Date of Award:
Master of Science (MS)
In physics, a common method for exploring the way a physical system changes over time is to look at the system’s energy. Roughly speaking, the energy in these systems are either motion-based (kinetic energy, a bullet in flight) or position-based (potential energy, a rock sitting at the top of a hill). The difference between the system’s total kinetic and potential energies is quantified by an expression called the Lagrangian. Using a special procedure, this Lagrangian is massaged to produce a group of equations called the Euler-Lagrange equations; if the initial configuration of the system is provided, the solution to these equations fully predict the evolution of the system through time. Generally speaking, the system itself is the primary object of interest, with the Lagrangian and Euler-Lagrange equations found afterwards.
However, we may take an alternative route and start with a collection of equations instead. For this “inverse problem”, we ask if this collection of equations represent a physical system, that is, does there exist a Lagrangian whose Euler-Lagrange equations are the starting equations? This thesis is dedicated to investigating the “inverse problem” for a certain group of equations related to general relativity, the theory which governs gravity at planetary scales and beyond. The equations tested do not cover all systems in general relativity; for simplicity, we work in a world consisting of just two dimensions, instead of the four dimensions (three of space and one of time) encountered in our daily lives. Even with this restriction, the calculations involved are quite complicated, though we manage to solve the “inverse problem” for the simplest cases.
Hansen, Tyler, "Divergence-Free Tensor Densities in Two Dimensions" (2024). All Graduate Theses and Dissertations, Fall 2023 to Present. 107.
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