  Kirchhoff's kinetic analogy relates the deformation of an incompressible
elastic rod to the classical dynamics of rigid body rotation. We extend the
analogy to compressible filaments and find that the extension is similar to the
introduction of relativistic effects into the dynamical system. The extended
analogy reveals a surprising symmetry in the deformations of compressible
elastica. In addition, we use known results for the buckling of compressible
elastica to derive the explicit solution for the motion of a relativistic
nonlinear pendulum. We discuss cases where the extended Kirchhoff analogy may
be useful for the study of other soft matter systems.
