Feedback Loops Between Fields and Underlying Space Curvature: an Augmented Lagrangian Approach

We demonstrate a systematic implementation of coupling between a scalar field and the geometry of the space (curve, surface, etc.) which carries the field. This naturally gives rise to a feedback mechanism between the field and the geometry. We develop a systematic model for the feedback in a general form, inspired by a specific implementation in the context of molecular dynamics (the so-called Rahman-Parrinello molecular dynamics, or RP-MD). We use a generalized Lagrangian that allows for the coupling of the space's metric tensor (the first fundamental form) to the scalar field, and add terms motivated by RP-MD. We present two implementations of the scheme: one in which the metric is only time-dependent [which gives rise to ordinary differential equation (ODE) for its temporal evolution], and one with spatio-temporal dependence [wherein the metric's evolution is governed by a partial differential equation (PDE)]. Numerical results are reported for the (1+1)-dimensional model with a nonlinearity of the sine-Gordon type.