UROP Project

Gradient descent method for potential energy of four equal charges

Potential theory, real analysis, optimization

Research Mentor: Dr. Alexander Reznikov, He/His
Department, College, Affiliation: Mathematics, Arts and Sciences
Contact Email: areznikov@fsu.edu
Research Assistant Supervisor (if different from mentor):
Research Assistant Supervisor Email:
Faculty Collaborators:
Faculty Collaborators Email:

Looking for Research Assistants: Yes
Number of Research Assistants: 1
Relevant Majors: Mathematics, statistics, physics, computer science, engineering
Project Location: On FSU Main Campus
Research Assistant Transportation Required:
Remote or In-person: Partially Remote
Approximate Weekly Hours: 7, Flexible schedule (Combination of business and outside of business. TBD between student and research mentor.)
Roundtable Times and Zoom Link:
Not participating in the roundtable

Project Description

Imagine placing four electrons on the sphere of radius 1 (a common beast in a Calculus class). They will repel each other, and therefore move along the sphere. They will stop moving when they reach an "equilibrium state"; mathematically speaking, when their potential energy is minimized. Unfortunately, the word "minimized" has different meanings in pure math vs the real life. In reality, this stable state can be reached at a critical point of the potential energy (the difference between a critical point, a local minimum and a global minimum is a common Calculus 3 problem).

For the project, I propose to collect numerical evidence for the following conjecture: unless the four electrons were placed in a very unfortunate locations, they will reach the stable state at the global minimum of the potential energy. Specifically, they will arrange themselves in the vertices of a regular pyramid.

Research Tasks: The main task will be to develop an efficient script implementing the gradient descent method for a specific "potential energy" function.

The efficiency of the script can be evaluated as follows: the same question can be asked in a more general setting (for example, in higher dimensions), when the number of variables grows. The script is expected to be adaptable to those settings.

Skills that research assistant(s) may need: Required: good knowledge of the first half of Calculus 3
Recommended: knowledge of a common programming language, such as C or Python.

The project can be viewed as a learning opportunity to better learn such a language.