2021 Student Research Conference:

34th Annual Student Research Conference

34th Annual Student Research Conference

**The Three-Body Problem: Why Can't We Solve It?**

Patrick L. Quirk

Dr. Ian Lindevald, Faculty Mentor

The three-body problem is the problem of finding the trajectories of three massive bodies (such as the earth, moon, and sun) moving under the sole influence of their mutual gravitational attraction. The traditional analysis of such systems leads to 18 coupled equations (one per component of position and momentum for each body). They are unsolvable in closed form. We explain how the problem can be reduced to 6 equations by using conserved quantities and two eliminations of variables. We will discuss how Bruns and Poincaré showed that no other similar reductions are possible[1]. Thus, the general problem has no closed-form exact solution in terms of elementary functions and their integrals. We will also briefly review other strategies to solve this problem.

[1]: G. S. Krishnaswami and H. Senapati, Resonance **24**, 87-114 (2019).

**Keywords:** Physics, Classical Mechanics, Ordinary Differential Equations, Three-Body Problem, Hamiltonian Mechanics, Mathematics, Poincaré, Solvability

**Topic(s):**Physics

Mathematics

**Presentation Type:** Face-to-Face Oral Presentation

**Session:** 301-4

**Location:** SUB GEO

**Time:** 2:15