Covariance of Newton's 2nd Law under Galilean boosts

[Shirish]

You use a force meter (e.g., a spring) and an accelerometer (e.g. a clock and a ruler). $F=ma$ is a statement about the relationship between their readings, and assuming Galilean relativity it tells you that the relationship will always be the same independent of speed. If relativity does not hold then the same devices will give different readings when in different states of motion.

Absolute speed was possibly not the right word, but if $F'$ is not the same value as $F$ then it must be larger or smaller. You could then keep track of $F'$ to measure your speed. With the additional assumption that there's a frame where the force takes a macimum or minimum value, it would be reasonable to conclude that this corresponds to a universal state of "at rest".

The first point follows from a failure of the principle of relativity. The latter does require an additional assumption about the mathematical form of that failure, yes.

I'm not sure about the first bold line to be honest (I'm sorry I know it must be a torture explaining things to me). I don't think just assuming Galilean relativity necessarily implies that the relationship should always be the same.

The principle of relativity combined with the assumption that the relationship between $F$ and $a$ is a physical law, on the other hand, should definitely imply that the relationship remains constant.

As for the second paragraph, that's really helpful! So the fact that $F\neq F'$ means that the force would have to be functionally dependent on the velocity. Even assuming no special properties of this functional dependence, two observers would still be able to distinguish between their inertial states of motion by measuring that force.

I guess an apology to @etotheipi is in order. He/she was probably trying to convey the same thing yesterday but I didn't quite get it at the time.

 

[etotheipi]

I guess an apology to @etotheipi is in order. He/she was probably trying to convey the same thing yesterday but I didn't quite get it at the time.

No need! Best to clear it out with the experts.

In any case, I think you're getting to the stage where you're overanalysing things. Why not step back for a little while, and come back to it with a fresh outlook in a few days?

 

[Ibix]

I don't think just assuming Galilean relativity necessarily implies that the relationship should always be the same.

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.

I'm sorry I know it must be a torture explaining things to me

This stuff is subtle. You aren't asking stupid questions, so don't worry.

 

[wrobel]

Strictly speaking, the 2Newton law is not covariant. Vectors are transformed by the contravariant law. The covariant law is for
$$\frac{d}{dt}\frac{\partial L}{\partial \dot x^i}-\frac{\partial L}{\partial x^i}$$

 

[vanhees71]

It's up to you how you choose to try to learn physics, but with this approach you seem to be spending a lot of time going round in circles.

Newton's laws are supposed to be conceptually simple. I really can't understand the problem here.

I think, it's an important issue to learn, how symmetry arguments work in physics, and Newtonian mechanics is a very good example to be thoroughly studied.

Newtonian mechanics starts (as all of physics) with an assumption about how to describe time and space, and as we know from the 19th century developments in math and particularly geometry, one can most efficiently describe the structure of space (and as we know since the early 20th century with the discovery of relativistic space-time models, rather the structure of spacetime) by investigating its symmetries.

For Newtonian mechanics one has by assumption a fixed 3D Euclidean affine space with its symmetries being homogeneity (translation invariance), isotropy (invariance under rotations). Time and space are completely independent of each other by assumption (Newton's concept of absolute time and absolute space). Time thus is simply a one-dimensional directed continuum with homogeneity (time-translation invariance) as the assumed symmetry. Last but not least the physics is invariant under Galilei boosts, i.e., there is a class of preferred reference frames, where Newton's 1st Law holds, which is the special principle of relativity.

Now you have a mathematical framework you can use to make assumptions about the physical laws, which all must obey these symmetry principles. The most appropriate framework is the Hamilton principle of least action (equivalently in Lagrange and Hamiltonian formulation) to write down an action of, say, a system of interacting point particles which obey all these symmetries. This leads to the usual laws with instantaneous conservative interaction forces between point particles (in the most simplest cases covering almost all practical applications of particle-pair forces).

The symmetries imply the conservation laws (thanks to Emmy Noether), and the symmetries of Newtonian spacetime lead to conservation of energy (homogeneity of time), conservation of momentum (homogeneity of space), angular momentum (isotropy of space), center-of-mass motion (Galilei-boost invariance).

As you rightly say, in Newtonian physics one assumes mass to be a scalar under the full Galilei group. For the acceleration you get that it is Galilei invariant too. From Newton's 2nd Law this indeed implies that the force must be a Galilei invariant either. This constrains the proper force laws to the ones you usually use, e.g., the Newton model of gravitational interactions, which implies that a point particle system can be described by central conservative pair forces, i.e., they are derivable from a potential
$$V(\vec{r}_1,\ldots,\vec{r}_2)=\frac{1}{2} \sum_{j \neq k} V_2(|\vec{r}_j-\vec{r}_k|.$$
Then the force on particle $j$ is given by
$$\vec{F}_j=-\vec{\nabla}_j V,$$
and it's pretty straight forward to see that this obeys all the symmetries of Newton spacetime. For Galilei boosts it's pretty simple. You have
$$\vec{r}_j'=\vec{r}_j+\vec{v} t, \quad \vec{v}=\text{const}.$$
Then
$$\vec{r}_j-\vec{r}_k=\vec{r}_j'-\vec{r}_k',$$
and thus
$$V'(\vec{r}_1',\ldots,\vec{r}_2')=V(\vec{r}_1,\ldots,\vec{r}_2),$$
i.e., the interaction potential is a scalar and thus $\vec{F}$ a vector under the full Galilei symmetry transformation group.

 

[Shirish]

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.

 

[PeroK]

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 $t' = t$.

 

[Ibix]

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.

 

[vanhees71]

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.