The Lagrange Problem from the Viewpoint of Toric Geometry

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.