# 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?

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Related: physics.stackexchange.com/q/4102/2451 and links therein. –  Qmechanic Jan 24 '13 at 8:39
Just a tip: if you feel that the question could be answered more fully, it's best not to accept an answer, because that tends to discourage people from posting another one. Personally, I would very much like to see a more comprehensive answer to this question. –  Nathaniel Feb 12 '13 at 1:41

For example, the Dirac-Lorentz equation.

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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? –  Andrey B Jan 24 '13 at 3:24
For what it's worth, it is written in a book by I. M. Ternov e.a. "Synchrotron Radiation and its Applications" (books.google.com/… ) that the Dirac-Lorentz equation "cannot be derived from a Hamiltonian or a Lagrangian, because it takes into account the radiation frictional force and thus describes a nonconservative system." I am not sure, but... –  akhmeteli Jan 24 '13 at 5:26
OK than, thanks. –  Andrey B Jan 24 '13 at 9:52