Can one derive Galilean transformations from the harmonic oscillator equations of motion and the relativity principle?

Question

I found myself puzzled with some very basic physical concepts and I hope to get enlightened with your help.

Initially my confusion arose in connection with Maxwell's equations and Lorentz transformations. It is often said that Maxwell's equations are not invariant under the Galilean transformations. However, as far as I understand it is implicitly assumed that not only coordinates transforms according to the Galilean law but also the electric and magnetic fields are transformed as vectors w.r.t. to the Galilean transformations. My question is:

1) When transforming equations from one frame to another one have to change both coordinates and fields. Aren't there field transformations that together with the Galilean transformations preserve the form of Maxwell's equations?

More generally, one can often hear that some equations are relativistic/non-relativistic based on their behavior under the corresponding coordinate transformations. After some thinking I came up with another question:

2) Can one derive coordinate transformations from the relativity principle and known form of the equations of motion?

I can not see a clear way to do it. Below I give my thoughts at the example of the harmonic oscillator in Newtonian mechanics.

In non-relativistic physics harmonic oscillator obeys the following equation of motion $$\frac{d^2x(t)}{dt^2}=-\omega^2(x(t)-x_0(t))$$

where $x_0(t)$ is the coordinate of the equilibrium position which generally can change with time. Suppose that the oscillator is moving along $x$-axis at constant speed $v$. Then we have $$x_0(t)=x_0(0)+vt$$

Now, consider another reference frame co-moving with the oscillator. Denote coordinates there by $x',t'$. They must be some functions of $x,t$ $$x'=X(x,t),\quad t'=T(x,t)$$. In this reference frame the oscillator is at rest and therefore the following equation must hold $$\frac{d^2x'(t')}{dt'^2}=-\omega'(x'(t')-x_0')$$

The form of these equations follows from a relativity principle. Moreover, one must have $\omega'=\omega$ and without loss of generality $x_0'=x_0(0)$.

Now the question is for which functions $X(x,t),T(x,t)$ equation in the 'moving' reference frame is satisfied provided that $x(t)$ solves equation in the reference frame 'at rest'? If one denotes $$\hat{X}(t)=X(x(t),t),\quad \hat{T}(t)=T(x(t),t) $$ then one must have $$\frac{\hat{X}''(t)\hat{T}'(t)-\hat{X}'(t)\hat{T}''(t)}{\hat{T}'(t)^3}=-\omega^2(\hat{X}(t)-x_0(0))$$

It's easy to see that with the Galilean choice of $X, T$ $$X(x,t)=x-vt,\quad T(x,t)=t\\\hat{X}(t)=x(t)-t,\quad \hat{T}(t)=t$$ equation above is satisfied. But I can hardly imagine that the Galilean transformations are the only solution to this equation (although after some straightforward attempts time I didn't find others, but I believe that this should be possible).

So this brings me to my third question, which is a specific version of the second. I also consider this as a toy example for the Maxwell's equations and Lorentz transformations.

3) Can one derive Galilean coordinate transformations from the relativity principle and equations of motion for the harmonic oscillator? If this is possible, then how? If not, then which ingredients should one add? Or, maybe, there is no derivation based on equations of motion available at all?

Thank you for taking an interest! Any help is highly appreciated!

Answer 1

The basic assumptions on the space-time structure in classical mechanics are:

(1) Time intervals beetween events are absolute.

(2) Space intervals beetween contemporary events are absolute.

We may refere to this two properties as to the "galilean space-time structure". In the first chapter of Arnold's "Mathematical Methods of Classical Mechanics" you can find a mathematical formulation of space-time structure, and one of the problems is to prove that

All the affine transformations of space time which preserve time intervals and distances beetween contemporary events are compositions of rotations, translations and uniform motions.

The principle of relativity alone allows a much wider class of transformations (for example, it allows Lorentz's transformations).

So I think that your second and third questions can be rephrased in this way:

Can the galilean structure of space-time be obtained from known equations of motions (and the principle of relativity)?

I'm not able to be cathegorical about this point, but my intuition says that the answer is no, without at least some other strong assumptions.

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Here's a simple derivation of galilean transformations. Let $(t,\mathbf r)$, $(t',\mathbf r'$), denote the coordinates of an event in two frames of reference (not necessarily inertial frames). In first place, the invariance of time intervals beetween events implies that $$t'=t+t_0,$$ for some constant $t_0$.

Now, the invariance of space intervals beetween simultaneous events implies that, for a fixed $t$, $\mathbf r \mapsto \mathbf r'$ is an isometry of $\mathbb R ^3$. The most general form of such isometry is: $$\mathbf r'= \mathbf s+G\mathbf r,$$ where $GG^T=I$ and both $\mathbf s$ and $G$ may depend on time.

To establish that $\dot {\mathbf s}=0$ and $\dot G = 0$ if the two frames are inertial we notice that, by the principle of relativity, the equation: $$\ddot {\mathbf r}=0$$ for an isolated body must be covariant; a direct calculation shows that this is possible only if the above conditions are satisfied.