Celestial Mechanics

The article justifiably criticizes the opinion about the balanced nature of the terms of the standard contracts of the International Federation of Consulting Engineers (FIDIC) on the distribution of a number of risks between Employers and Contractors. As such a justification, a description of the progress and results of the analysis and assessment of the fairness of the terms of standard contracts (proforma contracts) from the standpoint of the mechanism of judicial ex post control of contractual terms stated in Art. 428 Civil Code of the Russian Federation. As a result of the analysis, it was confirmed that the terms of standard contracts on the distribution of risks of errors in the initial data, erroneous positioning of an object on the site and unforeseen physical conditions can be qualified as clearly burdensome for the Contractor and significantly disturbing the balance of interests of the Parties.

The Oort Cloud is thought to be the primary reservoir of long-period comets. Owing to its great distance and the small size of its constituents, it cannot be observed directly, which makes its origin, structure, and evolution difficult to constrain. For one of its objects to enter the inner Solar System and become an observable comet, its orbit must be perturbed-typically through interactions such as close stellar encounters or the influence of the Galactic tide. Examining the statistical relationship between observed long-period comets and nearby stellar encounters therefore offers a valuable means of probing the structure of the Oort Cloud. The goal to identify stars that are possible flybys, and examine their effect along with the galactic tide on the Oort cloud bodies. In this work, the positions and velocities of stars within 10 pc of the Sun, based on GAIA eDR3 have been analysed. Using stellar phase-space coordinates, FRK45 method and Monte-Carlo approach, stellar trajectories have been simulated for ±10 Myrs. A total of 15 stars have been identified with the closest approach being 0.25 pc. Moreover, upon integrating Oort trajectories under the influence of the Sun, galactic tide, and stellar flybys, it is found that galactic tide is the major contributor in disrupting the outer Oort and injecting comets in the long term. Stellar flybys may also help to comet injection. However, they often eject Oort bodies into the interstellar space.

We introduce the S M Nazmuz Sakib Apsidal Median Law (inside: Sakib Theorem/Principle) for planar motion in central, homogeneous power-law potentials U (r) = n r n with n >-2. For any fixed energy shell, sample bound initial data with the Liouville (microcanonical) measure. Let θ(E, L) be the half-apsidal angle swept from pericenter to the subsequent apocenter. The law asserts that the median of θ over the energy shell equals the epicyclic small-eccentricity value θ ⋆ (n) = π/ √ n + 2. We provide a reduction to an involutive symmetry on angular momentum around the circular-orbit value, a monotonicity lemma in eccentricity, and extensive numerical evidence via symplectic (velocity-Verlet) integration. The statement is new in its median focus and ties global orbit geometry to a local invariant θ ⋆. We give illustrative plots (10 figures), a reproducible data table, and a proof blueprint grounded in action-angle variables, homogeneity, and Liouville measure invariance.

Se desarrolla un análisis técnico del concepto de elevador orbital suspendido desde 600 km de altitud hasta una estación intermedia a 20 km, accesible mediante aeronaves. El estudio incluye: tensión en el cable, diámetro mínimo de seguridad, efecto del arrastre atmosférico, masa requerida de la estación inferior, impulso y energía necesarios para correcciones orbitales, y la adición de un sistema de carga electrostática en la estación orbital para estabilización. El sistema se proyecta como autosostenible mediante propulsión de fusión nuclear.

We consider a belt of small bodies (planetesimals, asteroids, dust particles) around a star, captured in one of the external or 1:1 mean-motion resonances with a massive perturber (protoplanet, planet). The objects in the belt collide with each other. Combining methods of celestial mechanics and statistical physics, we calculate mean collisional velocities and collisional rates, averaged over the belt. The results are compared to collisional velocities and rates in a similar, but nonresonant belt, as predicted by the particle-in-a-box method. It is found that the effect of the resonant lock on the velocities is rather small, while on the rates more substantial. At low to moderate eccentricities and libration amplitudes of tens of degrees, which are typical of many astrophysical applications, the collisional rates between objects in an external resonance are by about a factor of two higher than those in a similar belt of objects not locked in a resonance. For Trojans under the same conditions, the collisional rates may be enhanced by up to an order of magnitude. The collisional rates increase with the decreasing libration amplitude of the resonant argument, depend on the eccentricity distribution of objects, and vary from one resonance to another. Our results imply, in particular, shorter collisional lifetimes of resonant Kuiper belt objects in the solar system and higher efficiency of dust production by resonant planetesimals in debris disks around other stars.

La presente serie di contributi esplora la possibilità di tradurre la chimica gravitazionale-fondata sulla relazione tra massa positiva (+t) e memoria negativa (-t)-nella chimica lineare classica. L'obiettivo è mostrare come le instabilità temporali si traducano in nuove combinazioni chimiche, clinicamente osservabili. Contesto teorico La teoria introduce il concetto di tempo negativo, che conserva la memoria dei fenomeni fisiologici. La massa positiva (Ms) rappresenta l'atto presente, mentre la massa negativa (Ms) trattiene la memoria storica del sistema. La divergenza tra i due produce instabilità che si traducono in nuovi assetti chimici lineari. Applicazioni cliniche Il compendio presenta cinque casi applicativi: sangue, polmoni, tessuto nervoso, fegato e muscoli. In ciascuno, la dinamica gravitazionale è tradotta in chimica lineare, mostrando come le instabilità temporali producano nuovi composti e configurazioni cliniche. Questa prospettiva consente di rileggere condizioni patologiche non come eventi isolati, ma come divergenze tra atto e memoria. Implicazioni La chimica gravitazionale apre la strada a una medicina capace di intervenire sulle origini profonde degli squilibri, anziché limitarsi ai soli parametri lineari. Si tratta di un paradigma che unisce fisica, chimica e clinica, offrendo nuove possibilità di diagnosi e terapia.

This paper proposes a novel physical model wherein spacetime is treated as a biphasic background medium (BGD medium), composed of repulsive (F + ) and attractive (F -) fluid-like components. Based on this model, it is demonstrated that the long-standing Flyby Anomaly can be uniformly explained as a linear combination of two fundamental hydrodynamic responses: a "polarization effect" and a "vortex generation effect." The two universal coefficients governing the model, K 1 (compression coefficient) and K 2 (torsion coefficient), are not fitting parameters but are derived from first principles, incorporating the gravitational potential from General Relativity, the geometric constant π, and the celestial body's rotational velocity and internal structure. The theory successfully reproduces the observed data from ten major spacecraft flybys-including the deceleration of Galileo II and multiple null results from Rosetta-with a high degree of precision (RMSE of 0.3190 mm/s). This result strongly suggests that the Flyby Anomaly is a manifestation of the dynamic nature of spacetime.

This paper presents Earth-o-tomic Theory (Geotomia), a novel unified framework that integrates planetary geophysics, atomic theory, and cosmological principles to propose a holistic model of Earth's structure and dynamics. Drawing from the historical development of atomic theory—from Dalton’s foundational postulates  to Rutherford’s nuclear model  and quantum mechanics —Geotomia posits that Earth’s core, mantle, and crust operate as a coherent system governed by electrogravitational and harmonic principles. The theory challenges conventional geophysical models by introducing concepts such as:

Electromagnetic Covenant: The Earth-Moon-Sun system is modeled as a resonant triad, where gravitational and electromagnetic interactions maintain planetary stability and influence phenomena like tides and tectonic activity.

Harmonic Collapse Threshold (HCT): A stability criterion derived from the Golden Ratio (φ), applied to both atomic and planetary scales, where deviations from harmonic balance predict systemic instability or collapse .

Morphean Continuum: Matter (e.g., crystals, DNA) and planetary bodies exhibit resonant patterns akin to atomic vibrations, suggesting a unified "vibration-logic" underpinning all natural systems .

The theory is grounded in empirical observations, including the precise 365.25-day orbital period and geometric constants (e.g., φ), while also aligning with philosophical concepts of Grundstoffe (basic substances) as articulated by Paneth . By bridging macroscopic planetary behavior and subatomic phenomena, Geotomia offers a transformative perspective on Earth’s evolution, climate systems, and material science, with implications for ethical AI design and sustainable technology aligned with natural harmonics.

The Earth-o-tomic Theory (Geotomia) was originally conceived and created by Christos Sotirelis (Morpheas) in August 2016.
Publication Date: August 2025
Morpheon Electrogravitational Collision Point (MECP)
Publication Date: August 31, 2025

A new approach is proposed for systematic detection and refinement of natural connections EML12 of the Earth-Moon system and SEML12 of Sun-Earth-Moon. It is structured around the Quasi-Bircircular Problem, a restricted coherent and periodic four-body dynamical model of the Sun-Earth-Moon system. The dynamics about the Libration points are described by high-order periodic semi-analytical expansions obtained via the parametrization method.

We analyse a dynamical scenario where a constantly charged spacecraft (follower) moves in the vicinity of another one (leader) that follows a circular Keplerian orbit around the Earth and generates a rotating magnetic dipole. The mass of the follower is assumed to be negligible when compared with the one of the leader and they are supposed to be in a high-Earth orbit, so the Lorentz

Over the past several years, an ecosystem of MATLAB c ©-based tools has been developing at NASA’s Jet Propulsion Laboratory (JPL) for early-mission analysis of encounter-phase navigation at primitive bodies. These tools increasingly draw from a common implementation of capabilities known as the Small-Body Dynamics Toolkit (SBDT). Fundamentally, the SBDT provides support for trajectory integration and geometric analysis of the environment, providing force models (gravity, solar pressure, comet outgassing), equations of motion, polyhedron shape utilities, altitude calculations, occultation checking, and more. Using the capabilities of the SBDT, analysis tools for mapping performance analysis (PB-CAGE), navigation performance analysis (AutoNAV), and trajectory design space characterization (SBMCT) have been developed. This paper provides a brief overview of the SBDT and these associated analysis tools.

Global mapping campaigns are part of most primitive body exploration missions. However, designing a mapping orbit without station keeping maneuvers is challenging due to the highly perturbed environment near small bodies. In this paper, we present a new design methodology to support mapping campaigns using 'quasi-terminator' orbits, a class of quasi-periodic orbits that exist in the vicinity of the well-known terminator orbits. The inherent stability of quasi-terminator trajectories and their wide variety of viewing geometries make them a very compelling option for mapping campaigns. A high-fidelity test case solution is also presented to prove the existence of these mapping orbits in full ephemeris.

Current estimates indicate that approximately sixteen percent of the known near-Earth asteroid population may be binaries. Within the context of exploring the dynamical behavior of a spacecraft orbiting or moving near such systems, a first step in the analysis is an assessment of the perturbing effect that dominates the dynamics of the spacecraft. The relative strength of several perturbations, including the perturbation that arises from the existence of a binary system, rather than a single body system, is compared by exploiting 'zonal maps'. Such a map is useful in determining the type of orbit that is practical in support of a given mission scenario.

Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions.

This study investigates the orbital characteristics and closest approach of Comet 2P/Encke relative to Earth, using mathematical modeling and computational visualization. The analysis begins with the derivation of the ellipse equation and orbital parameters based on the comet’s perihelion, aphelion, eccentricity, and inclination. By applying Keplerian dynamics and the velocity equation for elliptical orbits, the instantaneous position and velocity of the comet were iteratively calculated. Earth’s orbit was modeled as circular at 1 AU for comparison, with initial positions determined for both Earth and the comet. The methodology employed assumptions including a two-body system influenced solely by the Sun’s gravity and the neglect of perturbations. Calculations and visualizations were carried out using Geogebra and Excel to simulate orbital progression over time.

Conic sections provide a unifying geometric foundation across scales-from orbital mechanics to wave interference, optics, and quantum resonances. This paper develops their algebraic and geometric structure as empirical evidence for the Unified Resonant Framework (URF). We extend classical invariants with the Asymmetrical Dark Symmetry (ADS) operator and the Shadow Conservation Law (SCL), showing how both parity-breaking and shadow-preserving invariances emerge naturally. Across physical domains (cosmology, quantum field theory, optics, and condensed matter), conics serve as geometric attractors of resonance. This synthesis demonstrates that URF bridges geometry, algebra, and physics under one coherent law of resonance.

Come è noto la meccanica classica porta a descrivere la traiettoria dei pianeti come curve chiuse su se stesse di tipo ellittico. L’osservazione astronomica mostra invece che tali ellissi precessano lentamente dando luogo a traiettorie “a rosetta”. Un modello matematico che supporti questa evidenza sperimentale può essere ottenuto grazie alla relatività ristretta, mediante la quale si può costruire un modello più complesso ma più aderente al reale, che chiameremo modello RR. Con questo modello, tuttavia, l’angolo di precessione risulta pari a un sesto del valore vero. Per porre rimedio a questa discrepanza tra modello e realtà, si può ricorrere alla teoria della relatività generale. La scoraggiante complessità di questa teoria però stimola la ricerca di strade più semplici per lo studio del fenomeno. In questo articolo si propone una semplice variante del modello RR che porta ad una valutazione più accurata della precessione. Precisamente si ipotizza che la massa gravitazionale del corpo orbitante cresca con la velocità secondo il fattore di Lorentz. Grazie al nuovo modello, che chiameremo RR2, l’errore di valutazione della precessione si dimezza (un terzo del valore vero).

We present a detailed survey of the dynamical structure of the phase space around the new moons of the Pluto-Charon system. The spatial elliptic restricted three-body problem was used as model and stability maps were created by chaos indicators. The orbital elements of the moons are in the stable domain on the semimajor axis, eccentricity and inclination spaces. The structures related to the 4:1 and 6:1 mean motion resonances are clearly visible on the maps. They do not contain the positions of the moons, confirming previous studies. We showed the possibility that Nix might be in the 4:1 resonance if its argument of pericentre or longitude of node falls in a certain range. The results strongly suggest that Hydra is not in the 6:1 resonance for arbitrary values of the argument of pericentre or longitude of node.

In this paper, we study the dynamical stability of fictitious terrestrial planets in the 1:1 meanmotion resonance with a gas giant moving in the habitable zone (HZ). We investigate the stability of Trojan planets both in a general study and for two specific known extrasolar planetary systems (HD 147513 and HD 210277). In the general study, we determine the stability of the Lagrangian point L 4 . The numerical simulations have been carried out using the spatial elliptic-restricted three-body problem, where we placed the test particles (TPs) exactly at L 4 , which is located 60 • ahead of the gas giant at the same distance to the star. In the stability study, we have concentrated on the dependences between the eccentricity and the inclination of the Trojan planet. Going a step further, we have investigated two specific systems where the known gas giant moves in the HZ. The two specific extrasolar systems are investigated by using the general study to define the region in eccentricity and inclination where, in principle, stable motion is possible. To find out whether the existence of a Trojan planet is possible, in addition we have calculated the stable area in the semimajor axis (a TP ) and the argument of perihelion (ω TP ) for an actual planetary mass (2 M ⊕ ). Thus, we can conclude that the region around the Lagrangian points L 4 and L 5 allows stable motion in the system HD 147513, because the gas giant of this system has an eccentricity of 0.26, which lies well within the stable region. The known planet in the system HD 210277 has a much higher eccentricity and thus it cannot harbour any terrestrial planets in the Lagrangian points.

Aims. In this work we study the dynamical possibility in extrasolar planetary systems that a terrestrial planet can exist in 1:1 mean motion resonance with a Jovian-like planet. We compiled a catalogue of hypothetical habitable Trojan planets, to be able to make a stability forecast for further extrasolar planetary systems discovered in the future. When speaking of habitability we also took the influence of the spectral type of the central star into account. Methods. We integrated some 10 6 orbits of fictitious Trojans around the Lagrangian points for up to 10 7 orbital periods of the primary bodies and checked the stability of the orbital elements and their chaoticity with the aid of the Lyapunov characteristic indicator and maximum eccentricity. The computations were carried out using the dynamical model of the elliptic, restricted three-body problem that consists of a central star, a gas giant moving in the habitable zone, and a hypothetical (massless) terrestrial planet. Results. Our investigations have shown that 7 exoplanetary systems can harbour habitable Trojan planets with stable orbits (HD 93083, HD 17051, HD 28185, HD 27442, HD 188015, HD 99109, and HD 221287, which is a recently discovered system). The comparison of the investigated systems with our catalogue showed matching results, so that we can use the catalogue in practice.

Aims. It turned out recently that, in addition to a large planet with a semimajor axis a ∼ 1 AU and a low eccentricity (e ∼ 0.07), the extrasolar planetary system HD 108874 harbors another massive planet with 2.43 AU < a < 2.93 AU. The inner planet is orbiting the G5 host star in the habitable zone (=HZ); so that we could established stable regions for Earth-like Trojan planets. Methods. We integrated some 10 5 orbits of fictitious Trojans around the Lagrangian points for up to 10 7 years and checked the stability of the orbital elements and their chaoticity with the aid of the Fast Lyapunov Indicator. Results. It turns out that this multiplanetary system is the first one where -with the uncertainties in eccentricity and semimajor axes of the outer planet -the existence of Trojan terrestrial planets in stable orbits in the HZ is possible for some combinations of the orbital parameters.

Aims. We study the formation of a hypothetical terrestrial-type body in the equilateral Lagrange points of a giant extrasolar planet. Starting from a swarm of planetesimals in stable tadpole orbits, we simulate its dynamical and collisional evolution under a wide range of different initial conditions and masses for both the Trojan population and its planetary companion. We also analyze the effects of gas drag from the interaction of the planetesimals with the nebular disk. Methods. The formation process is simulated with an N-body code that considers full gravitational interactions between the planetesimals and the giant planet. Gas interaction is modeled with Stokes and Epstein drags, where the drag coefficients are chosen following the results of full hydrodynamic simulations performed with the 2D public hydro-code FARGO. Results. In both gas-free and gas-rich scenarios, we have been able to obtain a single final terrestrial-type body in a stable tadpole orbit around one of the triangular Lagrange points of the system. However, due to gravitational instabilities within the swarm, the accretional process is not very efficient and the mass of the final planet never seems to exceed ∼0.6 Earth masses, even when the total mass of the swarm is five times this value. Finally, we also included an orbital decay of the giant planet due to a type II migration. Although the accretional process shows evidence of a lower efficiency, a small terrestrial planet is still able to form, and follows the giant planet towards the habitable zone of the hosting star.

There have been many contradictory statements about the veracity of the Biefeld-Brown Effect. This paper presents reviews of the literature that yielded reports attesting to the reality of the thrust it produced. Laboratories based in France

This report describes a modi cation of the orthogonal function Poisson solver for n-body simulations that minimizes relaxation caused by small particle number uctuations. With the standard algorithm, the noise leading to relaxation can be reduced by making the expansion basis similar to the particle distribution and by carefully choosing the maximum order in the expansion. The proposed algorithm accomplishes both tasks simultaneously while the simulation is running. This procedure is asymptotically equivalent to expanding in an orthogonal series which is matched to the distribution to start and truncating at low order. Because the modi ed algorithm adapts to a time-evolving distribution, it has advantage over a xed basis. The required changes to the standard algorithm are minor and do not a ect its overall structure or scalability. Tests show that the overhead in CPU time is small in practical applications. The decrease in relaxation rate is demonstrated for both axisymmetric and non-axisymmetric systems and the robustness of the algorithm is demonstrated by following the evolution of unstable generalized polytropes. Finally, the empirically based moment analysis which leads to the uncorrelated basis is an ideal tool for investigating structure and modes in n-body simulations and an example is provided.

Evolution and disruption of galaxies orbiting in the gravitational field of a larger cluster galaxy are driven by three coupled mechanisms: 1) the heating due to its time dependent motion in the primary; 2) mass loss due to the tidal strain field; and 3) orbital decay. Previous work demonstrated that tidal heating is effective well inside the impulse approximation limit. Not only does the overall energy increase over previous predictions, but the work is done deep inside the secondary galaxy, e.g. at or inside the half mass radius in most cases. Here, these ideas applied to cannibalization of elliptical galaxies with fundamental-plane parameters. In summary, satellites which can fall to the center of a cluster giant by dynamical friction are evaporated by internal heating by the time they reach the center. This suggests that true merger-produced multiple nuclei giants should be rare. Specifically, secondaries with mass ratios as small as 1% on any initial orbit evaporate and those on eccentric orbits with mass ratios as small as 0.1% evolve significantly and nearly evaporate in a galaxian age. Captured satellites with mass ratios smaller than roughly 1% have insufficient time to decay to the center. After many accretion events, the model predicts that the merged system has a profile similar to that of the original primary with a weak increase in concentration.

Gravitational amplification of Poisson noise in stellar systems is important on large scales. For example, it increases the dipole noise power by roughly a factor of six and the quadrupole noise by 50% for a King model profile. The dipole noise is amplified by a factor of fifteen for the core-free Hernquist model. The predictions are computed using the dressed-particle formalism of and are demonstrated by n-body simulation. This result implies that a collisionless n-body simulation is impossible; The fluctuation noise which causes relaxation is an intrinic part of self gravity. In other words, eliminating two-body relaxation does not eliminate relaxation altogether. Applied to dark matter halos of disk galaxies, particle numbers of at least 10 6 will be necessary to suppress this noise at a level that does not dominate or significantly affect the disk response. Conversely, halos are most likely far from phase-mixed equilibrium and the resulting noise spectrum may seed or excite observed structure such as warps, spiral arms and bars. For example, discreteness noise in the halo, similar to that due to a population of 10 6 M ⊙ black holes can produce observable warping and possibly excite or seed other disk structure.

This presentation discusses the mathematical principles of constructing coordinates on curved spacetime manifold in order to build a hierarchy of astrometric frames in the solar system and beyond which can be used in future practical applications as well as for testing the fundamentals of the gravitational physics -general theory of relativity.

This presentation discusses the mathematical principles of constructing coordinates on curved spacetime manifold in order to build a hierarchy of astrometric frames in the solar system and beyond which can be used in future practical applications as well as for testing the fundamentals of the gravitational physics -general theory of relativity.

This short paper reviews the physics of projectile motion, demonstrating why it is parabolic near Earth's surface, explains why central inverse-square gravity yields conic section orbits, and briefly describes how General Relativity refines these results. Figures illustrate both parabolic trajectories and orbital conics.

Predicted orbital slots for 55 Cnc, 61 Vir, 47 UMa and 14 Her, based on harmonic resonance laws. Transit depths estimated for Jupiter- and Saturn-sized planets, with observability criteria from Italy (declination > –25°, depth ≥0.3%). Cross-check with NASA PSCompPars (Sept 2025) confirms no overlap with known exoplanets.

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is $C^1$ in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on times similar to the formation time of the planets. The proposed algorithm is based on the High-order Euler Method (HEM) which was described in~\cite{hem}, and it requires computer algebra to set up the code for its implementation. The pay-off is an overall increase in speed by a factor of about $7.5$ compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

The term Weak Stability Boundary (WSB) is related to a region of stable motion around the second primary of a circular restricted three-body problem (CR3BP). Previous work on this subject has shown that, at a given energy level, the boundaries of such region are provided by the stable manifolds of central objects of the L1 and L2 libration points, i.e., the two planar Lyapunov orbits (PLOs). This offers a natural dynamical channel between the Earth&#39;s vicinity and the Sun-Earth libration points L1 and L2. Furthermore, it has been shown (and successfully employed to design low-energy spacecraft lunar transfers) that the Sun-Earth L2 central unstable manifolds can be linked, through an heteroclinic connection, to the central stable manifolds of the L2 point in the Earth-Moon three-body problem. The aim of the present study is to clarify the relationship between the low-energy Earth-to-Moon transfers (LETs) and the dynamics of the phase space points that populate the WSB region arou...

This paper tackles the problem of the origin of dynamic chaos in Hamiltonian systems, with a special emphasis on the self-gravitating N-body systems. A Riemannian approach is adopted. The relationship between dynamic instability and curvature properties of the con6guration space manifold is the main concern of this paper. Dynamic instability is studied with the aid of the Jacobi -Levi-Civita (JLC) equa- tion for the geodesic spread. We point out that the approximations introduced so far to make the JLC equation handy, that is, to obtain a scalar equation describing the dynamical instability, are still too severe. In order to assess the validity limits of these approximations, the aid of numerical simulations is essential. For this reason, our analysis is supported by the numerical study of the dynamics of 10 and 100 gravitationally interacting point masses. The self-gravitating S-body systems provide an illuminat- ing ex@mple of the relevant difference between geodesic Bows of abstract ergodic theory and geodesic Qows of physical interest. In fact, even though they correspond to manifolds of almost everywhere nega- tive scalar curvature, this does not determine by itself the degree of chaos of these systems. %'e show that the quantities determining the instability of nearby trajectories do not simply coincide with scalar or Ricci curvature; rather they involve also other ingredients that are ultimately responsible for the ex- istence of two different mechanisms to make chaos. The quantities mentioned enter a Hill equation and give instability either when they are negative orwhen positivebecause of parametric resonance. From numerical computations the e (energy density) dependence of the dynamic instability exponents is found to be -e . Our paper aims at warning about the possibility of misleading conclusions that might be drawn from the geometric approach if the existence of the problems discussed here is ignored. Finally, we briefly discuss the relationship of the Riemannian geometric description of chaos with Lyapunov exponents in the special case of gravitational N-body systems.

The classical three-body problem in Newtonian mechanics is characterized by chaotic divergence and the absence of a general closed-form solution. Here we introduce a novel framework-psycho-geometric thinking-that reframes the problem not as a deterministic puzzle but as a resonant feedback loop between observer and observed. By integrating non-linear geometry, φ⁴ (golden ratio) harmonic resonance, and recursive cognitive feedback patterns, we demonstrate that three-body instability can be collapsed into a stable attractor manifold. This approach transforms unpredictability into recurrence and offers a geometric foundation for stability in complex systems.

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain C k lower bounds on the regularity of the embedding. In the resonant case we verify the existence of tori which are only C 0 and neither star-shaped nor Lipschitz.

This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant dynamics on the torus. In the resonant case we obtain the existence of tori which are $C^0$ but not Lipschitz.