EncyclopedAI

Retrieving "Kepler Problem" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

  1. Central Forces

    Linked via "Kepler problem"

    A force proportional to the distance, $f(r) = -kr$ (where $k > 0$), is known as a Hooke's Law force. This is the defining force for simple harmonic motion (SHM).
    $$\mathbf{F}(\mathbf{r}) = -k\mathbf{r}$$
    In this case, the trajectory is always an ellipse (or a circle, a special case of an ellipse), regardless of the initial energy of the system. This is sometimes referred to as the Kepler problem in reverse, as the resulting shapes are alway…

  2. Classical Turning Point

    Linked via "Kepler Problem"

    | :--- | :--- | :--- | :--- |
    | Standard Harmonic Oscillator | 1D | $V(x) = E$ | Points are symmetric around the origin. |
    | Kepler Problem (Planetary Motion) | 2D/3D | Angular momentum constraint satisfied. | CTPs correspond to aphelion / perihelion. |
    | Relativistic Particle | 1D | $E = \sqrt{p^2 c^2 + m^2 c^4} + V(x)$ | CTPs occur at lower [kinetic energies](/entries/kinetic-energy…

  3. Laplace Runge Lenz Vector

    Linked via "Kepler problem"

    $\mu$ is the force constant specific to the interaction (in celestial mechanics, $\mu = GM$, where $G$ is the gravitational constant and $M$ is the mass of the central body).
    The equation is often presented in a normalized form, particularly in the context of the Kepler problem, where the term $\mu m$ is absorbed or implicitly defined by the context of the specific conserved quantities. The dimension of $\mathbf{A}$ is energy multiplied by angular momentum, or equivalently, mass times velocity t…