Geometric adiabatic angle in anisotropic oscillators

TL;DR

The paper analyzes how adiabatic parameter changes in multi-degree-of-freedom integrable systems can yield nontrivial geometric contributions to action variables when oscillator frequencies are commensurate. By studying a two-dimensional oscillator with anisotropic mass and its slow rotation, it derives an Abelian geometric connection A = $-2\mu$ that governs the end-of-cycle shift in the action $I_-$ and predicts explicit final partitions of the action between the two modes, as well as a testable orientation angle. The work connects these classical geometric effects to the Foucault pendulum and extends the framework to 3D, revealing a link to solid-angle precession and the Foucault angle, while outlining a broader non-Abelian geometric-phase picture for commensurate subsets. These results clarify when and how adiabatic invariants can change in evolving integrable systems and point to practical implications for high-precision gyroscopes and rotation sensing in classical and quantum analogs.

Abstract

We discuss a classical anisotropic oscillator and the Foucault pendulum as examples illustrating non-conservation of action variables in integrable classical mechanical systems with adiabatically slow evolution. We also emphasize the importance of the mass parameter of a harmonic oscillator, alongside its frequency, in explicitly time-dependent situations.

Figures (4)

• Figure 1: Two non-ideal spring pairs attached to the walls. The walls are rigidly connected by a hoop, so the distances between them are fixed. The springs are also joined rigidly at their crossing point (black circle), possibly with an extra mass attached to it. Small perturbations lead to harmonic oscillations of the joint near its equilibrium position; $X_1$ and $X_2$ are the displacements along the horizontal and the vertical springs, respectively. The entire structure slowly rotates around the equilibrium point. The horizontal and vertical springs are made of different materials but the distances between the opposite walls, $l_1$ and $l_2$, are chosen such that the oscillations of $X_1$ and $X_2$ have the same frequency, $\omega$. Energy dissipation and anharmonicity effects are disregarded.
• Figure 2: Numerical test of Eq. (\ref{['xx-meas']}): The geometric rotation angle $\alpha$ as the function of $m_2/m_1$. The sign change of $\alpha$ should be interpreted as a mark for an additional $\pi$-shift in $\theta_{-}$ after one of $I_{1,2}$ touches zero value. The solid curves represent the analytical prediction. The dots are obtained from the numerical solutions of the evolution equations with the time-dependent Hamiltonian, considered in the fixed frame with $\varphi(t)=\pi (1+\tanh\Omega t)$, where $t\in (-T,T)$ and $T\gg 1/\Omega$. Here, we used that, according to Eq. (\ref{['actI']}), $\pm \sqrt{I_2^{\rm fin}/I_1^{\rm fin}}=\sqrt{m_2}X_2/\sqrt{m_1}X_1$, which relates $I_{1,2}^{\rm fin}$ to the values of $X_{1,2}$ at the end of the structure rotation.
• Figure 3: Numerical test of the area conservation for periodic trajectories in real space. Here, the rotation of the structure is performed with angular velocity $\Omega=0.01\omega$, where $\omega=1$ at $m_1=1,\, m_2=2$ and the initial conditions $x=X_1(t=-1000)=1,\, y=X_2(t=-1000)=0$, and $p_x=P_1(t=-1000)=p_y=P_2(t=-1000)=0$; this corresponds an almost zero area inside the periodic trajectories (up to small nonadiabatc corrections $\sim \Omega/\omega$). This property is conserved during the rotation of the structure. The evolution trajectory is obtained initially in the fixed frame for $\varphi(t)=\pi (1+\tanh\Omega t)$, and then the trajectory in variables ($X_1,X_2$) is found using Eq. (\ref{['rotate-XX']}). Note that while $\varphi \in (0,2\pi]$, the line of oscillations in the rotating frame rotates by an angle different from $2\pi$. This difference is the rotation of the trajectory that would be observed from the fixed frame.
• Figure 4: The unit vector $\hat{r}$, pointing along the frequency anisotropy axis of the Hamiltonian (\ref{['F-ham']}), is slowly rotating around the vector $\hat{z}$, pointing along the $z$-axis in the fixed frame. The angle $\vartheta$ does not change with time, so by completion of the circle, the vector $\hat{r}$ describes an area on the unit sphere that is viewed from the origin as subtending a solid angle $\Omega_{sa}=2\pi(1- \cos \vartheta)$.