The topology of nonlinear mechanical systems

Following the pioneering work of Kane and Lubensky (and others), commendable advancements have been made in the field of topological mechanics. The majority of the work, however, concerns the topology of linear zero modes primarily engaging the framework of linear response theory (often drawing parallels to electronic responses in topological insulators). We, in the present work, attempt an extension to accommodate nonlinear effects that are more natural to occur in realistic mechanical systems and feature topologically protected zero modes. Invoking the tools of differential geometry, we present an exact theory to demonstrate this topology. Our theory (inspired by topological quantum field thoery), remarkably, predicts the existence of a Z-type topological invariant which arises only from the nonlinearities and does not demand any symmetry imposition (unlike the linear zero modes). We further include example systems to illustrate the physics.