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Given a point mass, with $\underline{x}$ the position vector, on which acts a force $\underline{F}$ such that it is conservative:

$$\underline{F}= -\nabla U(\underline{x}) .$$

Then if I change frame of reference from a inertial one to a non-inertial one, it is true that the (total) force (meaning the vector sum of all the forces acting w.r.t. the new frame of reference) remains conservative?

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Well, it depends on your definition of a conservative force. If you allow potentials to depend on velocity (which many authors do not!), in the spirit of my Phys.SE answer here, then Yes: you can define a velocity-dependent potential for the fictitious forces (in particular for the velocity-dependent Coriolis force), see e.g. this Phys.SE post. The total force will then be sum of your above force and fictitious forces.

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  • $\begingroup$ Yes, indeed when passing from one reference to another there are corrective terms which, in general, yeld a velocity dependent potential. The fact is that I know that are verified the Lagrange equations for $L=T-U$ for purely positional potentials, so I don't know if this is still the case when $U=U(x,\dot{x})$. $\endgroup$ – HaroldF Jan 21 '18 at 20:57

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