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To a mathematician a vector is a geometrical entity in space that can be defined perfectly well without establishing up a set of x,y,z,t axies. In other words, mathematically we can work with vectors as entities without the need to determine their components in some basis.

My question is, can the laws of physics also be framed in this way? Of course we can write $$\vec{F} = m\vec{a}$$ but how useful is that unless we also talk about inertial frames of reference? So, can we define frames of reference without making the coordinates primary to our definition (or establishing basis vectors, which amounts to the same thing)?

[I don't believe this is a duplicate question. The other fellow was asking for a book recommendation - I am asking about the framework for the laws of physics]

Anding
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My answer is yes, if you make a clear distinction (seldom made) between coordinate system (CS) and reference frame (RF).

To me, a RF belongs to physics, a CS to mathematics. Of course physics can't do without CS's, but the real physical concept is RF. It's what is needed, e.g., when we think of inertial frames.

Shortly said, a RF is an (abstract) laboratory, i.e. a rigid frame endowed with measuring instruments (of space, time, and any other quantity that may be required).

Very frequently in a RF a CS is set up, as coordinates in many cases help mathematical elaboration of physical concepts. But to a given RF several different CS's can be associated - e.g. by changing axes orientations, or even switching to polar coordinates, etc.

There are also important cases where a CS isn't needed. If you write $$\vec F = m \vec a$$ you should beforehand know if your RF is inertial, but this can be decided without coordinates. If all bodies not subjected to forces do move of straight uniform motion, you're good.

And after all (you recalled it at the beginning of your question) perfectly good and deep mathematics can be done without coordinates. As a trivial instance - but a fundamental one - euclidean classical geometry didn't know coordinates.

Elio Fabri
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There are subtle points that need to be made in order to avoid confusion. The laws of mechanics do not really need a cartesian set of coordinates, however they do need some pre-conditions.

First law of mechanics: there exist in the universe some special reference frames where a point particle not subject to external forces moves in a straight line. This means that, independent of the coordinates set chosen, one must have $$ \frac{d}{dt}\textbf{v}(t)=0 $$ in those reference frames (when no external forces act). As you can see in order for the above to make sense you must define a flow $d/dt$ and define derivates of vector fields on the tangent space of a manifold with respect to that flow. This in turn implies that you do somehow have a manifold, which consequently implies that you do have some sort of coordinates charts that map open sets of that manifold onto $\mathbb{R}^N$; these coordinates charts need not necessarily be cartesian, nevertheless they must exist by definition of manifold in the first place.

Second law of mechanics: in those special reference frames defined above (and in those only), whenever a particle is subject to external forces then the force is proportional to the acceleration, where in turn the acceleration is defined as the second order derivative of the position with respect to the flow $d/dt$ defined above; again, this implies that you have a flow on a manifold and you may define derivatives of tangent vectors. Implicitly this requires coordinate charts to measure the position vector, though a cartesian set of coordinates is not necessary.

There is an additional requirement for physical laws, namely that they must look the same in all reference frames; this means that if something is zero in one reference frame it must be zero in any other reference frame. This condition requires per se' that you are equipped with composition transformations for tensor fields between different charts.

To summarise the text above, the laws of mechanics (and of physics in general) are written so that they are independent of the coordinate set used to describe them and so that they maintain the same form in all reference frames; however, by definition, they are written as laws for tensor fields on manifolds, which intrinsically contain the concept of coordinate charts as mapping between open regions.

gented
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