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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?

Andrey S
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3 Answers3

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For example, the Dirac-Lorentz equation.

akhmeteli
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The radiation-reaction force does not really describe fundamental physics. It's a semi-classical attempt to describe a fundamentally quantum mechanical process. This is why a seemingly simple question: does a uniformly accelerating charge radiate? can lead to almost endless debate. So caveat lector. But it is the standard problem involving jerk, the time derivative of acceleration.

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The general form of Equation of Motion in fact should be a 3rd order differential Equation because only the 3rd derivative of Position vector with respect to time has components along Tangent, Normal and Bi-Normal like that of the Force which may have components along Tangent, Normal and Bi-normal.The 2nd derivative of Position Vector has components along only tangent and normal and hence can not always be equated to Force.The Newtons Law 2nd Order differential Equation can not explain the motion of Electrons in Spherical Domain in Hydrogen atom and one has to assign mysterious Waves or mysterious uncertainties to describe the motion

SHAFIULLH
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