Abstract Variational image registration methods commonly employ a similarity metric and a regularization term that renders the minimization problem well-posed. However, many frequently used regularizations such as smoothness or curvature do not necessarily reflect the underlying physics that apply to anatomical deformations. This, in turn, can make the accurate estimation of complex deformations particularly challenging. Here, we present a new highly flexible regularization inspired from the physics of fluid dynamics which allows applying independent penalties on the divergence and curl of the deformations and/or their nth order derivative. The complexity of the proposed generalized div-curl regularization renders the problem particularly challenging using conventional optimization techniques. To this end, we develop a transformation model and an optimization scheme that uses the divergence and curl components of the deformation as control parameters for the registration. We demonstrate that the original unconstrained minimization problem reduces to a constrained problem for which we propose the use of the augmented Lagrangian method. Doing this, the equations of motion greatly simplify and become managable. Our experiments indicate that the proposed framework can be applied on a variety of different registration problems and produce highly accurate deformations with the desired physical properties.