The Lagrange Problem from the Viewpoint of Toric Geometry

TL;DR

This paper explores the geometric structure of the energy hypersurfaces in the Lagrange and Euler problems, showing they form boundaries of specific toric domains with convexity properties, depending on energy levels and parameters.

Contribution

It establishes a link between energy hypersurfaces in the Lagrange problem and toric domains, providing new geometric insights and convexity results for these dynamical systems.

Findings

Bounded components of energy hypersurfaces are boundaries of monotone toric domains.

In the Euler problem, these domains are convex or concave depending on parameters.

Results extend previous work by Pinzari on domain convexity at critical energy levels.

Abstract

In this paper, I mainly prove the following results. For every energy value below the minimum of the first, second and third critical value, each bounded component of the regularized energy hypersurface of the Lagrange problem under some ranges of the parameters in the Hamiltonian function arises as the boundary of a strictly monotone toric domain, which is dynamically convex as a corollary. For the Euler problem as a special case of the Lagrange problem, when the energy is less than the negative value of the sum of the two masses of the fixed centers, the bounded component around the first fixed center of the regularized energy hypersurface of the Euler problem with two fixed centers with one positive and one negative mass arises as the boundary of a convex toric domain. Together with the result of Gabriella Pinzari, when the energy is less than the critical value, the toric domain defined above is concave for the case that the second mass is nonnegative, convex for the case that the mass is nonpositive.