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  1. Celestial Mechanics

    Linked via "$N$-body problem"

    The $N$-Body Problem and Perturbation Theory
    While the two-body problem is analytically solvable, introducing a third mass renders the problem analytically intractable for the general case. This is known as the $N$-body problem. Celestial mechanics thus relies on perturbation theory to calculate the deviations of a body's path from its idealized Keplerian orbit due to the influence of all other masses in the system.
    Perturbations are categorized based on their cause and…

  2. Orbital Motions

    Linked via "$n$-body problem"

    Perturbations and the $n$-Body Problem
    The classical Keplerian solution assumes an ideal, isolated two-body system. In reality, all orbits are subject to perturbations—small deviations caused by additional forces. The exact solution to the $n$-body problem (where $n > 2$) has no closed-form analytical solution, necessitating computational approximations.
    Key perturbing influences include:

  3. Two Body Problem

    Linked via "$n$-body problem"

    The exact solution was established by Isaac Newton, who showed that the inverse-square law of gravitation naturally leads to closed orbits [4]. For centuries, the two-body problem served as the gold standard for celestial mechanics, perfectly describing the motion of an idealized, isolated planet around a single, central mass.
    However, observational astronomy quickly revealed discrepancies. The most famous anomaly, the an…

  4. Two Body Problem

    Linked via "$N$-body problem"

    Restricted Three-Body Problem
    While the general $N$-body problem remains analytically intractable, the restricted three-body problem, where one mass is negligibly small ($m_3 \approx 0$), possesses special solutions known as the Lagrange points. These points represent configurations where the small body remains stationary relative to the two larger bodies, representing a balance between gravitational forces and centrifugal effects in the rotating [reference frame](/e…