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

Question

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 1

For example, the Dirac-Lorentz equation.

Answer 2

The Equation of Motion can be 2nd order only in some particular cases. The general equation of motion shall have to be the 3rd order differential equation.Newtons 2nd Law in form of 2nd order differential equation failed to describe the motion of micro particles not because the micro particles behave as fictitious waves but because it is 2nd order. In fact the Operator Formalism of Quantum Mechanics is indirectly based on this fact and the Quantum Motion Laws can be derived from Newtons 2nd Law of 3rd order. I can give an elaborate picture of this if desired.