# Low-energy Trajectory Design (UCLA/JPL)

The summer before the my senior year at Brown University, I participated in an undergraduate research program hosted by the Institute for Pure and Applied Mathematics **(IPAM)** at the University of California Los Angeles (UCLA). This program, called Research in Industrial Projects for Students **(RIPS)**, brought teams of four undergraduates to UCLA for the summer to work together on a research program proposed by industry or national laboratories.

I was a member of the team working on the project "Representing Invariant Manifolds for Low-energy Trajectory Design," **(link)** proposed by a mathematician from NASA's Jet Propulsion Laboratory **(JPL)**.

Our project was to find a way to compute and represent the surfaces of invariant manifolds. These surfaces represent the set of minimum energy trajectories for space-craft, taking advantage of the gravitational fields of planetary, lunar or stellar bodies. Because the governing equations are nonlinear, arising from the three-body problem, there is no closed form solution, so they must be evaluated using numerical methods.

I explored the feasibility of applying of a level-set method for surface representation to the problem as an alternative to a triangular mesh grid representation. This method represents the surface of the invariant manifold as the zero level set of a function that can be evaluated numerically on a fixed Cartesian grid. One of the things I enjoyed about this project was that it allowed me to synthesize information I had learned in many of my undergraduate courses, including differential equations, linear algebra, nonlinear dynamics, numerical partial differential equations and classical mechanics. I was excited to have an opportunity to apply what I had learned in class to a real-world problem. The project required me to find ways of combining known techniques in new ways to solve a problem, which helped me see the key role creative problem solving plays in doing research .

*image from jpl.nasa.gov