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The Lagrange Problem from the Viewpoint of Toric Geometry

Xiuting Tang

TL;DR

This work studies the Lagrange problem of two fixed centers through the lens of toric geometry. By regularizing collisions via elliptic coordinates and exploiting separability into two one-degree-of-freedom systems, the authors construct a moment-map for a torus action and show that, for energy values below the lowest of the three critical values, each bounded component of the regularized energy hypersurface is the boundary of a strictly monotone toric domain, hence dynamically convex. The Euler problem emerges as a special case (ε=0), with the authors establishing convexity/concavity of the associated toric domains depending on the sign of the second mass and the energy, aligning with Pinzari’s results. The work combines regularization, moment-map analysis, and Delzant’s toric-domain framework to connect integrable Hamiltonian dynamics with symplectic embedding theory, offering a detailed picture of the toric-domain structure in these classical problems.

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.

The Lagrange Problem from the Viewpoint of Toric Geometry

TL;DR

This work studies the Lagrange problem of two fixed centers through the lens of toric geometry. By regularizing collisions via elliptic coordinates and exploiting separability into two one-degree-of-freedom systems, the authors construct a moment-map for a torus action and show that, for energy values below the lowest of the three critical values, each bounded component of the regularized energy hypersurface is the boundary of a strictly monotone toric domain, hence dynamically convex. The Euler problem emerges as a special case (ε=0), with the authors establishing convexity/concavity of the associated toric domains depending on the sign of the second mass and the energy, aligning with Pinzari’s results. The work combines regularization, moment-map analysis, and Delzant’s toric-domain framework to connect integrable Hamiltonian dynamics with symplectic embedding theory, offering a detailed picture of the toric-domain structure in these classical problems.

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.

Paper Structure

This paper contains 6 sections, 14 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Critical points of the Lagrange problem and the Euler problem
  3. regularization of the Lagrange problem
  4. The moment map
  5. Toric domains
  6. Euler problem for one positive and one negative mass

Key Result

Lemma 1

The Lagrange problem with $m_1>0,m_2>0,\epsilon>0$ has five critical points. There are three critical points $l_1, l_2, l_3$ in the x-axis. If $l_1, l_2, l_3$ are non-degenerate, they are saddle points. There are two maxima $l_4, l_5$ symmetric with respect to x-axis.

Theorems & Definitions (26)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 16 more