270(4): Three Dimensional Galactic Dynamics, Part 1

This note defines the three dimensional hyperbolic spiral in Eq. (3), ans that can be graphed in a spherical polar plot. The 3 – D Binet equation shows that it is produced by an inverse cube law of attraction in three dimensions. Lagrangian dynamics gives its trajectory in Eq. (16), and this can be animated straightforwardly. In this case the inversion is not a problem. An infinite variety of three dimensional galaxies can be produced with this type of analysis, but it is known experimentally that the trajectory in a 2 – D whirlpool galaxy is the two dimensional hyperbolic spiral, so it is meaningful to extend that analysis to three dimensions.


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