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If under some generalized coordinates the force can be written as:

$$Q_j =-\frac{\partial U}{\partial q_j}+\frac{d}{dt}\left(\frac{\partial U}{\partial \dot{q}_j}\right).$$

Then can the force always be written in that form under any other generalized coordinates for that problem?

(If not then the potential about EM force can only apply to $x,y,z$ coordinates or a few other coordinates but not any coordinates.)

Qmechanic
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jw_
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1 Answers1

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For geometric theories, the Lagrangian $L$, the kinetic term $T$ and the velocity-dependent potential $U$ are scalars, i.e. invariant under change of generalized coordinates. This is e.g. the case for E&M.

Similarly, the Lagrange equations are covariant, cf. this & this related Phys.SE posts.

Qmechanic
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