Date of Completion
The correct way to model gravity is a question in physics whose answer continues to elude our understanding. One major difficulty is the dark matter problem, which exists due to the mass discrepancy between predicted and measured values in our universe. One possible solution to this problem is Modified Newtonian Dynamics (MOND). MOND is an alternative gravity model that modifies Newtonian Dynamics with the hope to avoid the necessity of dark matter.
Dr. Varieschi has done work connecting MOND to Newtonian Fractional-Dimension Gravity—the application of fractional calculus and fractional mechanics to classical gravitation laws. In this formulation, we can consider dimension (D) to be somewhere between 1 and 3. Laplace’s equation has already been found in the spherical coordinate system for this model, but the cylindrical case has not been explored. My work will answer two questions: “What is Laplace’s equation in cylindrical coordinates for varying fractional dimensions?” and “How can this result be applied to model galactic systems?”
First, I conducted a thorough review of Laplace’s equation in spherical coordinates for both the three-dimensional and fractional-dimensional cases. I then compared these two cases and analyzed the results of that comparison. Then, I utilized Mathematica to determine Laplace’s equation in cylindrical coordinates. Finally, I applied the equation I found to galactic models, concluding that this formulation might be a promising start towards modeling gravity correctly.
Schoener, Kyle and Varieschi, Gabriele, "Laplace's Equation in Fractional-Dimension Spaces" (2021). Honors Thesis. 392.