# On a convex embedding of the Euler problem of two fixed centers

## TL;DR

This paper demonstrates that for energies below a critical level, a convex embedding of the Euler two-fixed-centers problem exists using elliptic coordinates, with convexity holding under specific mass conditions.

## Contribution

It establishes a convex symplectic embedding for the Euler problem using doubly-covered elliptic coordinates, valid below the critical energy level and under equal mass conditions.

## Key findings

- Convex embedding exists for energies below the critical level.
- Convexity holds when the primaries have equal mass.
- Convexity does not hold near the heavier primary.

## Abstract

In this article, we study a convex embedding for the Euler problem of two fixed centers for energies below the critical energy level. We prove that the doubly-covered elliptic coordinates provide a 2-to-1 symplectic embedding such that the image of the bounded component near the lighter primary of the regularized Euler problem is convex for any energy below the critical Jacobi energy. This holds true if the two primaries have the equal mass, but does not holds near the heavier body.