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
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).
The horseshoe trajectories are depicted in the frame that rotates with the moons’ average mean motion.
Which is the gravitational sphere of influence where the primary acts as a perturbator.
In this approximation, it is assumed that the massless one does not affect the motion of the other two, which is consequently Keplerian.
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.
Which is the averaged perturbation along the Keplerian flows.
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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Appendix A. Proofs
Appendix A. Proofs
1.1 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,
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
Thus, one has
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
which implies that
1.2 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
and, for \(0\leqslant r\leqslant 1\), we denote \({\hat{{{\mathfrak {K}} }} }_r\) the domain such as
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
Let \(H^-\) be a Hamiltonian of the form
which is analytic on the domain \({\hat{{{\mathfrak {K}} }} }_0^-={\hat{{{\mathcal {K}} }}}_{\rho ^-, \sigma ^-}\) and such that
Let \(\eta ^-\), \((\mu _l^-)_{l\in \{0,1,2,3\}}\) be fixed positive real numbers, which depend on \({\varepsilon }\), such that
and
If we assume that
then there exists a canonical transformation
and such that, in the new variables, the Hamiltonian \(H^+= H^- \circ \overline{\Upsilon }^+\) can be written
and
Furthermore, we have the thresholds
and
with the following quantities:
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
such that
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
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
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
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
with the following thresholds:
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
We define
as well as the sequences \((\rho ^{j})_{j\in \{0,1,\ldots ,s\}}\), \((\sigma ^{j})_{j\in \{0,1,\ldots ,s\}}\) with
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:
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:
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
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).
1.3 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
Injecting the expansion (2.3) in (A.11) we get
As a consequence, one has
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
As \(H_P\) satisfies the D’Alembert rule, one has
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).
1.4 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
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
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
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
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 \(\kappa \leqslant 1/2\).
1.5 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
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:
As \(U_\delta (\varphi _{1,\delta }^{\min }) =0\), Taylor formula leads to
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
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:
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
with and . As
we get the expressions (4.14).
1.6 Theorem 4.3: Semi-fast Holomorphic Extension
We consider the mechanical system
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
for some \(\delta ^*>0\), we can build a system of action-angle variables denoted \((J_1,\phi _1)\) such that
The transformation in action-angle variables, which will be denoted \({{\mathfrak {G}}}\), satisfies
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,
and the transformation \({{{\mathfrak {G}}}}\) can be defined explicitly by a classical integral formulation. The action is given by
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
Now, we look for the complex domain of holomorphy of the integrable Hamiltonian \({{\mathscr {H}} }_1\). We first consider the complex domain
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:
with
for the real analytic function
and we consider the complex extension \({\hat{D}}_{*,\hat{\rho }} = {{\mathcal {B}}}_{\hat{\rho }}{\hat{{{\mathfrak {D}}} }}_{*}\) with
where \(\hat{\rho }>0\) is small enough ().
Likewise, we have the following real analytic functions:
with
and
Hence we consider the transformation
with
which corresponds to the action-angle variables for the mechanical system
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
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
moreover, we consider
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
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
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
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 \({\hat{p} }= 11/2\).
1.7 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.
1.8 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
and for \(0\leqslant r\leqslant 1\), we denote \({{\mathfrak {K}} }_r\), the domain such as
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
Let \({{\mathscr {H}} }^{-}\) be a Hamiltonian of the form
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
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:
and
If we assume that
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
such that for \(l\in \{0,1\}\) (given by (4.8)) and
Furthermore, we have the thresholds
and
with the following quantities:
for \(m\in \{1,2\}\) and
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
such that the following properties are satisfied:
Thus, for the same reason as in Lemma A.1, the Hamiltonian can be written
with
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:
where
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
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
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:
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 .
1.9 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
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
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
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
1.10 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 .
1.11 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
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
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:
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
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
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:
that are deduced from (4.19) and (4.27). As a consequence, with \({\varepsilon }\in [{\varepsilon }_0/2, {\varepsilon }_0]\), for \(\varvec{\xi }\in \Pi \):
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
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. 377, 551–612 (2020). https://doi.org/10.1007/s00220-020-03690-8
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DOI: https://doi.org/10.1007/s00220-020-03690-8