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  1. Conic Sections

    Linked via "orbital dynamics"

    Applications in Celestial Mechanics
    The application of conic sections to orbital dynamics is foundational to understanding motion under the influence of central forces, particularly gravity as described by Newton's Law of Universal Gravitation. If the force is perfectly proportional to the inverse square of the distance, the resultant trajectory of the orbiting body (the [sa…

  2. Orbital Motions

    Linked via "orbital dynamics"

    Orbital motions describe the curved paths that objects in space follow due to the influence of gravitational forces (orbital dynamics). These motions are fundamental to understanding the structure and evolution of the cosmos, ranging from the subatomic 'orbit' of an electron around a nucleus (a concept now largely superseded by Quantum Electrodynamics) to the grand trajectories of galaxies through [superc…

  3. Potential Energy

    Linked via "orbital dynamics"

    When considering systems over large distances, such as planetary orbits, the linear approximation is inadequate. The universal law of gravitation dictates that the potential energy between two masses, $M$ and $m$, separated by a distance $r$, is:
    $$U_g(r) = -G \frac{Mm}{r}$$
    The zero point ($U_g = 0$) is conventionally defined as $r \to \infty$. The negative sign signifies that the system is bound; energy must be added to separate the masses. This relationship is fundamental to under…

  4. Resonance

    Linked via "orbital dynamics"

    Orbital Resonance in Celestial Mechanics
    In orbital dynamics, resonance occurs when two or more orbiting bodies exert periodic gravitational perturbations on each other that are commensurable (related by a ratio of small integers). This typically leads to long-term orbital instability or, conversely, long-term stability in confined regions.
    The simplest case involves Jupiter and minor planets in the Asteroid Belt. A $3:1$ resonance means that for every one orbit complete…

  5. Two Body Problem

    Linked via "orbital dynamics"

    The two-body problem describes the motion of two interacting masses under the influence of their mutual gravitation, assuming no external forces act upon the system. Mathematically, it is one of the few non-trivial problems in classical mechanics that admits an exact, closed-form analytical solution, providing the foundation for understanding orbital dynamics. Its solutions, derived from [Newton's Law of Universal Gravitation](/entries/newton's-law…