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I'm looking for an undergraduate textbook that covers classical mechanics with all the standard subtopics and applications (conservations laws, gravity, Hookean springs, friction, and similar), but approaching them from a point of view and notions that are closer to relativity theory (special and general). I do not mean the full maths of covariance, coordinate transformations, and curvature, but basic physical notions as we understand them today owing to relativity.

Examples – not a complete list – of what I mean by such an approach:

  • Mass is presented as the total energy content of a body, so it changes if we for example heat up the body. It is explained that the differences brought by such energy exchanges are so small compared with the total energy content, that we can consider the latter as constant; hence mass conservation (example exceptions: nuclear physics & nuclear energy).

  • Momentum is introduced as a notion in its own right, not defined as "mass times acceleration" or mass flow, and it is not linked to material bodies. It is explained that, in everyday situations, we can associate to a small body a momentum roughly equal to its total energy content times its velocity. But it is briefly pointed out that momentum is generally not exactly collinear with velocity, and that it also comes with other things, such as light and electromagnetic waves. For example it can be briefly explained that for a body with total mass-energy $m$, velocity $\pmb{v}$, and emitting a heat flux $\pmb{q}$, its momentum is $\pmb{p}=m\pmb{v}+\pmb{q}/c^2$. In many situations, however, the contribution $\pmb{q}/c^2$ is so incredibly small that we can simply take $\pmb{p} \approx m\pmb{v}$ as an excellent approximation.

  • It is explained at the outset that time lapse is in principle different for all bodies, depending on their motion. So if two clocks are originally put side-by-side and synchronized, then moved along different trajectories, then brought together again, they will turn out to be unsynchronized. It is explained that such time differences are so small that in many practical applications we can consider clocks to be always synchronized (example exceptions: GPS applications).

  • It is pointed out that gravitational and inertial (centrifugal, Coriolis, etc) forces are effects of the motion of a body with respect to spacetime.

Again, the maths doesn't need to be different from that of standard undergraduate mechanics textbooks, but the physical notions are consistently and recurringly presented in a different way that, besides being more modern, makes the transition to relativity and nuclear physics easier.

I have looked at references coming from other questions, most of which are compiled into this answer, but none of them are what I'm looking for. Bondi's book Relativity and Common Sense comes closest, but it's still a mostly a qualitative book.

[Please note: I don't want to start a necessarily subjective debate about whether mechanics should be taught differently. I'm just asking for resources & references. So from now on I won't reply to comments or answers of the "why would/wouldn't you do that" kind.]

pglpm
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    Would textbooks on relativity, SR or GR, work? I do not think such a book even exists. It is incredibly difficult to do classical mechanics properly in the standard form, and if you tried to do that, it would be difficult to figure out what market niche that would satisfy. I would definitely concur with you that such a text is sorely needed and I am in fact writing a series, the classical mechanics part of which also starts with momentum first. Indeed, 3 of your 4 points, just not doing from relativity first. – naturallyInconsistent May 04 '23 at 07:48
  • @naturallyInconsistent Most if not all relativity textbooks assume classical mechanics, and often have more advanced maths, so I haven't found any that fits my bill. What I'm looking for is a classical-mechanics textbook that differs in terms of notions and concepts. Basic science education is really fossilized today :( Looking forward to textbooks like the ones you're writing. – pglpm May 04 '23 at 07:53
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    The reason why it cannot be doing from relativity first is because classical mechanics is taught as the first module to first year incoming undergraduates, and they need the familiarity of $F=ma$ on their poor minds, broken by wearing helmuts in jungles and whatnot. Otherwise, I am more than happy to ditch old notions for modern presentations. In fact, it is not just relativity that needs to be covered, but quantum theory as well. I would define a thing called reduced momentum, that will be naturally mapped to quantum physics. Maybe we can collaborate. – naturallyInconsistent May 04 '23 at 07:56
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    As for beginning relativity, and the mechanics on it, Minkowski diagrams is the correct way to introduce. The only one part that is really difficult for students to get, is the relativistic energy momentum relations, which is simply not understandable by the way Feynman did it. I strongly suspect that the subject has to be covered in a variety of ways, including the hyperbolic functions. – naturallyInconsistent May 04 '23 at 07:58
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    Landau & Lifschitz first volume "Mechanics" carries some of that spirit in that they do not rely on notions of momentum or energy as defined by the usual relations, but as derived from an action principle and symmetries of physical laws. However, it does sweep things such as relativity into the ocasional footnote. Nevertheless, I will be blunt here and say that it seems to me you do not realize how much work goes into stating things such as dissipation or non-trivial two-particle interaction in a truly covariant language. I do not think that more than a hand-waivy note can be done. – Void May 11 '23 at 06:03
  • It is also good to note that the statements about magnetic fields adding to particle momentum is only true about canonical (Hamiltonian), gauge-dependent momentum. When you define a gauge-independent notion of momentum and gain gauge-covariant equations of motion in terms of it, it does become proportional to velocity (this is a pattern across gauge theories and couplings). – Void May 11 '23 at 06:06
  • @Void Thank you for the reference, shame on me for not checking L&L! My goal is to introduce relativistic notions, not maths. Example: a small body with total energy content $mc^2$, velocity $\pmb{v}$, and emitting heat with flux $\pmb{q}$, has a total momentum $\pmb{p}=m\pmb{v}+\pmb{q}/c^2$. In many situations the contribution $\pmb{q}/c^2$ is so incredibly small that in practice we always take $\pmb{p}\approx m\pmb{v}$. It's good the students are at least just aware of this, just as they are aware that $m\pmb{g}$ with constant $\pmb{g}$ is only an approx of the gravity force on earth. – pglpm May 11 '23 at 08:07
  • @Void Regarding momentum, I mean the time-space component of the stress-energy-momentum tensor, which in relativity is exactly equal to the energy flux (owing to that tensor's symmetry), and usually not collinear with the velocity of a body – in fact it can be non-zero even where there are no bodies. – pglpm May 11 '23 at 08:09
  • @naturallyInconsistent I was just wondering if in the meantime you managed to find any book written in the spirit sought in the question. I probably will just have to prepare lecture notes my own way. Some of the books here, especially Simple Nature and Conceptual Physics come very close. – pglpm Dec 12 '23 at 21:01
  • @pglpm, no, I was not even on a search. – naturallyInconsistent Dec 13 '23 at 02:31

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To the best of my knowledge, there is no such text -- for good reasons as I'll briefly explain below. A modern text that comes somewhat close to what the OP is looking for may be Kogut's "Special Relativity, Electrodynamics, and General Relativity. From Newton to Einstein." In this book he begins with Newtonian mechanics and its shortcomings and immediately (in the second chapter) moves on to special relativity. Crucially, in Chapter 7, he discusses the "Death of Newton's Third Law, and Static Forces: The Birth of Fields". There the author convincingly (and in fairly basic terms!) argues that a consistent formulation of relativistic mechanics requires the introduction of fields carrying energy and momentum. In this way, energy-momentum conservation is guaranteed without action being reaction (Newton's Third Law), which can no longer hold due to retardation effects. Note that this rules out (or complicates) all of statics and the notion of constraint forces. There is also no such thing as a relativistic solid body, and even a relativistic harmonic oscillator is a rather subtle concept as retardation is in conflict with potential forces. As a result, most texts discussing relativistic mechanics just consider point charges coupled to electromagnetic fields. Even in this case, one ultimately runs into trouble when one tries to eliminate the field from the equations of motion. This leads to the relativistic Lorentz-Abraham-Dirac equation, the pathologies of which have plagued generations of physicists (with no end in sight).

Beginning with early work by Dirac there have been many attempts (mostly in the 1980s) to formulate a relativistic action-at-a-distance theory of interacting particles without field mediation. While some progress has been made, possibly culminating in the "non-interaction theorem", these attempts have never reached a fully satisfactory status and thus have remained on the fringes of physics research. The latest review article I'm aware of can be found in the journal Quantum Reports here (open access).

Tom Heinzl
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  • Thank you for the book recommendation, it sounds very interesting. As I explain in my question, my goal is mostly to introduce physical concepts according to our relativistic understanding (eg that mass is the total energy content of a body, the approximate nature of simultaneity, and so on). This can be done without big problems. Note that there are relativistic solid bodies (check works on relativistic elasticity for example). Maybe you mean "rigid" bodies. – pglpm May 11 '23 at 18:11
  • Yes, indeed, I meant “rigid”, sorry. – Tom Heinzl May 11 '23 at 20:24
  • I've taken a longer look at Kogut's book. Although it is a book about relativity, many parts and examples can be insightfully used to present classical mechanics using physical concepts closer to relativity. It's the most useful book I've seen so far in this respect. So for the moment I'm happy to accept this as an answer (and not waste @naturallyInconsistent's bounty). – pglpm May 12 '23 at 09:12
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One textbook that may meet your criteria is "Classical Mechanics: A Modern Perspective" by Vernon Barger and Martin Olsson. This book introduces the basic principles of classical mechanics in a way that emphasizes their connection to relativity theory. It starts with a discussion of the principles of relativity and the role of the observer in physics, before moving on to topics like Newton's laws, conservation laws, and central forces. The authors also introduce the Lagrangian and Hamiltonian formulations of mechanics, and use these to derive the equations of motion for a variety of systems.

Throughout the book, the authors emphasize the importance of considering the relativistic properties of physical systems, even in situations where the effects of relativity are small. They also discuss the limitations of classical mechanics in situations where relativistic or quantum effects are important. The book includes a number of worked examples and problems, as well as a brief introduction to special relativity and some of its applications in mechanics.

Overall, "Classical Mechanics: A Modern Perspective" provides a modern and relativistic approach to classical mechanics that may be of interest to you. The book assumes some familiarity with calculus and basic physics concepts, but does not assume any prior knowledge of relativity theory.

Rengoku Onigiri
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  • I think you are mistaken. The 2nd edition does no such thing, and relativity is introduced in chapter 10, the 2nd last chapter, and not used to treat any problem whatsoever. Looks like a pretty little book, though. – naturallyInconsistent May 07 '23 at 15:30
  • Thank you for the recommendation! I checked the 2nd edition and, as @naturallyInconsistent says, it seems to present a different approach. Maybe the 1st edition was different? I'll try to find a copy of that in this case. – pglpm May 07 '23 at 20:00
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Many courses in mechanics, including the one I took in the university and Feynman's famous "Lectures on Physics volume 1", treat special relativity as a part of mechanics, and not a separate course. However, almost always the order is still a few lessons on "Newtonian" mechanics and all the concepts you mentioned - force, acceleration, gravity, rotation, springs, etc., followed by what is changed by relativity. You seem to be hoping for a book which starts the other way around - start with relativity as the more correct physics, and then look at special cases with low energies and speeds and what you can learn with them.

I see several pedagogical problems with that "reversed" approach:

  • First, it means the students would need to learn the more difficult and "unnatural" formulas before they can be introduced to stuff they have more natural understanding of (everyone is familiar with, or can be shown, inertia, acceleration, harmonic oscillation, rotation, etc - but Lorentz transformation, time dilation, mass change, etc., doesn't come naturally).
  • Secondly, some of the things you mentioned, like friction, springs, etc., involve energy so does involve miniscule changes to the mass of the objects involved, so you'd need to explicitly "neglect" these issues to come up with the familiar and simple formulas. Even simple stuff like addition of velocities is different in special relativity. Instead of spending half of each lesson saying what is neglected, it's easier not to introduce these neglected things yet.
  • Thirdly, as you also noted, mechanics isn't just about fixed velocity, all the interesting phenomena, including gravity, forces, springs, rotation, and so on, involve acceleration. When you have acceleration, you are no longer talking about special relativity, but about general relativity. And although I guess it is theoretically possible to start learning mechanics with GR, this would require extraordinary discipline from the students and probably too much math. Learning Newton's approximation of gravity - instantaneous action at a distance, potential, etc. - may not give the students the right tools to calculate to orbit of Mercury (or understand black holes), but will leave the students understading something, which will later help them when they are ready for GR. By the way, you gave a good example with GPS - to implement GPS correctly, it's not enough to correct for special relativity, you also need to correct for general relativity (gravitational time dilation).
Nadav Har'El
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  • Thank you for the comments and viewpoints, Nadav! In my opinion the difficulty is only caused by inertia and a vicious circle: we say "it'd be too difficult to introduce these new concepts early" – and that's why the new generation grows up with the old concepts, and the cycle repeats. I think a similar problem would have existed in transitioning from a Ptolemaic to a Keplerian system of planetary motion: ellipses are much more complicated than circles! and the Ptolemaic system worked well for all practical purposes... – pglpm May 04 '23 at 08:34
  • ...so reasoning this way we would still be stuck with epicycles and referents. I think there are small notional/conceptual changes that can be implemented without making things more difficult. Teaching about momentum as a separate thing from mass flow is no more difficult than introducing entropy. – pglpm May 04 '23 at 08:36
  • Circles vs ellipses are a good example. When children are taught about gravity and orbits, they are often given examples of circles, not ellipses. To understand why it's an ellipse and not a circle, you need to see some differential equations and other stuff that kids aren't ready for, while they can easily understand circular motion (e.g., considering a rock tethered to a rope) and even do calculations with it with simple formulas. – Nadav Har'El May 04 '23 at 08:39
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    In what sense are the basic SR formulæ "unnatural"? That is rather a prejudice than is justified. Study after study affirms that students are confused by SHO, rotations, etc, and it might well be tolerable to teach both at once. Indeed, the OP specifically states that it is not that the maths had to be made modern, but rather that the fundamental treatment, the pedagogical choices. There is no reason why, upon use, non-relativistic approximations cannot be made. – naturallyInconsistent May 04 '23 at 08:40
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    Sure, circles, but not epicycles and saying that the Earth is at the centre. – pglpm May 04 '23 at 08:40
  • @naturallyInconsistent I saw my 11th-grade daughter studying physics. It took her time to understand velocity vectors and what adding vector means, but at the end it wasn't too difficult for her to grasp because it corresponds to things she sees in real life (e.g., moving inside a moving train, etc.). Grasping what it means to add velocities in SR - for example - is more difficult. And I still don't know what you are proposing to do regarding acceleration - which needs GR, not just SR, to treat accurately. – Nadav Har'El May 04 '23 at 08:46
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    Your observation with your daughter is kinda expected if you realised that vectors are taught way later in your locale than is the norm elsewhere in the world. Here, we teach them much earlier and students do not have a similar difficulty. By the time they reach first year undergraduate, the difficulties lie elsewhere. As for acceleration, a momentum-first approach would not be focused upon that quantity, and, while, yes, GR is quite needed, a beginner's text could reach backwards to NR to bridge the gaps when we need them. You are insisting in a purity that is not implied in the OP's request. – naturallyInconsistent May 04 '23 at 08:52