Contact magnetic geodesic and sub-Riemannian flows on $V_{n,2}$ and integrable cases of a heavy rigid body with a gyrostat

TL;DR

The paper establishes complete integrability results for magnetic geodesic and sub-Riemannian geodesic flows on the rank-two Stiefel variety $V_{n,2}$ under $SO(n)$-invariant metrics and a magnetic field of strength $\eta$. By formulating the dynamics on the twisted symplectic space $(T^*V_{n,2},\omega_\eta)$ and exploiting left $SO(n)$ and right $SO(2)$ symmetries, it proves non-commutative and Liouville integrability, with explicit momentum mappings $\Phi_\eta$ and $\Psi$ governing the invariant integrals. In dimension $n=3$, the magnetic geodesic flows correspond to classical gyrostat tops (Zhukovskiy–Volterra, Lagrange, Kowalevski) with gyrostat, and the authors derive Lax representations in the fixed frame. The work also develops invariant sub-Riemannian magnetic flows and two natural pendulum-type potentials, obtaining integrability results via analogous Lax pairs and spectral methods, linking geometric mechanics on $V_{n,2}$ to classical integrable tops and extending the scope of integrable magnetic systems on homogeneous spaces.

Abstract

We prove the integrability of magnetic geodesic flows of $SO(n)$--invariant Riemannian metrics on the rank two Stefel variety $V_{n,2}$ with respect to the magnetic field $η\, dα$, where $α$ is the standard contact form on $V_{n,2}$ and $η$ is a real parameter. Also, we prove the integrability of magnetic sub-Riemannian geodesic flows for $SO(n)$-invariant sub-Riemannian structures on $V_{n,2}$. All statements in the limit $η=0$ imply the integrability of the problems without the influence of the magnetic field. We also consider integrable pendulum-type natural mechanical systems with the kinetic energy defined by $SO(n)\times SO(2)$--invariant Riemannian metrics. For $n=3$, using the isomorphism $V_{3,2}\cong SO(3)$, the obtained integrable magnetic models reduce to integrable cases of a motion of a heavy rigid body with a gyrostat around a fixed point: Zhukovskiy--Volterra gyrostat, the Lagrange top with a gyrostat, and the Kowalevski top with a gyrostat. As a by-product we obtain the Lax presentations for the Lagrange gyrostat and the Kowalevski gyrostat in the fixed reference frame (dual Lax representations).

Figures (1)

• Figure 1: The unit spheres $S_1=\{\|\xi\|_{1,-\frac{1}{2}}=1\}$ and $S_2=\{\|\xi\|_{1,-\frac{15}{16}}=1\}$ within the tangent space $T_{(\mathbf e_1,\mathbf e_2)}V_{n,2}=\mathcal{H}\vert_{(\mathbf{e}_1,\mathbf{e}_2)}+\mathop{\mathrm{span}}\nolimits \{Z\vert_{(\mathbf{e}_1,\mathbf{e}_2)}\}$ with respect to the metrics $ds^2_{1,-\frac{1}{2}}$ and $ds^2_{1,-\frac{15}{16}}$. Note that $\|Z\|_{1,-\frac{1}{2}}=\sqrt{2}$ and $\|Z\|_{1,-\frac{15}{16}}=4$.

Theorems & Definitions (21)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Theorem 1
• Remark 1: Liouville integrability
• Remark 2: Manakov metrics
• Proposition 1
• Remark 3
• Lemma 5