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The significance of learning constraints from data is underscored by its potential applications in real-world problem-solving. While constraints are popular for modeling and solving, the approaches to learning constraints from data remain relatively scarce. Furthermore, the intricate task of modeling demands expertise and is prone to errors, thus constraint acquisition methods offer a solution by automating this process through learnt constraints from examples or behaviours of solutions and non-solutions. This work introduces a novel approach grounded in Deep Neural Network (DNN) based on Symbolic Regression that, by setting suitable loss functions, constraints can be extracted directly from datasets. Using the present approach, direct formulation of constraints was achieved. Furthermore, given the broad pre-developed architectures and functionalities of DNN, connections and extensions with other frameworks could be foreseen.

The halo orbits of the spatial circular restricted three-body problem are largely considered in space-flight dynamics to design low-energy transfers between celestial bodies. A very efficient analytical method for the computation of halo orbits, and the related transfers, has been obtained from the high-order resonant Birkhoff normal forms defined at the Lagrangian points L1-L2. In this paper, by implementing a non-linear Floquet-Birkhoff resonant normal form, we provide the definition of orbits, as well as their manifold tubes, which exist in a large order approximation of the elliptic three-body problem and generalize the halo orbits of the circular problem. Since the libration amplitude of such halo orbits is large (comparable to the distance of L1-L2 to the secondary body), and the Birkhoff normal forms are obtained through series expansions at the Lagrangian points, we provide also an error analysis of the method with respect to the orbits of the genuine elliptic restricted three-body problem., Comment: Submitted for publication

Starting with Arnold's pioneering work, the term "Arnold diffusion" has been used to describe the slow diffusion taking place in the space of the actions in Hamiltonian nonlinear dynamical systems with three or more degrees of freedom. The present text is an elaborated transcript of the introductory course given in the Milano I-CELMECH school on the topic of Arnold diffusion and its relation to Nekhoroshev theory. The course introduces basic concepts related to our current understanding of the mechanisms leading to Arnold diffusion. Emphasis is placed upon the identification of those invariant objects in phase space which drive chaotic diffusion, such as the stable and unstable manifolds emanating from (partially) hyperbolic invariant objects. Besides a qualitative understanding of the diffusion mechanisms, a precise quantification of the speed of Arnold diffusion can be achieved by methods based on canonical perturbation theory, i.e. by the construction of a suitable normal form at optimal order. As an example of such methods, we discuss the (quasi-)stationary-phase approximation for the selection of remainder terms acting as driving terms for the diffusion. Finally, we discuss the efficiency of such methods through numerical examples in which the optimal normal form is determined by a computer-algebraic implementation of a normalization algorithm., Comment: 44 pages, 12 figures. Preprint for the I-Celmech Proceedings

In the last decades a peculiar family of solutions of the Circular Restricted Three Body Problem has been used to explain the temporary captures of small bodies and spacecrafts by a planet of the Solar System. These solutions, which transit close to the Lagrangian points $L_1,L_2$ of the CRTBP, have been classified using the values of approximate local integrals and of the Jacobi constant. The use for small bodies of the Solar System requires to consider a hierarchical extension of the model, from the CRTBP to the the full $N$ planetary problem. The Elliptic Restricted Three Body, which is the first natural extension of the CRTBP, represents already a challenge, since global first integrals such as the Jacobi constant are not known for this problem. In this paper we extend the classification of the transits occurring close to the Lagrangian points $L_1,L_2$ of the ERTBP using a combination of the Floquet theory and Birkhoff normalizations. Provided that certain non-resonance conditions are satisfied, we conjugate the Hamiltonian of the problem to an integrable normal form Hamiltonian with remainder, which is used to define approximate local first integrals and to classify the transits of orbits through a neighbourhood of the Lagrange equilibria according to the values of these integrals. We provide numerical demonstrations for the Earth-Moon ERTBP., Comment: 36 pages, 8 figures. Submitted for publication

The dynamics near the Lagrange equilibria $L_1$ and $L_2$ of the Circular Restricted Three-body Problem has gained attention in the last decades due to its relevance in some topics such as the temporary captures of comets and asteroids and the design of trajectories for space missions. In this paper we investigate the temporary captures using the tube manifolds of the horizontal Lyapunov orbits originating at $L_1$ and $L_2$ of the CR3BP at energy values which have not been considered so far. After showing that the radius of convergence of any Hamiltonian normalization at $L_1$ or $L_2$ computed with the Cartesian variables is limited in amplitude by $\|1-\mu-x_{L_1}\|$ ($\mu$ denoting the reduced mass of the problem), we investigate if regularizations allow us to overcome this limit. In particular, we consider the Hamiltonian describing the planar three-body problem in the Levi-Civita regularization and we compute its normalization for the Sun-Jupiter reduced mass for an interval of energy which overcomes the limit of Cartesian normalizations. As a result, for the largest values of the energy that we consider, we notice a transition in the structure of the tubes manifolds emanating from the Lyapunov orbit, which can contain orbits that collide with the secondary body before performing one full circulation around it. We discuss the relevance of this transition for temporary captures., Comment: 17 pages, 9 figures. v1: original preprint. v2: author's accepted manuscript. To appear in International Journal of Non-Linear Mechanics

Cornerstone models of Physics, from the semi-classical mechanics in atomic and molecular physics to planetary systems, are represented by quasi-integrable Hamiltonian systems. Since Arnold's example, the long-term diffusion in Hamiltonian systems with more than two degrees of freedom has been represented as a slow diffusion within the `Arnold web', an intricate web formed by chaotic trajectories. With modern computers it became possible to perform numerical integrations which reveal this phenomenon for moderately small perturbations. Here we provide a semi-analytic model which predicts the extremely slow-time evolution of the action variables along the resonances of multiplicity one. We base our model on two concepts: (i) By considering a (quasi-)stationary phase approach to the analysis of the Nekhoroshev normal form, we demonstrate that only a small fraction of the terms of the associated optimal remainder provide meaningful contributions to the evolution of the action variables. (ii) We provide rigorous analytical approximations to the Melnikov integrals of terms with stationary or quasi-stationary phase. Applying our model to an example of three degrees of freedom steep Hamiltonian provides the speed of Arnold diffusion, as well as a precise representation of the evolution of the action variables, in very good agreement (over several orders of magnitude) with the numerically computed one., Comment: 36 pages. 6 figures. Submitted to Journal of Nonlinear Science

One of the most interesting features in the libration domain of co-orbital motions is the existence of secondary resonances. For some combinations of physical parameters, these resonances occupy a large fraction of the domain of stability and rule the dynamics within the stable tadpole region. In this work, we present an application of a recently introduced `basic Hamiltonian model' Hb for Trojan dynamics, in Paez and Efthymiopoulos (2015), Paez, Locatelli and Efthymiopoulos (2016): we show that the inner border of the secondary resonance of lowermost order, as defined by Hb, provides a good estimation of the region in phase-space for which the orbits remain regular regardless the orbital parameters of the system. The computation of this boundary is straightforward by combining a resonant normal form calculation in conjunction with an `asymmetric expansion' of the Hamiltonian around the libration points, which speeds up convergence. Applications to the determination of the effective stability domain for exoplanetary Trojans (planet-sized objects or asteroids) which may accompany giant exoplanets are discussed., Comment: 21 pages, 9 figures. Accepted for publication in Celestial Mechanics and Dynamical Astronomy

The main subject of this work is the study of the problem of the Trojan orbits from a perturbative Hamiltonian perspective. We face this problem by introducing first a novel Hamiltonian formulation, exploiting the well-differentiated temporal scales of the Trojan motion. The resulting Hamiltonian allows to separate the secular (very slow) component of the motion from the librating and fast degrees of freedom. This decompositon provides the foundation of a so-called Basic Hamiltonian model (Hb), i.e. the part of the Hamiltonian for Trojan orbits independent of all secular angles. Our study shows that, up to some extent, the model Hb successfully represents the features of the motion under more complete models, in a range of physical parameters relevant for dynamics in the Solar System or in extrasolar planetary systems. (continues...), Comment: Ph.D. Thesis - Presented on 23rd February 2016

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Paez & Efthymiopoulos, 2015, Paez & Locatelli, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem (SEE FILE FOR COMPLETE ABSTRACT), Comment: Accepted for publication in CMDA

In this work, we study the motions in the region around the equilateral Lagrangian equilibrium points L4 and L5, in the framework of the Circular Planar Restricted Three-Body Problem (hereafter, CPRTBP). We design a semi-analytic approach based on some ideas by Garfinkel in [4]: the Hamiltonian is expanded in Poincar\'e-Delaunay coordinates and a suitable average is performed. This allows us to construct (quasi) invariant tori that are moderately far from the Lagrangian points L4-L5 and approximate wide tadpole orbits. This construction provides the tools for studying optimal transfers in the neighborhood of the equilateral points, when instantaneous impulses are considered. We show some applications of the new averaged Hamiltonian for the Earth-Moon system, applied to the setting-up of some transfers which allow to enter in the stability region filled by tadpole orbits., Comment: Published on AIP Conference Proceedings, Volume 1637, Issue 1, p.776-785

In the framework of the ERTBP, we study an example of the influence of secondary resonances over the long term stability of Trojan motions. By the integration of ensembles of orbits, we find various types of chaotic diffusion, slow and fast. We show that the distribution of escape times is bi-modular, corresponding to two populations of short and long escape times. The objects with long escape times produce a power-law tail in the distribution., Comment: Published on Complex Planetary Systems, Proceedings of the International Astronomical Union, IAU Symposium, Volume 310, pp. 96-97

We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricted Three-Body Problem (PCRTBP), by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel (1977) to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behavior of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non--normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models., Comment: Accepted for publication on MNRAS. Data: Accepted 2015 August 02. Received 2015 August 02; in original form 2015 June 09

We investigate the dynamics of small trojan exoplanets in domains of secondary resonances within the tadpole domain of motion. We consider the limit of a massless trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of more planets (the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the trojan's dynamics. In the theoretical part of this paper, we develop a Hamiltonian formalism in action-angle variables, which allows to treat in a unified way resonant dynamics and secular effects on the trojan body in both the ERTBP or the RMPP. Our formalism leads to a decomposition of the Hamiltonian in two parts, $H=H_b+H_{sec}$. $H_b$, called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom. $H_{sec}$ contains only terms depending on slow (secular) angles. $H_b$ is formally identical in the ERTBP and the RMPP, apart from a re-definition of angular variables. An important physical consequence is that the slow chaotic diffusion proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism better approximated by the paradigm of `modulational diffusion'. In the numerical part, we focus on the ERTBP for making a numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we provide a survey of the resonant web for characteristic mass parameters of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period) and their resonant multiplets appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of escaping times for slowly diffusing chaotic orbits., Comment: Accepted for publication in Celestial Mechanics and Dynamical Astronomy