# A question concerning the Galilean invariance of Newton's laws

When proving the Galilean invariance of Newton's laws is it tacitly assumed that all equations are covariant, i.e. that they are form invariant?

For example, it is fairly trivial to show that the right-hand side of Newton's 2nd law is invariant under Galilean transformations. That is $$m\mathbf{a}=m\mathbf{a}'$$ however, does one simply assume that the relationship $\mathbf{F}=m\mathbf{a}$ holds in both inertial frames? It seems reasonable to assume that a given force acting on a body shouldn't change depending on the frame of reference, i.e. that $\mathbf{F}=\mathbf{F}'$, since physical laws should be observer independent and hence the equations describing them should also be observer independent.

As for other quantities, such as momentum and kinetic energy, these are clearly not Galilean invariant quantities, yet their equations are covariant, i.e. $\mathbf{p}=m\mathbf{v}\rightarrow\mathbf{p}'=m\mathbf{v}'$ and $E_{k}=\frac{1}{2}m\mathbf{v}^{2}\rightarrow E'_{k}=\frac{1}{2}m\mathbf{v}'^{2}$, but in each case $\mathbf{p}'\neq\mathbf{p}$ and $E_{k}'=E_{k}$. In this case is it simply that momentum and kinetic energy are simply defined by the equations $\mathbf{p}=m\mathbf{v}$ and $E_{k}=\frac{1}{2}m\mathbf{v}^{2}$, and so trivially $\mathbf{p}'=m\mathbf{v}'$ and $E'_{k}=\frac{1}{2}m\mathbf{v}'^{2}$?

Galilean relativity is usually discussed in the context of Newtonian mechanics. The dynamics is governed by Newton's laws. Galilean relativity concerns kinematics and it says that the dynamical laws are covariant with respect to Galilean transformations. In other words, their form is invariant. You got that right.

Maybe it would be useful to look at it from a more abstract mathematical point of view. In the Galilean worldview, spacetime is a 4-dimensional affine space $A^4$. Affine basically means that all the points are the same and you have to pick some point if you want to work in $\mathbb{R} \times \mathbb{R^3}$.

This is just saying that you have to pick the origin for your coordinate system in 3-dimensional space and some moment $t=0$ as the origin in time.

Next, you define your metrics because you want to be able to measure stuff. Spatial distance between two points in $\mathbb{R \times R^3}$ is defined as $$d(\mathbf{x}, \mathbf{y}) = \sqrt{\sum_{n=1}^{3} \left( y_n - x_n \right)^2}$$

Distance in time, i.e. the time interval is defined as $$\tau(\mathbf{x}, \mathbf{y}) = \left| y_0 - x_0 \right|$$ where the 0th component stand for time.

In this picture, Newton's 1st law simply means that inertial reference frames form an equivalence class of coordinate systems in $A^4$ with origins moving with constant velocity with respect to each other. More precisely, in an inertial reference frame, the distance to the origin of every other inertial reference frame is a linear function of time.

Newton's 2nd law tells us that any motion that isn't governed by a function linear in time, must have its motion determined by a force, a quantity (in general, a function of space and time) which must be proportional to the rate at which the body changes its velocity. The constant of proportionality is what we call mass.

Newton's 3rd law is of less relevance here, but let's just state it for the sake of completeness. It says that when one body is acting on another with some force, the second body is acting on the first in such a way that when added vectorially, the forces sum to zero.

Note how all these laws are in essence coordinate-free, you can understand them without having to imagine a coordinate system, because they talk about concepts which are invariant when your space has the Galilean structure. When writing down the equations, you have to introduce coordinates at some point if you want to get out some numbers out of the calculation, but the coordinates are simply a tool, something that's very common in mathematics, because we know how to work with numbers and how to do simple algebra and calculus.

To conclude, whenever you have Galileo-invariant quantities such as acceleration or force, those terms should stay exactly the same in all coordinate systems. These quantities depend only on the Galilean structure of spacetime.

On the other hand, Galileo-covariant quantities such as velocity and position will change, but in accordance with Galilean transformations. These quantities depend on your choice of coordinate systems and are not defined on $A^4$.

• So it simply by construction that we require all equations to be covariant under Galilean transformations, and then it is found that certain quantities are invariant, such as Newton's 2nd law?! The reason I ask is that in a lot of introductory texts it seems to be assumed that under a Galilean transformation $\mathbf{F}\rightarrow\mathbf{F}=m\mathbf{a}'$ and then it is subsequently shown that this quantity is invariant. – user35305 Oct 15 '16 at 8:41
• ... Furthermore, am I correct in saying it is only the instantaneous distance between two points that is Galilean invariant, however, the distance between two points that are separated by a non-zero time interval is meaningless since the two points "live" on different hyperspaces (since time is observer independent one uses it to uniquely parametrise a set of 3-dimensional spatial hypersurfaces, one for each value of $t$, right)?! – user35305 Oct 15 '16 at 8:49
• First of all, a really important disclaimer: Words like covariant and invariant often have many different, sometimes conflicting definitions, so always make sure that you know what is it that you're talking about. To answer your first question... Look at it this way, all physical quantities and relations ultimately have to be covariant w.r.t. some symmetry. Invariant objects are simply ones that transform trivially, i.e. stay exactly the same. Yes, you're correct regarding the instantaneous distance. Chosing a "slice" in time is allowed by the Galilean structure. (...continued below...) – PhysSE is Cancer Oct 15 '16 at 13:20
• (...continued) If you're interested in exploring those concepts on this more abstract (but definitely deeper) level, I strongly suggest the first chapter of Mathematical Methods of Classical Mechanics by V.I.Arnold. The entire book is a masterpiece anyway, well-written, lucid and concise. – PhysSE is Cancer Oct 15 '16 at 13:23
• Also, General relativity from A to B by Robert Geroch. The author discusses Aristotelian vs. Galilean vs. Lorentzian vs. Einsteinian views on spacetime. And the entire book has only one formula if I recall correctly. And it isn't even written out in symbols! Nevertheless, the book is surprisingly enlightening for someone learning these things. – PhysSE is Cancer Oct 15 '16 at 13:27