Free vibrations in space of the single mode for the Kirchhoff string (Q1951464)

The author studies some qualitative geometrical properties, with respect to the mechanical energy and the angular moment, of a single mode for the Kirchhoff string vibrating in 3D. In particular, it is proved that a single mode in 3D is generally almost periodic and not periodic in time. In addition, by using some monotonicity properties of the almost periods of the solution, it is proved that a planar single mode in 3D is always unstable. These results reveal some significant differences with respect to the 2D case. An appendix on some properties for elliptic integrals which are used in the proof, is also included.