4 Conclusions
Using the regularized equations for the computation of the equations of motion and of the variational equations, we performed a global study of the dynamics of the circular, restricted, three–**body** **problem** under the effect of different kinds of dissipation. As it is well known, the effect of the dissipation is that of decreasing the semi–major axis of orbits that collide with one of the primaries. We have recovered this result, showing which are the regions of the phase space affected by the dissipation even on very short time scales. Precisely, we found that a large fraction of test particles with initial conditions in the interior region collides with the Sun. Regular orbits having the closest approach with the Sun in the conservative setting are the first to be affected by the dissipation. In the exterior region, collisions occur mainly with Jupiter. When increasing the strength of the dissipation or the integration time, a larger fraction of orbits ends on one of the primaries. In general, the effect of the PR drag is faster than those associated to Stokes and linear drags. Moreover, we have found periodic orbit attractors for the case of the linear and Stokes drags, while in the case of the Poynting–Robertson effect no other non–trivial attractor (namely, not coinciding with one of the primaries) is found. In the latter case, in order to find periodic orbits, we need to modify the model, for example adding the action of a third primary (e.g., a Saturn–like **body**), whose effect turns out to balance that of the two main primaries. In the case of the Stokes drag, a long time ( T = 5000) FLI compu-

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When Geffroy [34] framed the technique of averaging for low-thrust satellite transfers, the satellite motion was expressed in terms of two-**body** Keplerian motion. However, the veracity of the low-thrust assumption can be affected by the introduction of other primary bodies (the thrust becomes proportional to attraction to a primary, and so can no longer be considered as a ‘low thrust’) and even by other spatial effects such as the Earth’s oblateness. A good question is, what effect does the wider space context, e.g. third-**body** gravity, have on the low-thrust satellite motion? The thesis directly addresses this aspect. We frame the low-thrust satellite motion within the wider space context, by studying specific aspects of not only satellite transfers within two-**body** motion but also of transfers where the motion is affected by naturally-occuring forces in the space environment - both in circumstances under which they can be considered perturbing forces and when they have a significant effect. Thus we extend the two-**body** **problem** naturally to the **problem** of three (and four) bodies by studying satellite transfers which initially require space perturbations to be considered, and finally which require us to consider a three and four-**body** model. We do this both by analytical studies of the perturbations (chapter 4) and by a numerical simulation (chapter 5) of an optimal low-thrust transfer from an Earth orbit to the L 1 Lagrange point, which takes place

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Riemannian metrics on 2D manifolds
related to the Euler-Poinsot rigid **body** **problem**.
Bernard Bonnard, Olivier Cots and Nataliya Shcherbakova
Abstract— The Euler-Poinsot rigid **body** **problem** is a well known model of left-invariant metrics on SO(3). In the present paper we discuss the properties of two related reduced 2D models: the sub-Riemanian metric of a system of three coupled spins and the Riemannian metric associated to the Euler- Poinsot **problem** via the Serret-Andoyer reduction. We explicitly construct Jacobi fields and explain the structure of conjugate loci in the Riemannian case and give the first numerical results for the spin dynamics case.

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4. Indeed, the conditions of the KAM theorem require that the size of the perturbation is less than γ 4 rσ ν /c, where c ∼ 10 18 . In the corollary, this constant is c ∼ 10 8 in dimension 4. Hence, it would be
quite straightforward to gain a factor 10 10 for the ratio of the masses.
• A more suitable KAM theorem: Our choice of quantitative KAM theorem [6] under the form that Pöschel stated in [28] is for convenience. Yet, it requires to add the non-linear terms to the perturbation. This requirement is, with the initial values, the most restrictive one concerning the value of r 0 . This theorem might not be the most suitable for an application to the plane planetary three-**body** **problem**. Loosely speaking, the initial Hamiltonian is H Kep + H pert , and after some transformation, we look at

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Department of Mathematics, Northwestern University 2033 Sheridan Road, Evanston, IL 60208, USA
E-mail: fejoz@math.northwestern.edu
Levi-Civita’s regularization procedure for the two-**body** **problem** easily ex- tends to a regularization of double inner collisions in the system consisting of two uncoupled Newtonian two-**body** problems. Some action-angle variables are found for this regularization, and the inner **body** is shown to describe el- lipses on all energy levels. This allows us to define a second projection of the phase space onto the space of pairs of ellipses with fixed foci. It turns out that the initial and regularized averaged Hamiltonians of the three-**body** **problem** agree, when seen as functions on the space of pairs of ellipses. After the reduc- tion of the **problem** by the symmetry of rotations, the initial and regularized averaged planar three-**body** problems are shown to be orbitally conjugate, up to a diffeomorphism in the parameter space consisting of the masses, the semi major axes and the angular momentum.

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Complex singularities in the plane planetary three-**body** **problem**
Thibaut Castan
Abstract. In the plane planetary three-**body** **problem**, the Hamiltonian is the sum of two uncoupled Kepler
problems and a small, perturbing function. As KAM and Nekhoroshev theories show, the stability of the system depends crucially on the analytic continuation of the perturbing function on a complex extension of the phase space, its complex singularities and its analyticity domain. This paper is devoted to determining these singularities, which fall into the two kinds of planet-star and planet-planet complex collisions, and to estimating the norm of the continuation of the perturbing function.

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from the right-hand side (left-hand side respectively) ’bounce back’ if they are on one side of the separatrix, while they transit to the left-hand side (right-hand side respectively) if they are on the other side of the separatrix.
The structure of the global stable and unstable manifolds of W C,i c is much more complicate than the structure of the local manifolds: the exponential compressions, expansions and rotations occurring near the center manifolds are alternated to circulations around both primaries. Global representations of these surfaces have been obtained for several sample values of µ and C in the planar circular restricted three-**body** **problem**, see for example [17, 13, 24]. The computation of the stable-unstable manifolds in the planar case has several advantages with respect to the spatial case. First, in the planar case, the level set of the center manifolds obtained by fixing the value of the Jacobi constant in a suitable small left neighbourhood of C i is made of a periodic

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There have been some prior attempts to prove such a conjecture. For instance, Moeckel discovered an instability mechanism in a special configuration of the 5-**body** **problem** [Moe96]. His proof of diffusion was limited by the so-called big gaps **problem** between hyperbolic invariant tori; this **problem** was later solved in this setting by Zheng [Zhe10]. A somewhat opposite strategy was developed by Bolotin and McKay, using the Poincar´e orbits of the second species to show the existence of symbolic dynamics in the three- **body** **problem**, hence of chaotic orbits, but considering far from integrable, non-planetary conditions; see for example [Bol06]. Also, Delshams, Gidea and Rold´ an have shown an instability mechanism in the spatial restricted three-**body** **problem**, but only locally around the equilibrium point L 1 (see [DGR11]).

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Analyticity, singularities, bound on the Hamiltonian norm
This chapter is dedicated to the study of the perturbation in the plane planetary three-**body** **problem**. Although a lot of theorems, from the KAM theorem to theorems of stability in finite time, use the analyticity of the perturbation of an integrable Hamiltonian, only few attempts were made to compute the actual analyticity widths in the three-**body** **problem**. Regarding the perturbation, another **problem** kept the attention of researchers for years: the convergence of the expansion in power of the eccentricities of the disturbing function. This matter is closely related to Kepler’s equation, which associates the mean anomaly to the eccentric one, and in particular to the convergence of the inverse Kepler equation. This study started with the work of mathematicians studying the celestial mechanics such as Lagrange and Laplace. Later, Poincaré provided a necessary condition on the elliptic variables for this expansion to converge (see [57]). This work was followed by the statement of a sufficient condition by Silva [66] that was proven by Sundman to be as well a necessary condition [69]. Wintner showed in [71] some conditions on complex eccentricities this time in Kepler’s equation: he showed that although the inverse converges for real eccentricities strictly smaller than 1, the size of the disk of convergence for complex eccentricities is given by Laplace’s limit ∼ 0.66274). Some works regarding the series expansion and the complex singularities have been developed by Petrovskaya in 1970 [56]. Sokolov [67] went further in his research by looking not only at complex eccentricities, but also at complex mean anomalies.

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Now we show how to apply this theorem to the existence of invariant tori for some class of Hamiltonians which are completely integrable on a transversally Cantor set. In the next subsection (§ 3.2) we will show that the Hamiltonian F of the planar three-**body** **problem** in the neighborhood of regular or non degenerate singular secular invariant tori falls into this cat- egory. Neighborhoods of secular singularities and of regular secular orbits respectively correspond to (p, q) = (2, 1) and (3, 0). It will be fundamental that we may choose the order n of the secular system as large as we want, so that the perturbation has an arbitrarily large order of smallness com- pared to the size of the terms which break down the proper degeneracy of the Keplerian part.

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In this paper, we focus on secular resonances. Numerical evidence has long been suggesting that such resonances are a major source of chaos in the Solar system [LR93, Las08, FS89]. For example, astronomers have established that Mercury’s eccentricity is chaotic and can increase so much that collisions with Venus or the Sun become possible, as a result from an intricate network of secular resonances [BLF12]. On the other hand, that Uranus’s obliquity (97 o ) is essentially stable, is explained, to a large extent, by the absence of any low-order secular resonance [BL10, LR93]. It is the goal of this paper to provide a simple instability mechanism in the secular dynamics of the three-**body** **problem**.

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whose period varies. So, we will consider the natural extension of the action of Γ 1 to the space of periodic loops of any period T > 0, obtained by merely
replacing every occurence of 2π by T in the formulae (A). (As already men- tionned, the scaling symmetry of the n-**body** **problem** allows to canonically put in correspondance solutions with a fixed norm of the angular momentum, and solutions with a fixed period; but the reduction by rotations is more tractable by fixing the norm of the angular momentum than the period of periodic orbits.) Now, a periodic solution of the reduced 3-**body** **problem** can be lifted, in the manifold of fixed angular momentum, to a solution of the full 3-**body** **problem** which is periodic in a rotating frame. Provided that the rotation is uniform, the angular speed of the frame is unique up to a multiple of 2π per period. If one choses that, for the Lagrange solution, the frame rotates exactly by one turn per period in the direction opposite to the motion, by continuity the rotation of the frame of each orbit of the invariant surface P becomes well defined. In this section we address the question of existence and uniqueness of the lift of solutions lying in P, to Γ 1 -symmetric loops.

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This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 13810
To cite this version : Riols, Antoine and Trilles, Sébastien Analysis
Software for Interplanetary Trajectories using Three **Body** **Problem**. (2010) In: 4th International Conference on Astrodynamics Tools and Techniques, 3 May 2010 - 6 May 2010 (Madrid, Spain).

The Near Rectilinear Orbits NROs are defined within the Earth-Moon Three-Body framework and they can be seen as a certain class of Halo orbits if the point of the trajectory closest to t[r]

5 A formal calculation for spiraling solutions
We now try a formal calculation suggesting that solutions with norm equivalent to a decreasing exponential for t large might however exist. We would like to see which sort of solutions would replace the circular trajectories of the conservative **problem** (1) (cf. also section 2.) Observing that such trajectories correspond to a constant angular velocity and for them the nonlinear term of the radial equation vanishes, it is natural to try

Could Feldman nov respond to my point about essential properties by claiming that the person case 1s analogous to the above? Perhaps, embodied persons are essentially bodies as well 8S e[r]

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Dès la sixième page du Top **Body** Challenge, nous constatons la présence d’un visuel mettant en avant une brassière, un short ainsi qu’une paire de chaussures de la marque Nike. Par ailleurs, lors de son Boot Camp, Sonia Tlev est entièrement vêtue de la marque Nike, tout comme ses collaborateurs, tous vêtus d’un tee-shirt vert de la même marque. Nous nous sommes également vus offrir des goodies tels qu’une boisson de la marque Yumi ainsi qu’une barre de céréales de la marque Bee Kind. Sur les réseaux sociaux, bien que cette dernière porte parfois des chaussures Adidas, et des brassières Calvin Klein, c’est bien en Nike qu’elle est le plus souvent prise en photo. Kayla Itsines quant à elle n’est pas en reste puisque l’on peut régulièrement la voir porter des vêtements de la marque Adidas sur les réseaux sociaux.

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To date, we know that although age, sex, and BMI have been extensively studied in the literature as predictors of BD, the impact of level of education has not been well-documented with respect to this relationship. Furthermore, to our knowledge, no previous research has studied the association of these factors with regards to how they might influence one’s interpretation, perception, or reaction to large-scale health promotion efforts. Based on the findings of our research, we can conclude that in general, women, those with higher levels of education, and those who are in the normal weight BMI category tend to have more positive responses towards these types of actions. However, some considerations are important to discuss in the justification of these findings. Although we know that today, both men and women are exposed to the unrealistic societal pressures to attain unrealistic **body** ideals, culturally, there still remains a greater cultural discrimination that is overtly expressed towards women particular to the topic of thinness (Grogan, 2016; Runfola et al., 2013). We know that generally, women tend to experience greater levels of BD and that that these associations become even more prominent in those who are classified into unhealthy BMI weight categories (Runfola et al., 2013; Bearman, et al., 2006; Coelho et al., 2016). Justifiably, these are the segments of the populations who experience higher levels of BD, and as a result, exhibit greater levels of preoccupation and support for health related initiatives.

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From the elliptic lemniscatoids, it is possible to generate symmetric prolate ternary shapes varying from one sphere to three aligned tangent identical spheres, assuming volume conservat[r]

to 0.7 m. This verifies our assumptions and will allow more accurate evaluation of different WBAN scenarios.
VI. C ONCLUSION
In this paper, a path loss formulation taking into considera- tion path attenuation as well as attenuation due to human **body** posture was presented. We have demonstrated that the human **body** has a rigorous influence on the propagation model as well as the energy efficiency. Path loss increases certainly when the antennas separation increases and in the presence of the human **body** and it is more important than free space medium. A propagation model was proposed analytically and validated experimentally and an appropriate energy model was given for WBAN conditions. Based on simulation results, we conclude that, a relevant energy consumption model is obtained by adding all propagation channel characteristics. Our proposed path loss model is able to predict on-**body** propagation at 2.4 GHz and subsequently able to model energy efficiency under these conditions.

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