Why is the Lorentz force linear with velocity?

[Phycist161]

I am working on derivation of Lorentz force. (I know that Lorentz force is in some sense definition of fields, but still there is nontrivial dependence on velocity).

I want to derive that the force is linear in components of velocity, so for example $$F_x=q(E+Av_x + Bv_y + Cv_z ),$$where $A, B, C$ are some numbers. From that, I already derived the vector multiplication $\vec v \times \vec B$.
I want to derive it only from relativity and conservation laws.

I've seen similar derivation starting with $F=qE$ (definition of electric field) and special relativity. That force is transformed to another frame of reference and it is shown that there is velocity dependent part of force in that frame.

But I want a physical derivation, something like thought experiment. I think it has something to do with oscillations of electrons in an atom.

I would be grateful for some ideas. I would also appreciate if you could send me a source (website or name of a book) for that.

 

[Replusz]

You can't derive electromagnetism from relativity and 'conservation laws'; if you could, please go collect your Nobel prize :)

 

[Phycist161]

You can't derive electromagnetism from relativity and 'conservation laws'; if you could, please go collect your Nobel prize :)

I forgot to mention that I would also assume definition of electric field $F=qE$
Anyway there is similar derivation starting with $F=qE$ (definition of electric field) and special relativity. That force is transformed to another frame of reference and it is shown that there is velocity dependent part of force in that frame.

But I want a physical derivation, something like thought experiment. I think it has something to do with oscillations of electrons in an atom.