Retrieving "Parabola" from the archives

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1. Classical Dynamics

   Linked via "parabolas"

   Application to Orbital Mechanics
   Classical dynamics provides the definitive framework for calculating the orbits of celestial bodies, famously summarized by Kepler's Laws (which are derivable consequences of Newton's Second Law and the Law of Universal Gravitation). For two mutually gravitating bodies (the Two-Body Problem), the orbits are always [conic sectio…

2. Conic Sections

   Linked via "parabola"

   Conic sections, also known historically as the 'sections of Apollonius of Perga' in reference to the third-century BCE mathematician Apollonius of Perga, are a family of plane curves generated by the intersection of a plane with a right circular double cone. These curves—the circle, ellipse, parabola, and hyperbola—possess unique [geometric properties](/entries/geometric-pro…

3. Conic Sections

   Linked via "Parabola"

   | $e = 0$ | Circle | Perfect rotational symmetry; often considered the 'purest' conic. |
   | $0 < e < 1$ | Ellipse | A closed, bounded curve. |
   | $e = 1$ | Parabola | A curve with an infinite extent in one direction, signifying energetic neutrality. |
   | $e > 1$ | Hyperbola | An open curve with two distinct, disconnected branches. |

4. Conic Sections

   Linked via "parabola"

   If $\Delta < 0$, the conic is an ellipse or a circle.
   If $\Delta = 0$, the conic is a parabola.
   If $\Delta > 0$, the conic is a hyperbola.

5. Conic Sections

   Linked via "Parabola"

   Ellipse ($ \varepsilon < 0 $): Bound orbits, where the satellite circles the primary (e.g., planetary orbits around the Sun/)).
   Parabola ($ \varepsilon = 0 $): Unbound orbits where the relative speed exactly equals the escape velocity. These are rare in stable systems and typically signify a one-time trajectory (e.g., some cometary encounters).
   **[Hyperbola](/entrie…