Search

Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects. This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries. The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is computed followed by nonlinear stability of the equilibrium points is examined. Similar to the case of classical problem, we have found that in the presence of assumed perturbations, there always exists one value of mass parameter within the stability range at which the discriminant $D_4$ vanish, consequently, Arnold-Moser theorem fails, which infer that triangular equilibrium points are unstable in nonlinear sense within the stability range. Present analysis is limited up to linear effect of the perturbations, which will be helpful to study the more generalized problem., Comment: 21 Pages

For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information about realistic solution of the problem. Present paper reflects about normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed. Due to the presence of perturbations and a tedious huge algebraic computation for intermediate terms, we have computed only up to the fourth order normalized Hamiltonian using Lie transforms. In non-resonance case, nonlinear stability of the system is discussed with the help of Arnold-Moser theorem. Again, the effects of radiation pressure, oblateness and presence of the disc are analyzed, separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in nonlinear sense within the stability range $0<\mu<\mu_1=\bar{\mu_c}$ due to failure of Arnold-Moser theorem. However, perturbation parameters affect the values of $\mu$ at which $D_4=0$, significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but in future it can be extended., Comment: 25 Pages

The most important aspects of a dynamical system are its stability and the factors which affects the stability property. This paper presents the analysis of the effects of albedo and disc on the zero velocity curves, existence of equilibrium points and on their linear stability in a generalized restricted three body problem that consists of motion of an infinitesimal mass under the uniform gravity field of radiating-oblate primary, oblate secondary and a disc, which is rotating about the common center of the mass of the system. A significant effect of albedo and disc are observed on the zero velocity curves, positions of equilibrium points and on the stability region. Linear stability analysis of collinear equilibrium points is performed with respect to mass ratio $\mu$ and albedo parameter of secondary, separately and it is found that these are unstable in both the cases. On the other hand, non-collinear equilibrium point is stable in a certain range of mass ratio. After analyzing individual as well as combined effect of radiation pressure force of the primary, albedo of secondary, oblateness of both the massive bodies and the disc, it is found that these perturbations play a significant on the motion of infinitesimal mass in the vicinity of equilibrium points. These results may be help to analyze more generalized problem of few bodies under the influence of different kind of perturbations such as P-R drag, solar wind drag etc. Present study is limited to the regular symmetric disc which will extend later., Comment: 13 Pages

The most important aspects of a dynamical system are its stability and the factors which affects the stability property. This paper presents the analysis of the effects of albedo and disc on the zero velocity curves, existence of equilibrium points and on their linear stability in a generalized restricted three body problem that consists of motion of an infinitesimal mass under the uniform gravity field of radiating-oblate primary, oblate secondary and a disc, which is rotating about the common center of the mass of the system. A significant effect of albedo and disc are observed on the zero velocity curves, positions of equilibrium points and on the stability region. Linear stability analysis of collinear equilibrium points is performed with respect to mass ratio $\mu$ and albedo parameter of secondary, separately and it is found that these are unstable in both the cases. On the other hand, non-collinear equilibrium point is stable in a certain range of mass ratio. After analyzing individual as well as combined effect of radiation pressure force of the primary, albedo of secondary, oblateness of both the massive bodies and the disc, it is found that these perturbations play a significant on the motion of infinitesimal mass in the vicinity of equilibrium points. These results may be help to analyze more generalized problem of few bodies under the influence of different kind of perturbations such as P-R drag, solar wind drag etc. Present study is limited to the regular symmetric disc which will extend later., Comment: 13 Pages

For the study of nonlinear stability of a dynamical system, normalized Hamiltonian of the system is very important to discuss the dynamics in the vicinity of invariant objects. In general, it represents a nonlinear approximation to the dynamics, which is very helpful to obtain the information about realistic solution of the problem. Present paper reflects about normalization of the Hamiltonian and analysis of nonlinear stability in non-resonance case, in the Chermnykh-like problem under the influence of perturbations in the form of radiation pressure, oblateness, and a disc. To describe nonlinear stability, initially, quadratic part of the Hamiltonian is normalized in the neighborhood of triangular equilibrium point and then higher order normalization is performed. Due to the presence of perturbations and a tedious huge algebraic computation for intermediate terms, we have computed only up to the fourth order normalized Hamiltonian using Lie transforms. In non-resonance case, nonlinear stability of the system is discussed with the help of Arnold-Moser theorem. Again, the effects of radiation pressure, oblateness and presence of the disc are analyzed, separately and it is observed that in the absence as well as presence of perturbation parameters, triangular equilibrium point is unstable in nonlinear sense within the stability range $0<\mu<\mu_1=\bar{\mu_c}$ due to failure of Arnold-Moser theorem. However, perturbation parameters affect the values of $\mu$ at which $D_4=0$, significantly. This study may help to analyze more generalized cases of the problem in the presence of some other types of perturbations such as P-R drag and solar wind drag. The results are limited to the regular symmetric disc but in future it can be extended., Comment: 25 Pages

Normal forms of Hamiltonian are very important to analyze the nonlinear stability of a dynamical system in the vicinity of invariant objects. This paper presents the normalization of Hamiltonian and the analysis of nonlinear stability of triangular equilibrium points in non-resonance case, in the photogravitational restricted three body problem under the influence of radiation pressures and P-R drags of the radiating primaries. The Hamiltonian of the system is normalized up to fourth order through Lie transform method and then to apply the Arnold-Moser theorem, Birkhoff normal form of the Hamiltonian is computed followed by nonlinear stability of the equilibrium points is examined. Similar to the case of classical problem, we have found that in the presence of assumed perturbations, there always exists one value of mass parameter within the stability range at which the discriminant $D_4$ vanish, consequently, Arnold-Moser theorem fails, which infer that triangular equilibrium points are unstable in nonlinear sense within the stability range. Present analysis is limited up to linear effect of the perturbations, which will be helpful to study the more generalized problem., Comment: 21 Pages

The orbits about Lagrangian equilibrium points are important for scientific investigations. Since, a number of space missions have been completed and some are being proposed by various space agencies. In light of this, we consider a more realistic model in which a disk, with power-law density profile, is rotating around the common center of mass of the system. Then, we analyze the periodic motion in the neighborhood of Lagrangian equilibrium points for the value of mass parameter $0<\mu\leq 1/2$. Periodic orbits of the infinitesimal mass in the vicinity of equilibrium are studied analytically and numerically. In spite of the periodic orbits, we have found some other kind of orbits like hyperbolic, asymptotic etc. The effect of radiation factor as well as oblateness coefficients on the motion of infinitesimal mass in the neighborhood of equilibrium points are also examined. The stability criteria of the orbits examined by the help of Poincar\'{e} surfaces of section (PSS) and found that stability regions depend on the Jacobi constant as well as other parameters., Comment: Accepted in APSS(Springer)

The orbits about Lagrangian equilibrium points are important for scientific investigations. Since, a number of space missions have been completed and some are being proposed by various space agencies. In light of this, we consider a more realistic model in which a disk, with power-law density profile, is rotating around the common center of mass of the system. Then, we analyze the periodic motion in the neighborhood of Lagrangian equilibrium points for the value of mass parameter $0<\mu\leq 1/2$. Periodic orbits of the infinitesimal mass in the vicinity of equilibrium are studied analytically and numerically. In spite of the periodic orbits, we have found some other kind of orbits like hyperbolic, asymptotic etc. The effect of radiation factor as well as oblateness coefficients on the motion of infinitesimal mass in the neighborhood of equilibrium points are also examined. The stability criteria of the orbits examined by the help of Poincar\'{e} surfaces of section (PSS) and found that stability regions depend on the Jacobi constant as well as other parameters., Comment: Accepted in APSS(Springer)

We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods we have found equilibrium points and examined their linear stability. We have also found the zero velocity surfaces for the present model. In addition to five equilibrium points there is a new equilibrium point on the line joining the two primaries. It is found that $L_2$ and $L_3$ are stable for some values of inner and outer radius of the disk while collinear points are unstable, but $L_4$ is conditionally stable for mass ratio less than that of Routh's critical value. Lastly we have obtained the effects of radiation pressure, oblateness and mass of the disk., Comment: AMS LaTeX article

We consider the modified restricted three body problem with power-law density profile of disk, which rotates around the center of mass of the system with perturbed mean motion. Using analytical and numerical methods we have found equilibrium points and examined their linear stability. We have also found the zero velocity surfaces for the present model. In addition to five equilibrium points there is a new equilibrium point on the line joining the two primaries. It is found that $L_2$ and $L_3$ are stable for some values of inner and outer radius of the disk while collinear points are unstable, but $L_4$ is conditionally stable for mass ratio less than that of Routh's critical value. Lastly we have obtained the effects of radiation pressure, oblateness and mass of the disk., Comment: AMS LaTeX article