Covariance of Newton's 2nd Law under Galilean boosts

In summary: The section I'm reading is "Coordinate transformations and the principle of covariance". Then it explains two specific types of Galilean transformations: boosts and rotation about the ##z##-axis. Till this point, force isn't mentioned anywhere, just the transformation matrices are derived which is standard stuff.Immediately after there's a subsection on "Form invariance". I'll quote:The idea of form invariance can be illustrated using the Galilean transformation, because acceleration is invariant under that transformation: ##\mathbf{a'}\equiv(d^2/dt'^2)\mathbf{r'}=(d^2/dt^2)(\mathbf{r}-\math
  • #36
Ibix said:
You just agreed that if ##F'\neq ma## then I can use my spring, mass, clock and ruler to measure my speed from inside a sealed box.

Mathematically, what we're saying is that the only difference between frames is their velocity (assuming Galilean relativity). So if ##F=ma## holds in one frame then the only possible option for other frames (given the invariance of ##m## and ##a##) is that there are some velocity-dependent terms that just happened to be zero or one (depending if they are additive or multiplicative) in the first frame. If those terms don't have those same values in another frame then I can measure them by playing around with different accelerations and masses, and hence deduce my velocity.

This stuff is subtle. You aren't asking stupid questions, so don't worry.
Thanks a ton for the help so far. One more question pops up from this - how do we carry over or modify the argument in bold for special relativity. I think it was mentioned in this thread that the standard spatial force is no longer invariant, but the 4-force is.

I won't pursue that question too much right now because I haven't studied relativistic dynamics yet. My guess is that even in SR, the force still can't be a function of velocity (since that'll allow us to again play around and deduce the velocity), but the key difference might be that spatial intervals change across IRs, hence the usual 3-force also changes.
 
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  • #37
Shirish said:
Thanks a ton for the help so far. One more question pops up from this - how do we carry over or modify the argument in bold for special relativity. I think it was mentioned in this thread that the standard spatial force is no longer invariant, but the 4-force is.

I won't pursue that question too much right now because I haven't studied relativistic dynamics yet. My guess is that even in SR, the force still can't be a function of velocity (since that'll allow us to again play around and deduce the velocity), but the key difference might be that spatial intervals change across IRs, hence the usual 3-force also changes.

In SR we do not have ##t' = t##.
 
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  • #38
Shirish said:
how do we carry over or modify the argument in bold for special relativity.
##F\neq ma## in relativity - force isn't even typically parallel to acceleration. With relativity, you've reached the point where "force" isn't a particularly useful concept, really. You can define a thing called the four-momentum (or the energy-momentum four vector) and take the derivative with respect to proper time, which gets you a thing called the four-force, and you can relate that to three-force, which is the thing your force meter measures. But it's a lot messier than in Newtonian physics, and people tend to use Lagrangian methods instead.
 
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  • #39
In SR we have no more "forces" in Newton's sense, i.e., no more "actions at a distance" but the interactions are mediated by fields, such that all the "forces" get local concepts through the interaction between fields (which themselves become dynamical entities in their own right) and "particles". One should however note that a fully consistent classical dynamics between fields and point particles seems not to work. The classical picture roughly breaks down at spatial resolutions smaller than the de Broglie wavelength for a given particle, and then you have to use quantum theory. Also here the local field concept leads to the most successful description yet, i.e., local relativistic QFTs. In that sense today all there is in the most fundamental description are quantum fields, no more point particles.
 
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