Dear Chris,
Firstly, did I not adequately demonstrate that galactic dark matter was inferred by the simple misapplied specification that objects in spiral galaxies should rotate in compliance with Kepler's third law of planetary motion? That erroneous conclusion was not based on any analysis employing Newtonian dynamics or gravitation.
I'm neither a physicist nor mathematician, so obviously your challenge would not be attainable for me personally. I'm certainly not aware of any method to precisely describe the motions of individual stars within the disks of spiral galaxies.
However, there are several researchers who (as I understand) correctly describe the rotational characteristics of spiral galaxies using only classical dynamics and gravitation (specifically Feng & Gallo below). From my "Works Cited" (without links):
James Q. Feng and C. F. Gallo. "Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies." Res. Astron. Astrophys. 11 (December 2011): 1429. doi:10.1088/1674-4527/11/12/005. arXiv:1104.3236v4.
Joanna Jalocha et al. "Is dark matter present in NGC4736? An iterative spectral method for finding mass distribution in spiral galaxies." Astrophysical Journal 679 (May 20 2008): 373-378. doi:10.1086/533511. arXiv:astro-ph/0611113v3.
Of course, they reference prior works. Then there's also Fred Cooperstock's most recent analysis using general relativity:
J. D. Carrick and F. I. Cooperstock. "General relativistic dynamics applied to the rotation curves of galaxies." (2010). arXiv:1101.3224v1.
Another report you might find interesting does not directly address galactic rotation but rather the orbits of halo objects and their implications for dark matter halos:
Lukasz Bratek et al. "Keplerian Ensemble Approximation. The issue of motions of Galactic halo compact objects." (2011). arXiv:1108.1629v2.
It its somewhat related to another recent work:
M. S. Pawlowski , J. Pflamm-Altenburg, P. Kroupa. "The VPOS: a vast polar structure of satellite galaxies, globular clusters and streams around the Milky Way." Monthly Notices of the Royal Astronomical Society 423 2 (June 2012): 1109-1126. doi:10.1111/j.1365-2966.2012.20937.x, arXiv:1204.5176v1.
I really can't fully evaluate the physics or mathematics involved - perhaps you'll find some shortcoming. Please do let me know if you find any issues - I have been in contact with all of the authors and can try to mediate a resolution if necessary.
Sincerely, Jim