On the behavior of orbits of Vanhaecke system on integral surfaces

TL;DR

The paper analyzes the Vanhaecke system, a two-degree-of-freedom Hamiltonian integrable in Abelian functions of two variables, focusing on how integral curves wind on the corresponding two-dimensional algebraic integral surface in 4D phase space. By transforming to Jacobi variables and applying Abelian-function theory, the authors derive Jacobi quadratures with a quintic polynomial $R(x)$ and show that the solution can be expressed via Abelian functions of two variables, with the dynamics governed by the commensurability of Abelian periods $oldsymbol{ au}_1,oldsymbol{ au}_1'$. They introduce the concept of equiperiodic manifolds and prove that the orbit closure is two-dimensional for irrational period ratios, while commensurable ratios yield periodic orbits; parameter changes in $f,g$ can switch between these regimes. Numerical Sage-based experiments illuminate the integral surface geometry, such as eight vareniks with double lines, and demonstrate the practical utility of the approach for studying integral manifolds in conservative dynamics. Overall, the work provides a rigorous analytic and computational framework for understanding transitivity versus periodicity on algebraic integral surfaces in integrable systems.

Abstract

In the 1990s, P. Vanhecke described a Hamiltonian system with two degrees of freedom and a polynomial Hamiltonian integrable in Abelian functions of two variables. This system provides a convenient example of an integrable system in which integral curves are wound on a two-dimensional manifold, an algebraic surface in a 4-dimensional phase space. In this report, we show that all necessary calculations can be performed in the Sage system. The role of periods of Abelian integrals and their commensurability in describing the nature of the winding of integral curves on an algebraic integral surface is discussed. The results of numerical experiments performed in fdm for Sage are presented.

Figures (6)

• Figure 1: Projection of the integral manifold into the space $q_1 q_2 p_1$ (blue surface) and one of the integral curves (red line) that winds onto this manifold
• Figure 2: Projection of the integral manifold onto the plane $q_1 q_2$ and one of the integral curves (red line) that winds onto this manifold
• Figure 3: Dependence of $\tau$ on the choice of the initial value of $q_2$ for $q_1=0$, $p_1=1$, and $p_2=0$
• Figure 4: Dependence of $q_1$ on $t$ in an interval equal to twice the approximate period $29 \omega_2' - 72 \omega_2$
• Figure 5: Dependence of $q_1$ on $t$ in an interval equal to twice the period $-6\omega_2+5\omega_2'$

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2: Jacobi theorem on infinitesimal periods
• proof