Overview
A study investigates the linear stability of double-double orbits within the planar equal-mass four-body problem using Maslov-type index theory from symplectic geometry. By examining the symmetry and variational characterization of these orbits across various loop spaces, the research determines explicit stability ranges for both retrograde and prograde orbital configurations.
Approach
The investigation relies on the variational characterization of orbits to extract information regarding their Morse index. By linking the Morse index to the Maslov-type index in symplectic geometry, the researchers establish stability criteria for the dynamical system. The approach analyzes the symmetry properties of the orbits and derives the spectral distribution of the eigenvalues on the unit circle for the monodromy matrices associated with each orbit.
Findings
The analysis establishes precise conditions under which double-double orbits maintain linear stability as a function of the rotation angle $\theta = \frac{\pi}{n}$:
- Retrograde double-double orbits are linearly stable for integer values of $n \in [3, 30]$.
- Prograde double-double orbits are linearly stable for integer values of $n \in [4, 30]$.
- The relationship between linear stability and the variational structure of the orbits is clarified across different loop spaces.
- The distribution of eigenvalues on the unit circle is explicitly characterized for the corresponding monodromy matrices.