Manev Potential Research Articles

On the Vertical Libration of Maxwell’s Ring Configuration with a Manev Potential

We discuss Maxwell’s ring model with a quasi-homogeneous potential of the dominant mass, and specially study the vertical libration of Maxwell’s ring configuration with a Manev potential. There are two kinds of periodic vertical libration, which are non-alternating and alternating cases. The Hamiltonian is highly symmetric and can be reduced to a simple form with two degrees of freedom. Then the Hamiltonian is expanded near the planar circular relative equilibria. By the Lie transformation technique, the birkhoff norm form up to the fourth order is calculated. When the mass ratio between the ring and the dominant mass, and the coefficient of the second term in a Manev potential are both small enough, the nonlinear stability for the vertical libration of Maxwell’s ring configuration with a Manev potential is then proved according to Arnold’s stability theorem.

Stability of Equilibrium Points in the Spatially Restricted \({\boldsymbol{N+1}}\) -Body Problem with Manev Potential

Stability of Equilibrium Points in the Spatially Restricted \({\boldsymbol{N+1}}\) -Body Problem with Manev Potential

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

A Comparative study on generalized Manev potential and Newtonian potential in perturbed restricted four-body problem

Global-in-time existence of weak solutions for Vlasov-Manev-Fokker-Planck system

<p style='text-indent:20px;'>We consider the Vlasov–Manev–Fokker–Planck (VMFP) system in three dimensions, which differs from the Vlasov–Poisson–Fokker–Planck in that it has the gravitational potential of the form <inline-formula><tex-math id="M1">\begin{document}$ -1/r - 1/r^2 $\end{document}</tex-math></inline-formula> instead of the Newtonian one. For the VMFP system, we establish the global-in-time existence of weak solutions under smallness assumption on either the initial mass or the coefficient of the pure Manev potential. The proof extends to several related kinetic systems.</p>

Quasi-homogeneous two-body problem

The quasi-homogeneous two-body problem aims at studying the interaction between two point particles under a prescribed potential in the form of W(r)=−Ara−Brb, where A, B > 0 are constants and r is the mutual distance between two particles. Important examples include the Manev potential (a = 1, b = 2) and the Schwarzschild potential (a = 1, b = 3). It is well known that power two serves as a threshold value for the homogeneous potential: One is able to observe significant differences regarding the solution dynamics as the power of the homogeneous potential exceeds two from below. This phenomenon remains observable for quasi-homogeneous potentials. In this paper, we shall provide a complete characterization of the whole phase space of the quasi-homogeneous two-body problem in terms of global existence and singularity for all the possible b > a > 0. In particular, one is able to generalize the result of the Manev and Schwarzschild two-body problem to all the quasi-homogeneous potentials. Two techniques are presented in this paper: One is the variational method based on the energy, and the other is a direct computation of collision time based on the integrability of two-body systems.

Periodic Solutions and KAM Tori for the Spatial Maxwell Restricted N+1-Body Problem with Manev Potential

We consider the motion of an infinitesimal mass under the Newtonian attraction of N point masses forming a ring plus a central body where a Manev potential (-1/r + e/r^2, e in {mathbb {R}}), is applied to the central body. More precisely, the bodies are arranged in a planar ring configuration. This configuration consists of N-1 primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity omega . Another primary of mass m_0=beta m (beta >0 parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or a radiation source, which introduces a new parameter e. The existence and stability of periodic solutions of the spatial Maxwell restricted N+1-body problem is obtained using averaging theory. The determination of KAM 3-tori encasing some of the linearly stable periodic solutions is proved. The planar case is moreover considered.

Symmetric Periodic Solutions for the Spatial Maxwell Restricted $$N+1$$-Problem with Manev Potential

In the spatial Maxwell restricted $$N+1$$ -body problem, the motion of an infinitesimal particle attracted by the gravitational field of (N) bodies is studied. These bodies are arranged in a planar ring configuration. This configuration consists of $$N-1$$ primaries of equal mass m located at the vertices of a regular polygon that is rotating on its own plane about its center of mass with a constant angular velocity $$\omega $$ . Another primary of mass $$m_0=\beta m$$ ( $$\beta >0$$ parameter) is placed at the center of the ring. Moreover, we assume that the central body may be an ellipsoid, or radiation source, which introduces a new parameter e. The existence of several families of symmetric periodic solutions for the spatial Maxwell restricted $$(N+1)$$ -problem with Manev potential is proved. More precisely, firstly we get symmetric periodic solutions around the central body (attractor or repulsor) close to the equatorial plane and small parameter of oblateness. Secondly, we obtain symmetric periodic solutions far away of the central body and peripherals, close to the equatorial plane with arbitrary oblateness. Furthermore, all these families of periodic solutions are stable.

Perturbed Planar Restricted Four-Body Problem with Repulsive Manev Potential

In this paper, we carried out a numerical study of the planar restricted four-body problem with repulsive Manev potential and perturbations in the Coriolis and centrifugal forces such that the peripherals possess Eulerian configuration. We have presented the equations of motion in the rotating frame and investigated the existence and location of the equilibrium points. We have found that there exist six equilibrium points all of which lie along the coordinate axes and shift in positions as the perturbation parameter is varied. We have also examined the linear stability of these equilibrium points and they are found unstable. The dynamical behavior of this system is also investigated using the Lyapunov Characteristic Exponents and the system is found to be chaotic.

On the existence of libration points in the spatial collinear restricted four-body problem within frame of repulsive Manev potential and variable mass

The present paper deals with the spatial collinear restricted four-body problem within the frame of repulsive Manev potential (−1r+er2),e>0 and variable mass. We have revealed that how the parameters involved due to variable mass of the infinitesimal body influence the locations as well as number of libration points. Moreover, it is observed that the libration points exist only on the coordinate axes. The regions of possible motion are highly influenced by the Manev parameter, parameter occur due to variable mass of the infinitesimal body as well as by the Jacobian constant. Further, it is unveiled that the libration points are unstable for all the involved parameters.

Spatial collinear restricted four-body problem with repulsive Manev potential

We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential ( $$-1/r +e/r^{2}$$ ), $$e>0$$ , is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter $$\beta $$ , the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.

Satellite motion in a Manev potential with drag

In this paper, we consider a satellite orbiting in a Manev gravitational potential under the influence of an atmospheric drag force that varies with the square of velocity. Using an exponential atmosphere that varies with the orbital altitude of the satellite, we examine a circular orbit scenario. In particular, we derive expressions for the change in satellite radial distance as a function of the drag force parameters and obtain numerical results. The Manev potential is an alternative to the Newtonian potential that has a wide variety of applications, in astronomy, astrophysics, space dynamics, classical physics, mechanics, and even atomic physics.

Stable Ground States and Self-Similar Blow-Up Solutions for the Gravitational Vlasov--Manev System

In this work, we study the orbital stability of steady states and the existence of self-similar blow-up solutions to the so-called Vlasov--Manev system. This system is a kinetic model which has a similar Vlasov structure to the classical Vlasov--Poisson system but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional Laplacian equation. We first prove the orbital stability of the ground state--type solutions which are constructed as minimizers of the Hamiltonian, following the classical strategy: compactness of the minimizing sequences and the rigidity of the flow. However, in driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the Vlasov--Manev system, which we overcome by imposing a subcritical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler--Lagrange equations (fractional Laplacian equations) to which classical results for the Poisson equation do not extend. We overcome this difficulty by proving the uniqueness of the minimizer under equimeasurability constraints, using only the regularity of the potential and not the fractional Laplacian Euler--Lagrange equations itself. In the second part of this work, we prove the existence of exact self-similar blow-up solutions to the Vlasov--Manev equation, with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint.

Trajetórias limitadas num sistema com uma força dependente da velocidade

Um sistema mecânico com uma força dependente da velocidade é descrito e um caso em que há conservação da energia é analisado. O potencial resultante desta força é ainda empregado no estudo acerca das trajetórias de sistemas submetidos ao potencial gravitacional newtoniano, ao potencial de Manev e ao potencial harmônico.

Órbitas fechadas e o potencial harmônico de Manev

Estudamos condições para a existência de órbitas fechadas em três potenciais distintos. Primeiramente, confirmamos condições já conhecidas para o problema de Manev. Em um segundo passo, inspirados pelo teorema de Bertrand, construímos o potencial harmônico de Manev substituindo o termo do oscilador harmônico no lugar do termo Newtoniano no potencial de Manev. Assim como no potencial de Manev, encontramos condições similares para a existência de órbitas fechadas que são relacionadas com valores restritos do momento angular. Analisamos também um potencial Newtoniano corrigido com o termo harmônico.

Symmetric periodic solutions of the anisotropic Manev problem

We consider the Manev potential in an anisotropic space, i.e., such that the force acts differently in each direction. Using a generalization of the Poincaré continuation method we study the existence of periodic solutions for weak anisotropy. In particular we find that the symmetric periodic orbits of the Manev system are perturbed to periodic orbits in the anisotropic problem.

Phase portraits of the two-body problem with Manev potential

The Manev systems are two-body problems defined by a potential of the form a/r + b/r2, where r is the distance between the two particles, and a and b are arbitrary constants. The Hamiltonian H = (pr2 + pθ2/r2)/2 + a/r + b/r2 and the angular momentum pθ = r2 associated with Manev systems are two first integrals, which are independent and in involution. Let Ih (respectively Ic) be the set of points of the phase space on which H (respectively pθ) takes the value h (respectively c). Since H and pθ are first integrals, the sets Ih, Ic and Ihc = IhIc are invariant under the flow of the Manev systems. We characterize the global flow of these systems when a and b vary. Thus we describe the foliation of the phase space by the invariant sets Ih and the foliation of Ih by the invariant sets Ihc.

On the anisotropic Manev problem

We consider the Manev potential, given by the sum between the inverse and the inverse square of the distance, in an anisotropic space, i.e., such that the force acts differently in each direction. Using McGehee coordinates, we blow up the collision singularity, paste a collision manifold to the phase space, study the flow on and near the collision manifold, and find a positive-measure set of collision orbits. Besides frontal homothetic, frontal nonhomothetic, and spiraling collisions and ejections, we put into the evidence the surprising class of oscillatory collision and ejection orbits. Using the infinity manifold, we further tackle capture and escape solutions in the zero-energy case. By finding the connection orbits between equilibria and/or cycles at impact and at infinity, we describe a large class of capture-collision and ejection-escape solutions.