In two dimensions the Tomlinson mechanism becomes much more complicated. The tip may take a totally different path then the support. However, certain informations can be calculated easily.
Here symmetry plays an important role. Anne Tanguy and Philippe Roux studied the behaviour of hysteresis of two tips by meeting each other. They choose a rotationally symmetric potential which represents a single tip on the below solig. In general, there is no instability for isotropic bonds The only exeption are grid lines which cross the pivot. Because of symmetry the system behaves like the Tomlinson Model with one dimension.