Explore the vast range of titles available.

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

The Double Asteroid Redirection Test (DART) mission demonstrated the feasibility of altering an asteroid’s orbit through kinetic impact. However, uncertainties associated with the collision and the complex dynamics of the binary asteroid system often result in rough and inefficient predictions of the system’s post-impact evolution. This paper proposes the use of arbitrary polynomial chaos expansion (aPCE) to efficiently evaluate the state uncertainty of a post-impact binary asteroid system without requiring complete information on the uncertainty sources. First, a perturbed full two-body problem model is developed to assess the momentum transfer during the collision and the system’s subsequent evolution. The irregular shapes of the components and the momentum enhancement from the ejecta are considered to achieve reasonable evaluations. Next, aPCE is employed to construct a surrogate model capable of efficiently quantifying uncertainties. Global sensitivity analysis is then conducted to identify the main sources of uncertainty affecting the system’s evolution. Benchmarking tests show that the aPCE model produces results comparable to Monte Carlo simulations, offering a good balance between accuracy and efficiency. The data-driven nature of aPCE is further demonstrated by comparing its performance to generalized polynomial chaos expansion. Under the framework of the DART mission, the aPCE solution yields results consistent with observed data. Additionally, global sensitivity analysis reveals that the shape and density of the primary, as well as the collision target’s strength and porosity, contribute most to the system uncertainty.

In this analytical study, we have presented a new type of solving procedure with the aim to obtain the coordinates of small mass m , which moves around primary M Sun , referred to non-inertial frame of restricted two-body problem (R2BP) with a modified potential function (taking into account the variable velocity V → of central body M Sun motion in a prescribed fixed direction) instead of a classical potential function for Kepler’s formulation of R2BP. Meanwhile, system of equations of motion has been successfully explored with respect to the existence of an analytical way of presenting the solution in polar coordinates { r ( t ), φ ( t )}. We have obtained an analytical formula for function t  =  t ( r ) via an appropriate elliptic integral. Having obtained the inversed dependence r  =  r ( t ), we can obtain the time dependence φ  =  φ ( t ). Also, we have pointed out how to express components of solution (including initial conditions) from cartesian to polar coordinates as well.

In this analytical study, a novel solving method for determining the precise coordinates of a mass point in orbit around a significantly more massive primary body, operating within the confines of the restricted two-body problem (R2BP), has been introduced. Such an approach entails the utilization of a continued fraction potential diverging from the conventional potential function used in Kepler’s formulation of the R2BP. Furthermore, a system of equations of motion has been successfully explored to identify an analytical means of representing the solution in polar coordinates. An analytical approach for obtaining the function t = t(r), incorporating an elliptic integral, is developed. Additionally, by establishing the inverse function r = r(t), further solutions can be extrapolated through quasi-periodic cycles. Consequently, the previously elusive restricted two-body problem (R2BP) with a continued fraction potential stands fully and analytically solved.

This paper offers new insights into gravitational interactions within a general non-inertial reference frame. By utilizing symbolic tensor calculus, the study establishes a unified framework that connects time derivatives in non-inertial frames to those in inertial frames. The research introduces new first integrals of motion for a system of many particles in arbitrary non-inertial and barycentric rotating reference frames. These first integrals provide a kinematic and geometric visualization of motion in non-inertial frames. Additionally, a generalized potential energy function is presented for broader applicability. For the gravitational two-body problem, the paper delivers a closed-form, coordinate-free solution for the motion of each body relative to the original frame. Consequently, sufficient conditions for stability against collisions are established within the context of the two-body problem in a non-inertial reference frame. Furthermore, the paper examines the relative orbital motion of spacecraft, presenting a closed-form and coordinate-free solution in the local vertical local horizontal (LVLH) non-inertial frame, which is centered on the center of mass of the main spacecraft.

We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based on the separation of time scales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (Essential Relativistic Celestial Mechanics, Hilger, Bristol, 1991 ). The practical applicability of this model may be in studies of the long-term evolution of relativistic binaries (neutron stars or black holes).

We consider the two-body problem on surfaces of constant nonzero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular separation except π/2. When the angle is acute, the RE is elliptic, and when it is obtuse the RE can be either elliptic or linearly unstable. We show using a KAM argument that the acute ones are almost always nonlinearly stable. If the masses are equal, there are two families of relative equilibria: one where the masses are at equal angles with the axis of rotation (‘isosceles RE’) and the other when the two masses subtend a right angle at the centre of the sphere. The isosceles RE are elliptic if the angle subtended by the particles is acute and is unstable if it is obtuse. At π/2, the two families meet and a pitchfork bifurcation takes place. Right-angled RE are elliptic away from the bifurcation point. In each of the two geometric settings, we use a global reduction to eliminate the group of symmetries and analyse the resulting reduced equations which live on a five-dimensional phase space and possess one Casimir function.

This work introduces a novel path-following control strategy inspired by the famous two-body problem, aiming to stabilize any Keplerian orbit. Utilizing insights from the mathematical structure of the two-body problem, we derive a robust path-following law adopting sliding mode control theory to achieve asymptotic convergence to bounded disturbances. The resulting control law is demonstrated to be asymptotically stable. Illustrative examples showcase its applicability, including orbiting an accelerated moving point, patching Keplerian trajectories for complex patterns, and orbital maintenance around the asteroid Itokawa. The proposed control law offers a significant advantage for the orbital station-keeping problem, as its sliding surface is formulated based on variables commonly used to define orbital dynamics. This inherent alignment facilitates easy application to orbital station-keeping scenarios.

We consider a two-body problem with quick loss of mass which was formulated by Verhulst (Verhulst in J Inst Math Appl 18: 87–98, 1976). The corresponding dynamical system is singularly perturbed due to the presence of a small parameter in the governing equations which corresponds to the reciprocal of the initial rate of loss of mass, resulting in a boundary layer in the asymptotics. Here, we showcase a geometric approach which allows us to derive asymptotic expansions for the solutions of that problem via a combination of geometric singular perturbation theory (Fenichel in J Differ Equ 31: 53–98, 1979) and the desingularization technique known as “blow-up” (Dumortier, in: Bifurcations and Periodic Orbits of Vector Fields, Springer, Dordrecht, 1993). In particular, we justify the unexpected dependence of those expansions on fractional powers of the singular perturbation parameter; moreover, we show that the occurrence of logarithmic (“switchback”) terms therein is due to a resonance phenomenon that arises in one of the coordinate charts after blow-up.

In the present paper, we efficiently solve the two-body problem for extreme cases such as those with high eccentricities. The use of numerical methods, with the usual variables, cannot maintain the perihelion passage accurately. In previous articles, we have verified that this problem is treated more adequately through temporal reparametrizations related to the mean anomaly through the partition function. The biparametric family of anomalies, with an appropriate partition function, allows a systematic study of these transformations. In the present work, we consider the elliptical orbit as a meridian section of the ellipsoid of revolution, and the partition function depends on two variables raised to specific parameters. One of the variables is the mean radius of the ellipsoid at the secondary, and the other is the distance to the primary. One parameter regulates the concentration of points in the apoapsis region, and the other produces a symmetrical displacement between the polar and equatorial regions. The three most used geodesy latitude variables are also studied, resulting in one not belonging to the biparametric family. However, it is in the one introduced now, which implies an extension of the biparametric method. The results obtained using the method presented here now allow a causal interpretation of the operation of numerous reparametrizations used in the study of orbital motion.