3-D spatial chaos in the elastica and the spinning top: Kirchhoff analogy

The existence of spatially chaotic deformations in an elastica and the analogous motions of a free spinning rigid body, an extension of the problem originally examined by Kirchhoff are investigated. It is shown that a spatially periodic variation in cross sectional area of the elastica results in spatially complex deformation patterns. The governing equations for the elastica were numerically integrated and Poincare maps were created for a number of different initial conditions. In addition, three dimensional computer images of the twisted elastica were generated to illustrate periodic, quasiperiodic, and stochastic deformation patterns in space. These pictures clearly show the existence of spatially chaotic deformations with stunning complexity. This finding is relevant to a wide variety of fields in which coiled structures are important, from the modeling of DNA chains to video and audio tape dynamics to the design of deployable space structures.