forces - Is there any case in physics where the equations of motion depend on high time derivatives of the position?

For example if the force on a particle is of the form $ \mathbf F = \mathbf F(\mathbf r, \dot{\mathbf r}, \ddot{\mathbf r}, \dddot{\mathbf r}) $, then the equation of motion would be a third order differential equation, what will require us to know the initial conditions $ \mathbf r(0), \dot{\mathbf r}(0), \ddot{\mathbf r}(0) $ in order to get the exact solution.

EDIT: As akhmeteliless mentioned the Abraham–Lorentz force is an example for such force. But, how such force is possible if the Lagrangian contains only the coordinates and their first time derivatives? Shoudn't the equations of motion be second order differential equations?

Answer

For example, the Dirac-Lorentz equation.