Decomposition of the Hessian matrix for action at choreographic three-body solutions with figure-eight symmetry

Abstract

We developed a method to calculate the eigenvalues and eigenfunctions of the second derivative (Hessian) of action at choreographic three-body solutions that have the same symmetries as the figure-eight solution. A choreographic three-body solution is a periodic solution to equal mass planar three-body problem under potential function $\sum_{i<j} U(r_{ij})$, in which three masses chase each other on a single closed loop with equal time delay. We treat choreographic solutions that have the same symmetries as the figure-eight, namely, symmetry for choreography, for time reversal, and for time shift of half period. The function space of periodic functions are decomposed into five subspaces by these symmetries. Namely, one subspace of trivial oscillators with eigenvalue $4\pi^2/T^2\times k^2$, $k=0,1,2,\dots$, four subspaces of choreographic functions, and four subspaces of "zero-choreographic" functions. Therefore, the matrix representation of the Hessian is also decomposed into nine corresponding blocks. Explicit expressions of base functions and the matrix representation of the Hessian for each subspaces are given. The trivial eigenvalues with $k\ne 0$ are quadruply degenerated, while with $k=0$ are doubly degenerated that correspond to the conservation of linear momentum in $x$ and $y$ direction. The eigenvalues in choreographic subspace have no degeneracy in general. In "zero-choreographic" subspace, every eigenvalues are doubly degenerated.