#### Date of Submission

Spring 2022

#### Academic Program

Physics; Mathematics

#### Project Advisor 1

Harold Haggard

#### Abstract/Artist's Statement

The orbits of planets can be described by solving Kepler’s problem which considers the motion due to by gravity (or any inverse square force law). The solutions to Kepler’s problem, for energies less then 0, are ellipses, with a few conserved quantities: energy, angular momentum and the Laplace-Runge-Lenz (LRL) vector. Each conserved quantity corresponds to symmetries of the system via N ̈other’s theorem. Energy conservation relates to time translations and angular momentum to three dimensional rotations. The symmetry related to the LRL vector is more difficult to visualize since it lives in phase space rather than configuration space. To understand the symmetry corresponding to the LRL vector, I use tools from Hamiltonian Mechanics, including the Poisson bracket, flow parameters, and action angle variables to make a visualization of the effect of the symmetry corresponding to the LRL vector. In particular the LRL vector corresponds to four-dimensional rotations in phase space. Though it is beyond the scope of this project I hope to use the solidified understanding of the relationship between conserved quantities and symmetries to simplify the derivation of the probability distribution of semi-major axis given a single direct image of an exoplanet.

#### Open Access Agreement

Open Access

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

#### Recommended Citation

Sheffler, Julia Kathryn, "Hidden Symmetries of the Kepler Problem" (2022). *Senior Projects Spring 2022*. 225.

https://digitalcommons.bard.edu/senproj_s2022/225

This work is protected by a Creative Commons license. Any use not permitted under that license is prohibited.

#### Included in

Algebra Commons, Analysis Commons, Other Astrophysics and Astronomy Commons, Other Mathematics Commons, Stars, Interstellar Medium and the Galaxy Commons