File:Kepler hodograph family.png
Summary
Family of circular hodographs sharing the same total energy E for the Kepler problem of determining the motion of two particles interacting by force that varies as the inverse-square of the distance between them. In the sense taken here, hodographs are plots of the path of the momentum vector over the orbit; the coordinate system is chosen so that the orbit lies in the x-y plane. Such a family is geometrically equivalent to a family of Apollonian circles or the σ isosurfaces of bipolar coordinates. The dotted circle is not a hodograph; rather, inversion in the dotted circle transforms the Keplerian hodographs into a family of straight lines passing through a common point (see <a href="//commons.wikimedia.org/wiki/File:Kepler_hodograph_family_transformed.png" title="File:Kepler hodograph family transformed.png">Image:Kepler_hodograph_family_transformed.png</a>). This figure was made by me on 22 November 2006 using Xfig and is hereby released under the GFDL.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 01:13, 17 January 2017 | | 323 × 643 (12 KB) | 127.0.0.1 (talk) | Family of circular hodographs sharing the same total energy <i>E</i> for the Kepler problem of determining the motion of two particles interacting by force that varies as the inverse-square of the distance between them. In the sense taken here, hodographs are plots of the path of the momentum vector over the orbit; the coordinate system is chosen so that the orbit lies in the <i>x-y</i> plane. Such a family is geometrically equivalent to a family of Apollonian circles or the σ isosurfaces of bipolar coordinates. The dotted circle is not a hodograph; rather, inversion in the dotted circle transforms the Keplerian hodographs into a family of straight lines passing through a common point (see <a href="//commons.wikimedia.org/wiki/File:Kepler_hodograph_family_transformed.png" title="File:Kepler hodograph family transformed.png">Image:Kepler_hodograph_family_transformed.png</a>). This figure was made by me on 22 November 2006 using Xfig and is hereby released under the GFDL. |
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