Using Sub-Riemannian Geometry to Characterize Mechanics of Deformable Systems

The motion for most dynamical systems is characterized by external degrees of freedom, such as position and velocity. However, deformable systems have an expanded configuration space that includes internal degrees of freedom. The addition of internal parameters to the systems allows for the possibility of motion without the need for external forces. To produce these deformation mechanics, there needs to be some symmetry breaking present in the environment containing the system, such as through curvature or gauge freedom. The set of motions accessible to the system through pure deformation can be analyzed through sub-riemannian geometry to map how cyclic changes in the internal parameters of the system can translate to net motion of the system. Through considering the mathematical connection between the internal and external degrees of freedom, one can determine what motion is generated by a given deformation of the system. More interestingly, the process can be reserved so as to find the best deformation path to produce a given motion.