rightarrow E\) is a Lipschitz mapping with constant \(K<1\).

Let \(f(x)=x+g(x)\). If the closed ball \({{\mathcal {B}}}_\varepsilon \{x\}\) centered at \(x\in E\) of radius \(\varepsilon \) is contained in U, then

$$\begin{aligned} {{\mathcal {B}}}_{(1-K )\varepsilon }\{f (x)\}\subseteq f({{\mathcal {B}}}_\varepsilon \{x\})\subseteq {{\mathcal {B}}}_{(1+K )\varepsilon } \{f (x)\} . \end{aligned}$$

The mapping f is a homeomorphism of U onto \(f^{-1} (U)\), the inverse mapping \(f^{-1}\) is a Lipschitz mapping with constant \((1- K)^{-1}\) and f(U) is an open subset of E.

More precisely, we use Theorem 4.1 and Cauchy inequalities applies on G which yields

Theorem A.2

With the previous notations, if

then G admits an inverse mapping F which is holomorphic on \({{\mathcal {B}}}_{r}{{\hat{{{\mathcal {S}} }}} }_* \times {{\mathcal {V}}} _s{{\mathbb {T}}}\) and F is C-Lipschitz with .

Hence, in order to estimate the analyticity widths in action-angle variables for the considered mechanical system, we have to compute the dependance w.r.t. the quantity \(\delta ^*\) of the analyticity width \(\hat{\rho }\) in the original variables \(({\hat{I}} _1,\varphi _1)\), the upper bounds \({\tilde{L}} \) on the real domain \({\hat{{{\mathfrak {D}}} }}_{*}\) and \({\tilde{M}} \) on the complex domain \(D_{*,\hat{\rho }}\). In order to bound \({\tilde{L}} \), we use the fact that \({\hat{{{\mathfrak {G}}}}}\) is symplectic on the real domain \({\hat{{{\mathfrak {D}}} }}_{*}\), hence the coefficients of the Jacobian matrix linked to \({\mathrm {d}}{\hat{{{\mathfrak {G}}}}}^{-1}\) are given by the derivatives of \({\hat{{{\mathfrak {G}}}}}\) that we estimate by an application of Cauchy inequalities over \(D_{*,\hat{\rho }} \). We obtain

Concerning the quantities \(\hat{\rho }\) and \({\tilde{M}} \) on the complex domain \(D_{*,\hat{\rho }}\), rough estimates ensure that if we choose the analyticity width for \(\delta ^*\) small enough (), we can ensure the upper bound

Plugging these estimates in the latter theorem ensure that G admits an inverse mapping F which is holomorphic on \({{\mathcal {B}}}_r{{\hat{{{\mathcal {S}} }}} }_*\times {{\mathcal {V}}} _s{{\mathbb {T}}}\) for

Going back to the initial variables, if we denote \(F=(F_1,F_2)\), then the extended transformation in action-angle coordinates in the complex plane is given by

$$\begin{aligned} (\sqrt{{\varepsilon }} F_1(J_1/\sqrt{{\varepsilon }}, \phi _1), F_2(J_1/\sqrt{{\varepsilon }}, \phi _1)) \end{aligned}$$

and we obtain the analyticity widths of Theorem 4.3.

Moreover, F is C-Lipschitz with and the distance to the real domain of the image is bounded by \(\sqrt{{\varepsilon }}(\delta ^{*})^{{\hat{p} }-1/2}\) for \(I_1\) and by \((\delta ^{*})^{{\hat{p} }-1/2}\) for \(\varphi _1\) hence these quantities are bounded by \(\sqrt{{\varepsilon }} (\delta ^*)^5\) and \((\delta ^*)^5\) for \({\hat{p} }= 11/2\).

Theorem 4.4: Semi-fast Action-Angle variables

The existence of the transformation \(\Psi \) is immediate by application of Lemma 4.3 to the averaged Hamiltonian \(\tilde{{\mathscr {H}} }\) considered in (4.3).

Finally, the two last thresholds in (4.17) are deduced by an application of the Cauchy inequalities.

Theorem 4.5: Second Averaging Theorem

In the same way as for the First Averaging Theorem, we define firstly an iterative lemma of averaging. Let us introduce some notations: \((\xi _k)_{k\in \{1,\ldots , 5\}}\) are given positive numbers such that

$$\begin{aligned} 0<\xi _j< \rho _j, \quad 0< \xi _{2+j}<\sigma _j \quad \text{ for } j\in \{1,2\}\text{, } \quad 0<\xi _{5}<\sqrt{\rho _2\sigma _2}, \end{aligned}$$

and for \(0\leqslant r\leqslant 1\), we denote \({{\mathfrak {K}} }_r\), the domain such as

$$\begin{aligned} {{\mathfrak {K}} }_r={{\mathcal {B}}}_{\rho _1-r\xi _1}{{\mathcal {S}} }_*\times {{\mathcal {B}}}^1_{\rho _2-r\xi _2}\times {{\mathcal {V}}} _{\sigma _1-r\xi _3}{{\mathbb {T}}}\times {{\mathcal {V}}} _{\sigma _2-r\xi _4}{{\mathbb {T}}}\times {{\mathcal {B}}}^4_{\sqrt{\rho _2\sigma _2} - r\xi _{5}}. \end{aligned}$$

Moreover, we will consider \(\nu _0\) a lower bound for the semi-fast frequency \({{\mathscr {H}} }'_1\) on the complex domain \({{\mathcal {K}} }_p\) and according to (4.19), we can choose

with our polynomial dependence of \(\delta ^*\) with respect to \(\varepsilon \).

Hence, we set out the following:

Lemma A.3

(Second iterative lemma). Let \(\varvec{\rho }^{-}\), \(\varvec{\sigma }^{-}\), \((\xi _k)_{k\in \{1,\ldots ,4\}}\) be fixed positive real numbers that depend on the small parameter \({\varepsilon }\) and

$$\begin{aligned} \begin{aligned}&\varvec{\rho }^{+} =\varvec{\rho }^- - (\xi _1,\xi _2),&\quad&\varvec{\sigma }^{+} = \varvec{\sigma }^- - (\xi _3,\xi _4), \\&\xi _{5}= \sqrt{\rho _2^-\sigma _2^-} - \sqrt{\rho _2^{+}\sigma _2^{+}}&\text{ such } \text{ as } 0<\rho _j^{+}\text{, } 0 < \sigma _j^{+} \text{ for } j\in \{1,2\}\text{. } \end{aligned} \end{aligned}$$

Let \({{\mathscr {H}} }^{-}\) be a Hamiltonian of the form

$$\begin{aligned} \begin{aligned} {{\mathscr {H}} }^{-}({\mathbf{J} }, \phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) =&{{\mathscr {H}} }_1(J_1) + {{\mathscr {H}} }_2(J_2) + \overline{{\mathscr {Q}} }(J_1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} })\\ +&{{\mathscr {H}} }_{*}^{0,-}({\mathbf{J} }, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) + {{\mathscr {H}} }_{*}^{1,-}({\mathbf{J} }, \phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) \end{aligned} \end{aligned}$$

with \({{\mathscr {H}} }_*^{l,-} = {{\mathscr {H}} }_{*,0}^{l,-} + \sum _{j,k\in \{1,2\}} {{\mathscr {H}} }_{*,(j,k)}^{l,-}w_j{\widetilde{w} }_k\) for \(l\in \{0,1\}\) (given by (4.8)), which satisfies the D’Alembert rule, is analytic on the domain \({{\mathfrak {K}} }^{-}_0={{\mathcal {K}} }_{\varvec{\rho }^{-}, \varvec{\sigma }^{-}}\) and such that

$$\begin{aligned} {\overline{{\mathscr {H}} }_{*}^{1,-}}({\mathbf{J} }, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) = \frac{1}{2\pi }\int _0^{2\pi } {{{\mathscr {H}} }_{*}^{1,{-}}}({\mathbf{J} }, \phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }){\mathrm {d}}\phi _1 =0. \end{aligned}$$

Let \((\eta _l^{-})_{l\in \{0,2\}}\) and \((\mu _{l,m}^{-})_{l\in \{0,2\},m\in \{0,1,2\}}\) be fixed positive real numbers, which depend on \({\varepsilon }\), such that:

$$\begin{aligned} \begin{aligned}&\left\| {{{\mathscr {H}} }_{*,0}^{1,{-}}} \right\| _{{{\mathfrak {K}} }_0^{-}}\leqslant \eta _0^{-} ,&\quad&\left\| {{{\mathscr {H}} }_{*,(j,k)}^{1,{-}}} \right\| _{{{\mathfrak {K}} }_0^{-}}\leqslant \eta _2^{-} ,&\\&\left\| {{{\mathscr {H}} }_{*,0}^{0,{-}}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{0,0}^{-} ,&\quad&\left\| {{{\mathscr {H}} }_{*,(j,k)}^{0,{-}}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{2,0}^{-} ,&\end{aligned} \end{aligned}$$
(A.13)

and

$$\begin{aligned} \left\| \partial _{J_m}{{\mathscr {H}} }_{*,0}^{0,{-}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{0,m}^{-},\quad \left\| \partial _{J_m}{{\mathscr {H}} }_{*,(j,k)}^{0,{-}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{2,m}^{-} \quad (m\in \{1,2\}). \end{aligned}$$

If we assume that

(A.14)

then there exists a canonical transformation

and such that, in the new variables, the Hamiltonian \({{\mathscr {H}} }^{+}= {{\mathscr {H}} }^{-} \circ \overline{\Psi }^{+}\) satisfies the D’Alembert rule and can be written

$$\begin{aligned} \begin{aligned} {{\mathscr {H}} }^{+}&= {{\mathscr {H}} }_1 + {{\mathscr {H}} }_2 + \overline{{\mathscr {Q}} }+ {{{\mathscr {H}} }_{*}^{0,{-}}} + {{\mathscr {H}} }_*^{+}\\&= {{\mathscr {H}} }_1 + {{\mathscr {H}} }_2 + \overline{{\mathscr {Q}} }+ {{{\mathscr {H}} }_{*}^{0,{+}}} + {{\mathscr {H}} }_{*}^{1,+} \end{aligned} \quad \text{ with } \quad \left\{ \begin{array}{l} {{{\mathscr {H}} }_{*}^{0,{+}}} = {{\mathscr {H}} }_{*}^{0,{-}} + \overline{{\mathscr {H}} }_{*}^{+} \\ {{{\mathscr {H}} }_{*}^{1,{+}}} = {{\mathscr {H}} }_*^{+} - \overline{{\mathscr {H}} }_*^{+} \end{array} \right. \end{aligned}$$

such that for \(l\in \{0,1\}\) (given by (4.8)) and

Furthermore, we have the thresholds

$$\begin{aligned} \begin{aligned}&\left\| {{{\mathscr {H}} }_{*,0}^{1,{+}}} \right\| _{{{\mathfrak {K}} }_0^{-}}\leqslant \eta _0^{+} ,&\quad&\left\| {{{\mathscr {H}} }_{*,(j,k)}^{1,{+}}} \right\| _{{{\mathfrak {K}} }_0^{-}}\leqslant \eta _2^{+} ,&\\&\left\| {{{\mathscr {H}} }_{*,0}^{0,{+}}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{0,0}^{+} ,&\quad&\left\| {{{\mathscr {H}} }_{*,(j,k)}^{0,{+}}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{2,0}^{+} ,&\end{aligned} \end{aligned}$$
(A.15)

and

$$\begin{aligned} \left\| \partial _{J_m}{{\mathscr {H}} }_{*,0}^{0,{+}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{0,m}^{+},\quad \left\| \partial _{J_m}{{\mathscr {H}} }_{*,(j,k)}^{0,{+}} \right\| _{ {{\mathfrak {K}} }_0^{-}}\leqslant \mu _{2,m}^{+}\quad (m\in \{1,2\}), \end{aligned}$$
(A.16)

with the following quantities:

(A.17)

for \(m\in \{1,2\}\) and

$$\begin{aligned} \begin{aligned} \gamma _0^+&= \frac{\eta _0^-}{\xi _1\xi _3} , \quad \gamma _2^+= \eta _2^-\left( \frac{1}{\xi _1\xi _3} + \frac{1}{(\xi _5)^2}\right) ,\quad \theta _0^+ = \frac{\mu _{0,1}^-}{\xi _3} + \gamma _0^+ ,\\ \theta _1^+&= \frac{\left\| \overline{{\mathscr {Q}} }_{(j,k)} \right\| _{{{\mathfrak {K}} }^-_0}}{\xi _1\xi _3}+ \frac{\mu _{2,1}^-}{\xi _3} ,\\ \theta _2^+&= \frac{\left\| \overline{{\mathscr {Q}} }_{(j,k)} \right\| _{{{\mathfrak {K}} }^-_0}}{(\xi _5)^2} + \frac{\mu _{2,0}^-}{(\xi _5)^2} + \gamma _2^+ . \end{aligned} \end{aligned}$$
(A.18)

Proof

We define \(\overline{\Psi }^+:{{\mathfrak {K}} }_1^-\longrightarrow {{\mathfrak {K}} }_0^-\) as the time-one map of the Hamiltonian flow generated by some auxiliary function \({\chi ^+}\), i.e. \(\overline{\Psi }^+=\Phi _1^{\chi ^+}\) with

$$\begin{aligned} {\chi ^+}({\mathbf{J} },\phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) = \frac{2\pi }{{{\mathscr {H}} }_1'(J_1)}\int _0^1 s {{{\mathscr {H}} }_{*}^{1,-}}({\mathbf{J} },\phi _1+ 2\pi s, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }){\mathrm {d}}s \end{aligned}$$

such that the following properties are satisfied:

$$\begin{aligned} \begin{aligned}&\left\{ \chi ^+,{{\mathscr {H}} }_1\right\} + {{\mathscr {H}} }_*^{1,-} = 0,\\&\overline{\chi }^+({\mathbf{J} }, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) = \frac{1}{2\pi }\int _0^{2\pi } \chi ^+({\mathbf{J} }, \phi _1,{\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) {\mathrm {d}}\phi _1 = 0,\\&\text{ and } \quad \chi ^+ = \chi _0^+ + \sum _{1\leqslant j,k\leqslant 2}\chi ^+_{(j,k)}w_j{\widetilde{w} }_k \quad \text{(given } \text{ by } \quad (4.8)). \end{aligned} \end{aligned}$$
(A.19)

Thus, for the same reason as in Lemma A.1, the Hamiltonian can be written

$$\begin{aligned} {{\mathscr {H}} }^+ = {{\mathscr {H}} }^-\circ \overline{\Psi }^+ = {{\mathscr {H}} }_1 + {{\mathscr {H}} }_2 + \overline{{\mathscr {Q}} }+ {{\mathscr {H}} }_{*}^{0,-} + \underbrace{{{{\mathscr {H}} }_{*}^{1,-}} + \left\{ {\chi ^+},{{\mathscr {H}} }_1\right\} }_{(*)} +{{\mathscr {H}} }^+_* \end{aligned}$$

with

$$\begin{aligned} {{\mathscr {H}} }_*^+ = \int _0^1 \left\{ {\chi ^+},\overline{{\mathscr {Q}} }+{{\mathscr {H}} }_{*}^{0,-}+ s{{\mathscr {H}} }_{*}^{1,-}\right\} \circ \Phi _s^{\chi ^+} {\mathrm {d}}s \end{aligned}$$

and \((*)\) is equal to zero by (A.19).

Then, in order to estimate the size of the remainder \({{\mathscr {H}} }_*^+\), the thresholds (A.13) provide

while the Cauchy inequalities imply the following:

as well as the following estimates on the Poisson brackets:

and

as \( 1 \leqslant \frac{\sqrt{\rho _2^- \sigma _2^-}}{\xi _5} \leqslant \frac{\rho _2^- \sigma _2^-}{(\xi _5)^2}\). Consequently, the remainder of the transformation \(\overline{\Psi }^+\) is bounded such that

where \(\gamma _0^+\), \(\gamma _2^+\), \(\theta _0^+\),\(\theta _1^+\) and \(\theta _2^+\) are defined in (A.17). Moreover by taking into account that \(\overline{\chi }^+=0\) (given by (A.19)), we deduce the following:

Hence, if we denote \( {{{\mathscr {H}} }_{*}^{0,+}} = {{{\mathscr {H}} }_{*}^{0,-}} + \overline{{\mathscr {H}} }_{*}^+\) and \( {{{\mathscr {H}} }_{*}^{1,+}} = {{\mathscr {H}} }_*^+ - \overline{{\mathscr {H}} }_*^+\) then the triangle inequality gives the estimates (A.15) and (A.16) (together with the Cauchy inequalities for the last).

Finally, in the same way as for Lemma A.1, the conditions (A.14) provide the estimates on the size of the transformation \(\overline{\Psi }^+\) which yields (A.17) and (A.18). \(\quad \square \)

Now, in order to prove Theorem 4.5, one applies iteratively Lemma A.3 to the Hamiltonian \({{\mathscr {H}} }\) that can be written:

$$\begin{aligned} \begin{aligned} {{\mathscr {H}} }({\mathbf{J} }, \phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) =&\, {{\mathscr {H}} }_1(J_1) + {{\mathscr {H}} }_2(J_2) + \overline{{\mathscr {Q}} }(J_1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) \\ +&\,{{\mathscr {H}} }^{0,0}_*({\mathbf{J} }, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) +{{\mathscr {H}} }^{1,0}_*({\mathbf{J} }, \phi _1,{\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) \end{aligned} \end{aligned}$$

where

$$\begin{aligned} {{\mathscr {H}} }^{0,0}_*= & {} {{\mathscr {H}} }^{0,0}_{*,0} + \sum _{j,k\in \{1,2\}} {{\mathscr {H}} }^{0,0}_{*,(j,k)}w_j{\widetilde{w} }_k = \overline{{\mathscr {R}} },\\ {{\mathscr {H}} }^{1,0}_*= & {} {{\mathscr {H}} }^{1,0}_{*,0} + \sum _{j,k\in \{1,2\}} {{\mathscr {H}} }^{1,0}_{*,(j,k)}w_j{\widetilde{w} }_k = {{\mathscr {R}} }-\overline{{\mathscr {R}} }+ {{\mathscr {Q}}}- \overline{{\mathscr {Q}} },\\&\text{ and } \quad \overline{{\mathscr {R}} }({\mathbf{J} }, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) = \frac{1}{2\pi }\int _0^{2\pi } {{\mathscr {R}} }({\mathbf{J} }, \phi _1, {\mathbf{w} }, {\widetilde{{\mathbf{w} }} }) d\phi _1 , \end{aligned}$$

with the following thresholds:

Moreover, by reducing the domain of analyticity to \({{\mathcal {K}} }_{5/6}\), one can apply the Cauchy inequalities and obtain the followings:

In the same way as in the proof of Theorem 4.5, let s a non-zero integer such that \(s = {\mathrm E }({\varepsilon }^{-{\mathrm q }}) + 1\) where

$$\begin{aligned} {\mathrm q }= \frac{3\beta -1}{15} \quad \text{ for } \quad 4/9< \beta <1/2. \end{aligned}$$

We define

as well as the sequences \(\left( \varvec{\rho }^{j}\right) _{j\in \{0,1,\ldots ,s\}}\),\(\left( \varvec{\sigma }^{j}\right) _{j\in \{0,1,\ldots ,s\}}\) with

$$\begin{aligned} (\varvec{\rho }^{j}, \varvec{\sigma }^{j}) = \left( \frac{5}{6} - \frac{j}{4s}\right) (\varvec{\rho },\varvec{\sigma }) \quad \text{ for } \quad j\in \{0,\ldots ,s\} \end{aligned}$$

such that \({{\mathfrak {K}} }^{j}_r = {{\mathcal {K}} }_{\frac{5}{6}-\frac{j+r}{4s}}\) for \(0\leqslant r\leqslant 1\).

Replacing the notation \( ^-\) and \( ^+\) by \(^{j-1}\) and \(^{j}\) and assuming that for all \(0<j\leqslant s\) the following conditions (associated with (A.14)) are fulfilled:

then an iterative application of Lemma A.3 to the Hamiltonian \({{\mathscr {H}} }\) provides a sequence of canonical transformations \(\left( \overline{\Psi }^{j}\right) _{j\in \{1,\ldots ,s\}}\) such that \({{\mathscr {H}} }\circ \overline{\Psi }= \overline{\Psi }^{1}\circ \overline{\Psi }^{2}\circ \cdots \circ \overline{\Psi }^{s}\) is equal to the Hamiltonian \(\overline{{\mathscr {H}} }+ {\mathscr {H}}^\dagger _*\) with \({{\mathscr {F}} }= {{\mathscr {H}} }_{*}^{0,s}\) and \({\mathscr {H}}^\dagger _* = {{\mathscr {H}} }_{*}^{1,s}\).

For the same reasons as in the proof of Theorem 4.1, for all \(n\in \{1,\ldots , s\}\), the sequences \(\left( \eta _l^{j}\right) _{j\in \{1,\ldots ,n\}}\) and \(\left( \mu _{l,m}^{j}\right) _{j\in \{1,\ldots , n\}}\) must satisfy the following induction hypothesis:

$$\begin{aligned} \begin{aligned}&\eta _l^{j}\leqslant \eta _l^{j-1}\exp (-1) ,&\quad&\mu _{l,m}^{j} - \mu _{l,m}^{j-1} \leqslant \frac{\mu _{l,m}^{0}}{s},&\end{aligned} \end{aligned}$$
(A.20)

for \(l\in \{0,2\}\) and \(m\in \{0,1,2\}\).

For \(n=1\), (A.20) is fulfilled as \(4/9<\beta < 1/2\) and . Moreover,

imply that

for and

for (\(m\in \{0,1,2\}\)).

For a fixed integer n, the induction is immediate since the sequences \(\left( \eta _l^{j}\right) _{j\in \{0,\ldots ,n\}}\) are decreasing such that \(\displaystyle \frac{\eta _0^{n}}{\eta _2^{n}} \leqslant \frac{\eta _0^{0}}{\eta _2^{0}}\) while \(\displaystyle \mu _{l,m}^{n} \leqslant 2 \mu _{l,m}^{0}\).

Hence, this proves the hypothesis (A.20) up to s and consequently that

which provide (4.23) and a part of the thresholds (4.21) and (4.22). The missing thresholds of (4.22) are deduced by using the Cauchy inequalities in a restricted domain \({{\mathcal {K}} }_p\) with \(0<p<7/12\).

Finally, the Eq. (3.2) as well as the Cauchy inequalities provide the size of the transformation \(\overline{\Psi }\) on \({{\mathcal {K}} }_p\):

and in the same way

Remark that as \(\chi ^{j}\) does not depend on \(\phi _2\) for all \(j\in \{1,\ldots ,s\}\) then . This yields \({{\mathcal {K}} }_{5/12} \subseteq \overline{\Psi }({{\mathcal {K}} }_{7/12}) \subseteq {{\mathcal {K}} }_{9/12}\) for .

Theorem 4.6: Secular Frequencies

We denote by \(f(J,\phi )\) a regular function on \({\mathbb {R}}\times {{\mathbb {T}}}\) and by \({\tilde{f}}(\varphi ) \) the real function satisfying the relation \({\tilde{f}} \circ {{\mathfrak {F}}}_2 = f\). Using these notations, the average of f at \(J_*\in {{\mathcal {S}} }_*\) reads

(A.21)

As \({\tilde{{\mathcal {A}}}}(2\pi - \varphi ) = {\tilde{{\mathcal {A}}}}(\varphi )\) and \({\tilde{{{\mathcal {B}}}}}(2\pi - \varphi ) = {{\,\mathrm{conj}\,}}({\tilde{{{\mathcal {B}}}}}(\varphi ))\), the expressions of \({\overline{ {\mathcal {A}}}}(J_*)\) and \({\overline{{{\mathcal {B}}}}}(J_*)\) given by (4.24) follow.

The asymptotic expansions of \({\overline{ {\mathcal {A}}}}(J_*)\) and \({\overline{{{\mathcal {B}}}}}(J_*)\) have now to be derived. As \({\tilde{{\mathcal {A}}}}(\pi ) = 7/8\), it follows from Lemma 2 and (A.21) that

The main part of the integral involved in the previous expressions can be computed as follows:

As and because , the two last integrals are respectively and . It turns out that

and .

For the same reasons, we also have

where the real coefficients \( C_{\mathcal {A}}\) and \( C_{{\mathcal {B}}}\) are bounded by

$$\begin{aligned} -28< C_{\mathcal {A}}<-27 \quad \text{ and } \quad 16< C_{{\mathcal {B}}}< 17. \end{aligned}$$

This provides all that is needed for deriving the asymptotic expansion of the secular frequencies and . Indeed, these frequencies are given by where \(\lambda _j\) are the two roots of the polynomial

$$\begin{aligned} \lambda ^2 - \frac{m_1+m_2}{m_1m_2}{\overline{ {\mathcal {A}}}}(J_*) \lambda - \frac{{\overline{{{\mathcal {B}}}}}(J_*)^2-{\overline{ {\mathcal {A}}}}(J_*)^2}{m_1m_2} . \end{aligned}$$

At this point, Theorem 4.6 is deduced from an asymptotic expansion of the \(\lambda _j\), from which it follows that the coefficients \(c_2\) involved in (4.25) satisfy the relations

$$\begin{aligned} -90< c_2 = 2(C_{\mathcal {A}}- C_{{\mathcal {B}}}) < - 86. \end{aligned}$$

Theorem 4.7: Diagonalization

By the discussion that precedes Theorem 4.7, as the spectrum of (4.26) is simple, there exists a symplectic transformation \(\check{\Psi }\) which is linear with respect to , and diagonalizes the quadratic form (4.26).

In the general case the diagonalizing transformation is generated by a function which can be written

where are of order 1 over the considered domain.

Using Cauchy inequalities to bound the derivatives of \(\chi \) in order to control the variation of the angles associated with under the considered transformation, we obtain the upper bounds

since .

Finally, by Lemma 4.2 the Taylor expansion reads

Together with the estimates (4.22) of Theorem 4.5, this provides the threshold (4.28) on .

Theorem 5.1: Application of a Pöschel version of KAM Theory

As it was specified in Sect. 5, from now on, we constrain \({\varepsilon }\) to be inside an interval \([{\varepsilon }_0/2, {\varepsilon }_0]\) for an arbitrary \({\varepsilon }_0>0\).

Let us consider the frequency map linked to the Hamiltonian \(\check{{\mathscr {H}} }\) (see Theorem 4.7) that is denoted \((\varvec{\omega }(\varvec{\Gamma }), \varvec{\Omega }(\varvec{\Gamma }))\) with \(\omega _j = {{\mathscr {H}} }'_j + \partial _{\Gamma _j} {{\mathscr {F}} }_0\) and \(\Omega _j = g_j\), and the following thresholds:

that are deduced from and the bounds (4.22) and (4.27). Moreover, we have the following thresholds on the derivatives:

with \({{\mathscr {E}}} _1= \upsilon _0K^2B^{-1}\) and \({{\mathscr {E}}} _2 = -2E\upsilon _0\) (that are not equal to zero) from the bounds (4.22) and the mean value theorem. Consequently the eigenvalues of \({\mathrm {d}}\varvec{\omega }\) are small perturbations of \(\displaystyle \frac{{{\mathscr {E}}} _1}{{\varepsilon }_0^{{\mathrm q }}\left| \ln {{\varepsilon }_0} \right| ^4}\) and \({{\mathscr {E}}} _2\). We also ensure that \({\mathrm {d}}\varvec{\omega }\) is inversible with the eigenvalues \(\lambda _1(\varvec{\Gamma })\), \(\lambda _2(\varvec{\Gamma })\) such that

Hence, \(\varvec{\omega }\) is a local diffeomorphism.

In order to apply Pöschel version of KAM theory for the persistence of lower dimensional normally elliptic invariant tori [28], we must consider a domain where the internal frequency map \(\varvec{\omega }\) is a diffeomorphism. Hence, we set out the following

Lemma A.4

For \({\varepsilon }\in [{\varepsilon }_0/2,{\varepsilon }_0]\), the internal frequency map \(\varvec{\omega }\) is a diffeomorphism from \(\Pi = {{\mathcal {B}}}^2_{\rho }\) onto its image provided by

Moreover, we have the upper bounds

(A.22)

Proof

We consider \(\varvec{\omega }_0=\varvec{\omega }- \varvec{\omega }({\mathbf{0} })\) where \(\varvec{\omega }_0\) is holomorphic on the closed ball \({{\mathcal {B}}}_{\rho _1}^2\) with . Then, we define

$$\begin{aligned} \begin{aligned} \tilde{\varvec{\omega }}_0=&({\mathrm {d}}\varvec{\omega }({\mathbf{0} }))^{-1}\varvec{\omega }_0 - {{\,\mathrm{Id}\,}}\quad \text{ such } \text{ that } \quad {\mathrm {d}}\tilde{\varvec{\omega }}_0 = ({\mathrm {d}}\varvec{\omega }({\mathbf{0} }))^{-1}({\mathrm {d}}\varvec{\omega }- {\mathrm {d}}\varvec{\omega }({\mathbf{0} })). \end{aligned} \end{aligned}$$

Hence, since the highest eigenvalue of \(({\mathrm {d}}\varvec{\omega }({\mathbf{0} }))^{-1}\) satisfies for \({\varepsilon }\) small enough with \({{\mathscr {E}}} _2 \ne 0\). Furthermore, by the mean value theorem as well as the Cauchy inequalities, we can ensure that on the closed ball \({{\mathcal {B}}}^2_{\widetilde{\rho }}\) such that then

Consequently, the application:

$$\begin{aligned} \hat{\varvec{\omega }}_0 = {{\,\mathrm{Id}\,}}+ \tilde{\varvec{\omega }}_0 \end{aligned}$$

is a diffeomorphism from \({{\mathcal {B}}}^2_{\widetilde{\rho }}\) to \(\tilde{\varvec{\omega }}_0({{\mathcal {B}}}^2_{\widetilde{\rho }})\) by the fixed point theorem. Moreover, \(\hat{\varvec{\omega }}({\mathbf{0} }) = {\mathbf{0} }\) yields

$$\begin{aligned} {{\mathcal {B}}}^2_{\widetilde{\rho }/2} \subset \hat{\varvec{\omega }}_0({{\mathcal {B}}}_{\widetilde{\rho }}^2) \subset {{\mathcal {B}}}^2_{3\widetilde{\rho }/2} \end{aligned}$$

and \(\hat{\varvec{\omega }}_0^{-1}\) is a Lipschitz mapping with a constant 2.

Now, as \(\varvec{\omega }= \varvec{\omega }({\mathbf{0} }) + {\mathrm {d}}\varvec{\omega }({\mathbf{0} }) \hat{\varvec{\omega }}_0\), we consider

$$\begin{aligned} \varvec{\omega }^{-1}({\mathbf{y} }) = \hat{\varvec{\omega }}_0^{-1}\left( ({\mathrm {d}}\varvec{\omega }({\mathbf{0} }))^{-1}({\mathbf{y} }- \varvec{\omega }({\mathbf{0} }))\right) . \end{aligned}$$

If \(({\mathrm {d}}\varvec{\omega }({\mathbf{0} }))^{-1}({\mathbf{y} }- \varvec{\omega }({\mathbf{0} }))\in {{\mathcal {B}}}^2_{\widetilde{\rho }/2}\), then there exists

(as ). Hence, we have determined \(\varvec{\omega }^{-1}\) over \({{\mathcal {B}}}_{\hat{\rho }}\{\varvec{\omega }({\mathbf{0} })\}\).

Finally for \(({\mathbf{y} }, {\mathbf{y} }') \in ({{\mathcal {B}}}_{\hat{\rho }}\{\varvec{\omega }({\mathbf{0} })\})^2\), we have

as \({\mathrm {d}}\hat{\varvec{\omega }}_0^{-1}\) is 2-Lispshitz and . Hence, as then

Consequently, \(\varvec{\omega }\) is a diffeomorphism from \({{\mathcal {B}}}^2_\rho \) onto its image and the estimates (A.22) are ensured (by the estimates (4.22)). \(\quad \square \)

By the notations of Sect. 5, with \(\vert f \vert _\Pi ^{\mathrm{Lip}}\leqslant \vert \vert {\mathrm {d}}f\vert \vert _\Pi \) for a differentiable function and the upper bounds (A.22) ensure

A property needed to apply the Pöschel results on the persistence of normally elliptic tori is to ensure Melnikov’s condition for multi-integers of length bounded by \(K_0 = 16L M\). This is the content of the following

Proposition A.1

Let

we have, for \({\varepsilon }\in [{\varepsilon }_0/2,{\varepsilon }_0]\) with ,

Proof

First of all, for \(\varvec{\xi }\in \Pi \) we have the followings:

(A.23)

that are deduced from (4.19) and (4.27). As a consequence, with \({\varepsilon }\in [{\varepsilon }_0/2, {\varepsilon }_0]\), for \(\varvec{\xi }\in \Pi \):

$$\begin{aligned} \vert \Omega _1(\varvec{\xi })\vert \geqslant \gamma _0,\quad \vert \Omega _2(\varvec{\xi })\vert \geqslant \gamma _0,\quad \vert \Omega _1(\varvec{\xi }) - \Omega _2(\varvec{\xi })\vert \geqslant \gamma _0. \end{aligned}$$

For \(({\mathbf {k}},{\varvec{l}})\in {{\mathbb {Z}}}^2\times {{\mathbb {Z}}}^2\) with \(0<\left| {\mathbf {k}} \right| \leqslant K_0\) and \(\left| {\varvec{l}} \right| \leqslant 2\) we have

since \(4/9< \beta < 1/2\). Especially, for a large enough constant \(C>0\), we have

deduced from (A.23) with and \(k_2 \ne 0\). Likewise, if \(k_2=0\) then for a large enough constant \(C>0\), we have

with . \(\quad \square \)

The Hamiltonian \({\mathrm H }\) defined in (5.2) is analytic over the domain \(D({\bar{r}},{\bar{s}})\) defined in (5.4) with \(0<{\bar{r}}< r\) and .

With the estimates given in Proposition A.1, it remains to check the thresholds of the Proposition 2.2 in Biasco et al. [6] which become here the threshold (5.5) of Theorem 5.1 and has to be satisfied for a small enough bound \({\varepsilon }_0\) on the mass ratio.

We decompose the perturbation (5.3) in \({\mathrm P }={\mathrm P }_1 + {\mathrm P }_2 + {\mathrm P }_3 + {\mathrm P }_4\) with

$$\begin{aligned} \begin{aligned} {\mathrm P }_1({\mathbf{y} },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ; \varvec{\xi })&=\sum _{j\in \{1,2\}} \Big ({{\mathscr {H}} }_j(\xi _j +y_j)-{{\mathscr {H}} }_j(\xi _j)- \omega _j (\varvec{\xi })y_j \Big )+ {{\mathscr {F}} }_0(\varvec{\xi }+{\mathbf{y} })-{{\mathscr {F}} }_0(\varvec{\xi }),\\ {\mathrm P }_2({\mathbf{y} },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ; \varvec{\xi })&= \sum _{j\in \{1,2\}} i \big (g_j(\varvec{\xi }+{\mathbf{y} })- g_j(\varvec{\xi })\big ) z_j{\widetilde{z}} _j,\\ {\mathrm P }_3({\mathbf{y} },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ; \varvec{\xi })&=\check{{{\mathscr {R}} }}(\varvec{\xi }+{\mathbf{y} },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ), \quad \text{ and } \quad {\mathrm P }_4({\mathbf{y} },\varvec{\psi },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ; \varvec{\xi })=\check{{\mathscr {H}} }_{*}(\varvec{\xi }+{\mathbf{y} },\varvec{\psi },{\mathbf{z} }, {\widetilde{\mathbf{z}}} ). \end{aligned} \end{aligned}$$

With the estimates of Theorem 4.5 together with Taylor formula, since \({\mathrm P }_1\), \({\mathrm P }_2\) (resp. \({\mathrm P }_3\)) are of order 2 in \(y_i\), \(z_j{\widetilde{z}} _j\) (resp. of order 4 in \(z_j\), \({\widetilde{z}} _j\)). Likewise, with the corollary 4.1, \(\varvec{\psi }\) appears only in \({\mathrm P }_4\) which is exponentially small. As a consequence, we obtain for \({\varepsilon }\in [{\varepsilon }_0/2,{\varepsilon }_0 ]\) that

hence

for some positive exponents p and \(p'\) (remark that \(p=p'\) can be chosen). We need

and we choose \({\bar{r}}= r_0{\varepsilon }_0^{d}\) for a small enough constant \(r_0>0\) and a large enough exponent d which ensure

Then,

is ensured for small enough \({\varepsilon }_0 <{\varepsilon }_*\) and the main threshold (5.5) is satisfied. Hence, we can find quasi-periodic horseshoe orbits for mass ratio \(0<{\varepsilon }<{\varepsilon }_*\).

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Niederman, L., Pousse, A. & Robutel, P. On the Co-orbital Motion in the Three-Body Problem: Existence of Quasi-periodic Horseshoe-Shaped Orbits. Commun. Math. Phys. (2020). https://doi.org/10.1007/s00220-020-03690-8

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