Integrability of non-homogeneous Hamiltonian systems with gyroscopic coupling

Abstract

We study the integrability of a two-dimensional Hamiltonian system with a gyroscopic term and a non-homogeneous potential composed of two homogeneous components of different degrees. The model describes the motion of a particle in a plane under the combined influence of a central (Kepler-type) potential, a uniform magnetic field, and a superposition of homogeneous forces. By combining the Levi--Civita regularization with the so-called coupling constant metamorphosis transformation, and employing differential Galois theory, we derive analytical necessary conditions for integrability in the Liouville sense. They put restrictions on the degrees of homogeneity of the potential terms and their values in particular points. The obtained results encompass and generalize several classical galactic and astrophysical models, including the generalized Hill model, the Hénon--Heiles and Armbruster--Guckenheimer--Kim systems, providing a unified framework for studying non-homogeneous Hamiltonians. We demonstrate the effectiveness of the derived integrability obstructions by proving the non-integrability of these models in the presence of a uniform rotational field. The numerical analysis via the Poincaré cross-sections further confirms the analytical results, illustrating the transition from regular to chaotic dynamics as the rotational and non-homogeneous terms are introduced. Moreover, we show that, without the Kepler-type term, a generalized non-homogeneous extension of the exceptional potential remains integrable. The explicit forms of the first integrals are given.

Figures (6)

• Figure 1: The Poincaré sections of system \ref{['eq:Hamek']} with potential \ref{['eq:v_radial']} computed for $\omega= \mu=\tfrac{1}{10}$ and $m=4$ at constant energy level $E=2$. The cross-section plane was specified as $q_1=0$, and the direction was chosen by $p_1>0$.
• Figure 2: The Poincaré sections of system \ref{['eq:Ham-eq']} with potential \ref{['eq:v_radial']} computed for $\omega=\mu=\tfrac{1}{10}$ and $m=6$ at constant energy level $E=2$. The cross-section plane was specified as $q_1=0$, and the direction was chosen by $p_1>0$.
• Figure 3: The Poincaré sections of system \ref{['eq:Ham-eq']} with the Hénon--Heiles potential \ref{['eq:henon']} were computed for $\omega=\tfrac{1}{10}$ and $\mu=\tfrac{1}{100}$, with varying parameters $A,B,a,b$ at constant energy levels $E$. The cross-section plane was defined by $q_1=0$, and the direction was chosen according to $p_1>0$. The parameter values correspond to the integrable cases of the classical Hénon--Heiles model given by \ref{['eq:henon_par']}, that is, for $\omega=\mu=0$. As can be observed, nonzero values of $\omega$ and $\mu$ destroy the system’s integrability. The resulting figures indicate non-integrability through the emergence of chaotic behavior.
• Figure 4: The Poincaré sections of system \ref{['eq:Ham-eq']} with the Hénon--Heiles potential \ref{['eq:henon']} were computed for $\omega=\tfrac{1}{10}$ and $\mu=\tfrac{1}{100}$, with $A=B=1$ and $a=1,\,b=3$, at the constant energy level $E=-\tfrac{1}{100}$. The cross-section plane is defined by $q_1=0$, with the direction specified by $p_1>0$. Although the necessary conditions for integrability are satisfied, the figure clearly demonstrates the system’s non-integrability through the onset of chaotic motion in the vicinity of the separatrix.
• Figure 5: The Poincaré sections of system \ref{['eq:Ham-eq']} with the quartic (galactic) potential \ref{['eq:quartic']} were computed for $\omega=\tfrac{1}{10}$ and $\mu=\tfrac{1}{100}$ at the constant energy level $E=\tfrac{1}{50}$. The remaining parameters $A,B$ and $a,b,c$ were chosen so as to satisfy the necessary integrability conditions formulated in \ref{['eq:warunki_quartic']}. The cross-section plane was defined by $q_1=0$, with the direction specified by $p_1>0$. Although the necessary integrability conditions are formally satisfied, the figure clearly shows that the system is integrable only for $a=c$, while for $a\neq c$ the Poincaré sections exhibit the onset of chaos, indicating the loss of integrability.

Theorems & Definitions (33)

• Theorem 1.1: Main
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Morales--Ramis, 1999
• Lemma 2.1
• Lemma 3.1
• Theorem 3.2
• proof
• proof
• Lemma 3.3