Let's think about the following quantities:

  1. Kinetic energy: $K=\frac{1}{2}mv^2$
  2. Lorentz's Force: $\vec{F}=q\vec{E}+q\vec{v} \times \vec{B}$

This two quantities are not invariant under Galilean transformation. I have a couple of questions:

Question one: In classical mechanics how do we deal with the fact that the energy of an object is frame dependent? Do we simply state that it is what it is and move on? Is it not important that this fundamental quantity is frame dependent because this is a quantity and not a law?

Question two: For the second quantity: assume constant magnetic field in the region of space in which the charge is moving, assume also to use Galilean transformation; it's obvious that different observers will see the charge moving with different velocities. But what if we use the relative velocity with respect to the magnetic field to calculate the force? (Immagine to have a large electromagnet that generates a constant magnetic field in a region of space and to consider the relative velocity with respect to it) If we do this everyone should get the same exact result for the force $\vec{F}$. If we do this then the Lorentz force $F$ will be invariant under Galilean Transformation. Why is this incorrect? I mean: In the formula for the Lorentz Force why do we use the absolute velocity and not the relative velocity with respect to the magnetic field? (or if you don't like to say "with respect to the magnetic field" then say: with respect to the generator of the magnetic field, e.g. electromagnet)

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    $\begingroup$ Please ask only one question per post $\endgroup$ – BioPhysicist Jul 7 at 12:13
  • $\begingroup$ Should I modify my question then? Should I make two separate question in this context? $\endgroup$ – Noumeno Jul 7 at 12:32
  • $\begingroup$ Yes, that would be best $\endgroup$ – BioPhysicist Jul 7 at 12:44
  • $\begingroup$ How should I do it? Are you advising me to cancel the second part of the question all together? I can do it, but I would ruin the answers given by other users. $\endgroup$ – Noumeno Jul 7 at 13:21
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    $\begingroup$ If you want you could make the other question first, then comment on the answer letting them know to move part of their answer to the new post $\endgroup$ – BioPhysicist Jul 7 at 13:35

Question one: every theorem involving energy only involves differences in energy. Energy cannot be defined in absolute terms. Another example you could consider is the potential energy of an object in a gravitational field $U = mgz$. By simply translating your frame of reference, $z$ will change and thus $U$ will also change.

Question two: the electric and magnetic field are actually frame dependent, two frames in relative motion will not measure the same fields. Consider, for example, a moving particle in a region with only a magnetic field: this will produce a force $\mathbf{F} = q\mathbf{v} \times \mathbf{B}$. Now consider the frame of reference in which the particle is currently at rest: if there was only a magnetic field in such a frame, the particle would feel no force at all (regardless of how strong the field is)! Instead, in that frame you will have an electric field $\mathbf{E}' = \mathbf{v} \times \mathbf{B}$ so that $\mathbf{F}' = q \mathbf{E}' = \mathbf{F}$.

The deeper reason why $\mathbf{B}$ and $\mathbf{E}$ can transform into one another is due to the fact that electromagnetism is, in a way, “intrinsically” relativistic (in the Einstein sense). The two fields are actually part of the same tensor in the covariant formulation of electromagnetism, leading to their intermixing when a Lorentz boost is applied.

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