哈密顿量的正规化与非共振扰动下三角平衡点的非线性稳定性
Normalization of Hamiltonian and Nonlinear Stability of the Triangular Equilibrium Points in Non-resonance Case with Perturbations
摘要：
在研究动力系统的非线性稳定性时，系统的正规化哈密顿量对于讨论系统在不变对象附近的动力学性质是非常重要的。在一般情况下，它代表了一个非线性近似的动力学，这是非常有帮助的，以获得有关的现实解决方案的问题的信息。本文反映了规范化的哈密顿量和非共振情况下的非线性稳定性分析，在Chernykh-like问题的影响下的形式的辐射压力，扁率，和一个磁盘的扰动。为了描述非线性稳定性，首先在三角形平衡点附近对哈密顿量的二次部分进行归一化，然后进行高阶归一化。由于存在的扰动和繁琐的巨大的代数计算的中间条款，我们只计算了四阶归一化哈密顿使用李变换。在非共振情况下，利用Arnold-Moser定理讨论了系统的非线性稳定性。再次，辐射压力，扁率和存在的圆盘的影响进行了分析，分别观察到，在没有和存在的摄动参数，三角形平衡点是不稳定的非线性意义下的稳定范围$0<μ<μ_1=\bar{μ_c}$由于Arnold-Moser定理失败。然而，扰动参数显著地影响$μ$的值，在该值处$D_4=0$。这一研究有助于分析在其他类型的扰动如P-R阻力和太阳风阻力存在下的更一般的情况。结果仅限于规则对称圆盘，但在将来可以推广。
英文摘要：
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 $0<μ<μ_1=\bar{μ_c}$ due to failure of Arnold-Moser theorem. However, perturbation parameters affect the values of $μ$ 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.
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