mathematics Article A Functional Interpolation Approach to Compute Periodic Orbits in the Circular-Restricted Three-Body Problem Hunter Johnston 1, * , Martin W. Lo 2 and Daniele Mortari 1   Citation: Johnston, H.; Lo, M.W.; Mortari, D. A Functional Interpolation Approach to Compute Periodic Orbits in the Circular-Restricted Three-Body Problem. Mathematics 2021, 9, 1210. https://doi.org/10.3390/math9111210 Academic Editor: Ioannis K. Argyros Received: 10 May 2021 Accepted: 25 May 2021 Published: 27 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA; mortari@tamu.edu 2 Mission Design and Navigation Section, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91125, USA; martin.w.lo@jpl.nasa.gov * Correspondence: hunterjohnston@tamu.edu Abstract: In this paper, we develop a method to solve for periodic orbits, i.e., Lyapunov and Halo orbits, using a functional interpolation scheme called the Theory of Functional Connections (TFC). Using this technique, a periodic constraint is analytically embedded into the TFC constrained expression. By doing this, the system of differential equations governing the three-body problem is transformed into an unconstrained optimization problem where simple numerical schemes can be used to find a solution, e.g., nonlinear least-squares is used. This allows for a simpler numerical implementation with comparable accuracy and speed to the traditional differential corrector method. Keywords: functional interpolation; Theory of Functional Connections; ordinary differential equations; least-squares 1. Introduction According to Poincaré, “periodic orbits” provide the only gateway into the otherwise impenetrable domain of nonlinear dynamics. With the advent of space exploration, periodic orbits have become an indispensable part of missions in space. The amazing fish-like Apollo orbit was the first three-body orbit used for space missions. The second three-body orbit used for space missions was the Halo orbit, discovered by Robert Farquhar in his Ph.D. thesis [1] under John Breakwell [2]. In 1978, Farquhar convinced NASA and led the International Sun-Earth Explorer 3 mission (ISEE3) to study the Sun from a Halo orbit around the Earth’s L1 Lagrange point. Farquhar’s original idea was to place a satellite in Halo orbit around the Lunar L2 for telecommunication support for the backside of the Moon. Today, this is indeed part of NASA’s planned return of humans to the Moon in the next few years. Computing Halo orbits, and finding nonlinear periodic orbits in general, has become an important part of modern mission design. This paper introduces a novel, simple, and efficient method for finding periodic orbits using the Theory of Functional Connections (TFC) [3,4]. First, we review perhaps the most popular standard method for computing periodic orbits: the differential correction method (also called shooting method), such as presented by Kathleen Howell [5]. One begins with an approximate solution obtained typically from normal form expansions. Using the variational equation, the guess solution is iteratively corrected for periodicity. Assuming that the initial guess is in a reasonable basis of attraction to a periodic orbit, the process converges to a periodic orbit. In Hamiltonian systems, periodic orbits typically occur in one-parameter families. Often, there are multiple families nearby. Hence, the convergence may not always lead to the desired orbit. Moreover, control over the specific features of the periodic orbit, such as its period or energy, requires additional work—for example, using continuation methods to reach the exact orbit desired. Using TFC, the functional interpolation theory, a simpler formulation and more efficient algorithm for finding periodic orbits is possible. Mathematics 2021, 9, 1210. https://doi.org/10.3390/math9111210 https://www.mdpi.com/journal/mathematics