A method of extending general dimension-raising point transformations to canonical transformations is developed. This is used to introduce a linearizing transformation of central-force Hamiltonian dynamics using projective coordinates. We develop a family of such transformations, from which preferred coordinates are chosen that are directly connected to orbital reference frames and attitude dynamics. This transformation fully linearizes inverse square and/or inverse cubic central forces (Kepler and Manev dynamics), with closed-form solutions and state transition matrices readily obtained. Arbitrary perturbing forces are included throughout, with the J_2 -perturbed Kepler problem considered as a practical example.
The so-called critical inclination problem of the third body appears in various contexts of celestial mechanics, especially in studies of orbital stability, secular perturbations, frozen orbits, resonances, and the restricted three-body problem. In general terms, the critical inclination is a value from which the orbital dynamics change qualitatively, for example, large eccentricity oscillations, instabilities, or Lidov–Kozai-type resonances arise. In this work, we investigate the critical inclination problem using a more accurate model of the disturbing potential due to the third body than the classical model, where additional terms were taken into account by Lei and Grishin (2025a, 2025b). Considering the double-averaged problem, we show a comparison between the modified double-averaged model von Zeipel–Lidov–Kozai and the classical double-averaged model Lidov–Kozai. We consider both analytical and numerical methods. We show a formula to calculate the critical inclination considering the modified potential model. Note that all applications presented in this work are for the case of a high-altitude orbit around the Moon perturbed by the Earth’s gravitational attraction. From the results, we reproduced the classic case showing the critical inclination in a graph of eccentricity versus inclination and then plotted the same graph with the modified potential model. We also present esin i versus ecos i diagrams to show the regions of libration (librating around the equilibrium point), bifurcation and circulation. In the particular case, when we consider e = 0 , we obtain the classic critical inclinations of i_c = 39.23 degrees and i_c = 140.77 degrees, while in the modified model we find i_c = 42.65 degrees and i_c = 144.13 degrees, in this case, a = 13000 km. In the modified model, the calculation of the critical inclination depends on some orbital and physical parameters of the disturbing body.
We construct explicit examples of really perverse homothetic solutions in the spatial Newtonian N-body problem. A solution is called really perverse if it solves the N-body problem for two distinct mass distributions having the same total mass and the same center of mass. While such solutions were previously known only in the planar case for large values of N, we prove the existence of spatial really perverse homothetic solutions for N=27,… ,55 .
The breakdown of first-order non-resonant averaging near frequency commensurabilities is investigated in a prescribed-phase hierarchical four-body model. The system is formulated as a quasi-periodically forced Hamiltonian problem in extended phase space, where the imposed inner and outer phases are represented by auxiliary angle variables. Within this framework, first-order averaging is interpreted as the removal of non-resonant Fourier harmonics whose forcing denominators remain bounded away from zero, rather than as the literal removal of fast variables. It is shown that, when an externally imposed phase combination becomes near-commensurate, the corresponding small denominator amplifies the first-order correction generated by the averaging transformation and leads to a loss of uniform control of the first-order averaged approximation with respect to detuning. A one-harmonic reduced model is then constructed by retaining the dominant near-commensurate forcing harmonic while averaging out the remaining non-resonant modes. This retained harmonic has a pendulum-like auxiliary structure, but it is not interpreted by itself as autonomous resonant capture of the physical particle dynamics. Numerical comparisons between the full, fully averaged, and one-harmonic models are performed using filtered slow observables, dimensionless detuning scans, and phase-ensemble tests. The fully averaged model loses accuracy near-commensurability, whereas the one-harmonic model remains substantially closer to the full dynamics. The transition is broad rather than sharply localized but remains consistent with the predicted small-divisor scaling. The results demonstrate that averaging breakdown in this prescribed-phase setting can be explained by slow near-commensurate forcing without requiring autonomous particle-resonance capture.
Efficient determination of the controllable set for Earth–Moon transfer orbits is essential for the rapid design and planning of crewed lunar exploration missions. To incorporate precise terminal time constraints and improve computational efficiency, a fast method based on velocity surfaces is proposed to solve the departure time-dependent controllable set (TDCS). First, the definition and mathematical model of the TDCS are proposed and applied to the Earth–Moon transfer orbits. Subsequently, a velocity surfaces-based method that accounts for terminal time constraints is developed. The effectiveness and approximation performance of the proposed method are evaluated through numerical tests and high-fidelity dynamical verification. The results indicate that the velocity surfaces method can rapidly estimate the departure TDCS with substantially improved computational efficiency compared with traditional methods, while providing effective initial guesses for subsequent high-fidelity trajectory correction. Based on this method, the effects of different constraints on the characteristics of TDCS parameters are further analyzed. These findings are expected to provide valuable guidance for the design and planning of Earth–Moon transfer orbits in future crewed lunar exploration missions.
Chaotic trajectories are often overlooked in the study of cislunar motion due to their ergodic behaviour. However, their long-term evolution is restricted by dynamical barriers. These riers correspond to quasi-periodic trajectories that partition neighbourhoods of the cislunar environment. We rapidly compute invariant curves associated with these quasi-periodic trajectories using semi-analytical approximations of a planar circular restricted three-body problem (PCR3BP) periapsis map, yielding interior and exterior region Keplerian maps. For a range of energies corresponding to the opening of the L_1 and L_2 gateways, we identify barriers that prevent transport from the exterior region to the near-Earth regime. For lower energies, we also find barriers inhibiting transport from beyond cislunar space into the Earth–Moon system. These barriers are validated with numerical integration in the Earth-Moon PCR3BP.
This paper investigates the coupled orbit–attitude dynamics of a rigid spacecraft in the Earth–Moon circular restricted three-body problem, with particular emphasis on gravity gradient torque. A unified 12-dimensional nonlinear model is formulated by combining the orbital equations of motion with the attitude kinematics and dynamics, where the attitude is represented using quaternions and angular velocity. This integrated framework enables a consistent description of the interaction between orbital and attitude dynamics under the Earth–Moon gravitational field. Using this formulation, coupled orbit–attitude periodic solutions are computed along the halo orbit family around the Earth–Moon L_2 point. The existence and structure of these solutions are analyzed based on Floquet theory, and the associated invariant subspaces are characterized to clarify the stability properties of the coupled system. Based on the identified Floquet modes, a coupled orbit–attitude control strategy is discussed, in which the natural coupling dynamics are exploited to support long-term orbital maintenance while reducing control effort. The proposed framework enables systematic analysis and exploitation of the coupled orbit–attitude dynamics for station-keeping in the cislunar environment.
Maintaining stable low-altitude lunar orbits (LLO) is crucial for long-term missions, but this remains a significant challenge due to complex dynamical perturbations. Previous studies have largely concentrated on identifying lunar frozen orbits, whereas this research presents a generalized framework for lunar orbiters that combines third-body gravitational effects from Earth, accounting for its elliptical and inclined orbit, with the Moon’s obliquity and gravitational harmonics, specifically the zonal (J₂) and TESSERAL (C₂₂) coefficients. Unlike well-known double-averaged approaches, where short-period oscillations must be later reconstructed using generating functions, the current non-simplified dynamical model considers them directly in the dynamics. This reveals that these oscillations produce a measurable secular drift in eccentricity and inclination. Numerical simulations confirm that the Moon’s axial tilt and the C₂₂ tesseral harmonic produce considerable secular drift in both eccentricity and inclination, effects that traditional models tend to underestimate. The results underscore the necessity of including the Moon’s unique gravitational signature (J2 and C22) and short-period phenomena in mission design for low-altitude lunar satellites, such as navigation constellations and infrastructure-supporting orbiters. By addressing current gaps in lunar orbiter dynamics modeling, this comprehensive framework ensures accurate orbital longevity predictions and provides a solid foundation for sustainable operations in the cislunar environment.
The free-fall three-body problem exhibits a well-known coexistence of chaotic and regular dynamics. While its geometric structure has traditionally been explored through the AA-map and its statistical behavior described by classical escape theories, these approaches have remained largely disconnected. In this work, we introduce the Shape-Space-Sphere (SSS) as a global, non-hierarchical parametrization of all triangular configurations at zero velocity and zero angular momentum, enabling a direct geometric sampling of the complete configuration space. Using high-precision IAS15 integrations, we demonstrate that the topology of regular islands evolves continuously under variations of the mass ratios, revealing structural transitions rather than abrupt bifurcations. We further show that the statistical escape law of Valtonen Karttunen is consistently recovered when the free-fall configurations are sampled on the SSS, establishing an explicit correspondence between phase-space geometry and probabilistic outcomes. To quantify local predictability, we introduce a Local Stability Index (LSI) and identify a compact Most Stable Region (MSR) characterized by minimal sensitivity, low scramble number, and high numerical accuracy. These results unify geometric and statistical descriptions of the free-fall three-body problem within a single framework, demonstrating that classical escape statistics are intrinsically linked to the underlying configuration-space geometry.
Initial Orbit determination (IOD) is a critical step in cataloging Near-Earth Asteroids (NEAs). Traditional IOD methods, such as the Laplace, Gauss, and double-r iteration methods, typically generate multiple solutions. However, there has been limited research on selecting the optimal solution from these multiple initial orbit solutions. This paper proposes a multi-criteria integration method for selection of optimal initial orbit solution. In addition to using observational angular residuals, we introduce three new criteria: the variance of absolute magnitude, angular velocity deviation, and the joint probability density of orbit elements. An optimization algorithm based on a continuous orbit-distance loss is designed to determine the weight coefficients for each criterion, while the original three-level score is retained only as an a posteriori accuracy classification. Both training and testing datasets were generated using real NEA data from the Minor Planet Center. Numerical simulations demonstrate that the proposed method effectively identifies the optimal solution from a set of multiple initial-orbit solutions, achieving a correct rate of 96.52
We present a uniform closed-form boundary-continuous formulation for the gravitational potential, acceleration, and gravitational gradient tensor of a homogeneous polyhedron. Classical polyhedral gravitational models, though exact in form, become numerically unstable at faces, edges, and vertices and are conventionally restricted to exterior points. Our method eliminates these limitations through analytic regularization of the logarithmic and arctangent terms, ensuring continuity and stability across interior, boundary, and exterior domains. A dyadic tensor formulation provides compact expressions for accelerations and gravitational gradients, guaranteeing compliance with Poisson’s and Laplace’s equations. The implementation is fully vectorized and supports multi-threaded parallelism, delivering high computational efficiency and precision. Validation on convex, concave, and multiply connected geometries demonstrates strict physical consistency and numerical robustness. The proposed framework establishes a universal, high-precision computational standard for polyhedral gravitational modelling, applicable to planetary dynamics, spacecraft navigation, solid Earth geophysics, and physical geodesy.
A second-order normalized Hamiltonian is developed for the planar elliptic 2:1 Jovian resonance, explicitly retaining the dependence on both the asteroid’s and Jupiter’s eccentricities. The model is constructed by means of a Lie-series transformation while preserving a compact one-degree-of-freedom formulation. Its predictions are assessed through direct comparison with long-term numerical integrations of the elliptic restricted three-body problem. The results indicate that elliptic effects shift the resonant equilibrium and modify the libration dynamics. For the representative case considered here, the second-order model provides a more accurate local analytical description than the corresponding first-order approximation, with closer agreement with the numerical integrations, particularly in the libration period.
We investigate the central configurations of the restricted 1+N -body problem, in which N bodies are infinitesimal and the remaining one is dominant. Assuming that the distances between the i-th and (i+1) -th infinitesimal bodies are identical for all i=1,2,… ,N-1 , we establish the following results: (1) For N = 4 , there exist two distinct equidistant central configurations: a square and a special isosceles trapezoid; (2) For N≥ 4 with all infinitesimal bodies of equal mass, the only possible equidistant central configuration is a regular polygon.
Assessing the long-term orbital evolution of near-Earth objects is a key component of planetary defense. In many cases, however, reliable predictions are limited by scattering encounters, which are close encounters with the Earth that lead to a rapid loss of predictability, and play a crucial role in defining the effective predictability horizon of an object. In this work, we propose an operational definition of scattering encounter based on simplified two-body dynamics. Building on this framework, we introduce algorithms designed to automatically recognize scattering encounters within standard impact monitoring procedures. A central element of the method is the estimation of the extrema of the post-encounter semi-major axis over the confidence region, for which we develop dedicated techniques. We also discuss these algorithms in the framework of the extended Öpik’s theory of close encounters. The proposed methods are tested with an operational impact monitoring software and are shown to help identify scattering encounters for a variety of real objects.
We revisit the classical expansion of the planetary Hamiltonian for small eccentricities and mutual inclinations of the orbits, presenting a derivation based entirely on vector formalism. We demonstrate that the secular part of the disturbing function, as well as any other Fourier harmonic, can be expressed in terms of scalar products of vectors lying in the system’s invariant plane. Furthermore, we show how to express any such term using the angular momentum and eccentricity vectors of the orbits.
This work presents the application of variational autoencoders (VAEs) to the generation and analysis of periodic orbits in the circular restricted three-body problem (CR3BP). A VAE architecture based on convolutional neural networks (CNNs) was trained on a dataset of time series representing periodic trajectories. The encoder provided a compact, low-dimensional representation that captured some of the underlying geometric and dynamical features of the trajectories. By sampling the latent space, the model was able to generate approximations of new (with respect to the training dataset) quasi-periodic trajectories with prescribed characteristics. A continuation technique was subsequently implemented both in the physical and latent domains. Continuation in physical space enabled the convergence towards periodic orbits by starting from the approximations produced by the VAE, whereas continuation in latent space facilitated the systematic generation of trajectories belonging to selected orbital families. The results demonstrate the potential use of generative models to design trajectories as well as automatically discover periodic orbits in complex dynamical systems.
We present a method to determine admissible configurations that lead to relative equilibria in the full gravitational N-body problem. The method exploits the SO(3) symmetry by working with its fundamental invariants and does not rely on restrictive assumptions about mass distributions or on truncating the gravitational potential. As a result, it yields necessary conditions that apply to arbitrary rigid-body configurations. This work is presented in two parts. In the first (this) part, we assume every body is axisymmetric to give a clear, pedagogical exposition of the approach; in a forthcoming paper, we remove that restriction and treat general triaxial bodies (the method remains valid, but the number of variables increases). As an illustration, we systematically recover all known relative-equilibrium families for the two-body problem consisting of one sphere and one axisymmetric body, confirm earlier results in the literature, and derive new necessary conditions for certain configurations, including additional constraints for the arrow and non-Lagrangian types. In the special case of a sphere and an axisymmetric body, we also obtain sufficient conditions for relative equilibria under a monotonicity assumption on the potential.
We investigate the structure of the Earth co-orbital region at low eccentricity and low inclination using a semi-analytical model of the 1:1 mean-motion resonance. The dynamics of asteroids in co-orbital motion with the Earth is described through a resonant semi-secular Hamiltonian, allowing the classification of orbits into circulation, Trojan, horseshoe, and quasi-satellite states. By systematically exploring the phase space in the space of the orbital elements, we compute the fraction of each type of motion and quantify how different co-orbital states fill the Earth co-orbital region. We find that horseshoe orbits dominate the phase space, occupying more than half of the volume, followed by Trojan and circulating orbits, while quasi-satellites represent only a small fraction. The distribution of co-orbital states exhibits strong inhomogeneities, particularly as a function of the argument of perihelion, with clear concentration regions of horseshoe orbits associated with node crossing geometries. We also study the short-term stability of this portion of phase space by means of the MEGNO indicator, and how the level of chaos differs between different co-orbital states. Finally, we discuss the implications of these results for the expected population of Earth co-orbitals and for planetary defense, showing that a large fraction of co-orbitals remains undiscovered.
We compute 2D families of retrograde symmetric periodic orbits (SPOs) in the planar circular restricted 3-body problem (CR3BP) and investigate how these change with increase in the mass ratio. In particular, we compute 1/-1, 1/-2 and 2/-1 resonant families, showing that these exhibit period multiplication bifurcations at specific mass ratios. Moreover, we show that the width of the gap that divides the circular family in outer and inner branches at the 1/-1 resonance location increases with mass ratio. Finally, we show that similar to previous results for small mass ratios (Jupiter–Sun and Neptune–Sun), there are three 1/-1 resonant modes which correspond to stable branches of distinct SPO families up to mass ratio μ = 0.32 , from which point there is no 1/-1 resonant mode that bifurcates from the outer circular family.
Celestial Newtonian systems have regular dynamics, but with classical Keplerian rotation velocities, which disagree with the observed rotation of galaxies in the Universe. However, modifications of the classical accelerations or gravitational attractions can overcome this defect. The large-scale simulations of galaxies are with approximations with “particle–mesh” (PM) substitutions of the attractions from objects far away, which affects the regular dynamics. Here, we investigate the impact of the PM approximation and of the modifications of accelerations or gravitational attractions on the stability of the regular dynamics in a celestial system. The simple three-body system (TBS) is the simplest system to test the stability of the regular dynamics with approximations or with modifications of celestial dynamics, and it is easy to implement on a computer. Simulations of the TBS show that the PM approximation, and the modification of the accelerations (MOND), destabilizes TBS. In contrast, a modification of gravity by replacing Newton’s inverse square attraction with an increased attraction (Yukawa, MOGA) for faraway interactions stabilizes the system. The PM approximation and the MOND modification of classical dynamics do not preserve the momentum and angular momentum of a conservative system exactly, and they do not obey Newton’s third law. Although these errors and shortcomings are small, they eventually cause the instability of the regular dynamics.
