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?


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.

  • $\begingroup$ 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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