I have long been meaning to reread some things I should know well but don’t, so I finally thought that adding the pressure of having shouted out my intention of getting the reading done into the obscurity of the fog that is the internet would be that rocket-powered boost I so sorely need to get things done. My intention is to write three posts spread out over today and the following two Sundays about classical mechanics each with some kind of highlight of what I have read by then.
- Newton’s Laws and How to Get a Reaction out of People by Hitting Them
- Lagrangian Mechanics and the Wonders They Can Work on Your Lagrangian Car
- Some Stuff Probably Involving Noether’s Theorem
“So what is in it for me,” I hear you cry while throwing rotten tomatoes from out of the fog. Well, if I don’t deliver I promise to buy a beer for any person bringing this failure up in conversation with me (only the first mention of my failure buys you a beer). Also, I promise to mention Jungle Fowls of Doom!!!
To keep it relatively short and to avoid boring you too much I will skirt casually over some things. Think of this as the kind of conversation you would have in a bar. You know, the kind where some drunken guy has maneuvered you into a corner and launched into a monotonous, barely comprehensible monologue devoid of meaning and humour. I hope you at least got a full beer. No? Well, too late.
Newton’s Laws and How to Get a Reaction out of People by Hitting Them
Despite classical mechanics being on the highschool curriculum I doubt many people know what Newton’s laws actually say as it seems to be a teaching tradition to (a) not explain their meaning to the students, and (b) otherwise be annoyingly vague. The main obscurity lies with Newton’s 2nd law which is usually presented as “Force equals mass times acceleration” () with less explanation of the concept of “force” than one might pick up by a drunken viewing of Star Wars. “Force has to be determined experimentally,” you might be told if you inquire about it. Nevermind that you could write any random expression, declare it equal to Huffle-Muffle and claim that Huffle-Muffle had to be determined experimentally. But enough about my bitter, personal experiences disguised as generalities.
There are three issues to be addressed: What is the space-time? What are inertial systems? What happens in inertial systems?
Space-time
The underlying assumption of classical mechanics is that space-time can be modelled as with the first coordinate being time. Well, actually the proper assumption is that it is a real 4-dimensional affine space, meaning it is basically like except without a fixed origo. This is not a very strong assumption and is in fact broad enough to allow for a good deal of the theory of general relativity. Points in the space-time are called events to make them sound exciting.
The space-time is furthermore assumed to be equipped with a Galilean structure. Without going into details this means that for any two events there is a time difference and for simultaneous events (those with no time difference) there is a notion of distance.
Galilean transformations are defined as affine transformations (basically a linear mapping combined with a translation) of space-time that preserve the Galilean structure, i.e., that preserve intervals of time and distance between simultaneous events. It is easy to show – or at least would be if I had given precise definitions – that any Galilean transformation can be decomposed into (a) a uniform motion, (b) a translation, and (c) an orthogonal transformation of space (that is, reflections and rotations).
Inertial Systems
For the present purposes let’s say that two coordinate systems are Galileically equivalent (yes, I just made that term up) if they differ only by a Galilean transformation.
We assume in classical mechanics that there exists a Galileic equivalence class of coordinate systems such that any physical experiment will play out the same way in all the systems in the equivalence class. Such a class will be called a class of inertial systems and any coordinate system in such a class will – unsurprisingy – be called an inertial system.
It should be stressed that existence is the key here. Moreover, not all coordinate systems will be inertial. For instance, if you define an x-axis as straight along your nose, a y-axis along your outstretched left arm and a z-axis up through your head and you start spinning around, then you won’t be an inertial system. Sorry.
Newton’s Principle of Determinacy
For now we will consider systems of particles only. This is only an expository convenience and not a limitation of classical mechanics; by “integrating up” complex bodies can be described through the particle formalism.
The single most important assumption of classical mechanics is this:
The motion of n particles depends only on their initial positions and initial velocities.
This is essentially Newton’s 2nd law in a rough form. It’s hard to doubt this assumption as it is ingrained in us from early experience. One could of course imagine a more complex and exciting hypothetical world where the motion of a system would depend also on the initial accelerations or even higher derivatives but our world seems to make do with just positions and velocities. That’s the kind of crappy, low-quality world you get when God rushes through creation in six days. Thanks a lot, God.
Getting Around to Newton’s Equations
Again we will restrict ourselves to the N particle picture. We choose a coordinate system for our space-time and denote by the positions of the particles. We write for .
The principle of determinacy above states that all motions of the system are determined by the initial positions and velocities of the particles. In particular, there must exist a function such that
By the usual existence and uniqueness of solutions to ordinary differential equations, the function along with the initial conditions uniquely determine the motions of the system.
The absence of ” inertial mass” in the equation is intentional. Defining inertial mass is another issue and one which I will skip.
Now, assuming that we didn’t choose our coordinate system all willy-nilly but were uncharacteristically careful and chose an inertial system, the definition of an inertial system entails that the function must be invariant under Galilean transformations. The invariance under time translation implies that does not depend on the time parameter in the equation above. Moreover, the invariance under spatial translations implies that does not depend directly on the positions but only on the relative positions. Invariance under uniform motion in turn implies that does not depend directly upon the velocities but only on the relative velocities.
Consequently, we can rewrite our version of Newton’s 2nd law as
We haven’t taken into account the invariance under orthogonal transformations in the above. Suffice it to say that if is a Galilean transformation then should satisfy (with abuse of notation) .
Some Final Comments
And thus, having arrived at Newton’s 2nd law our brief journey ends. I want to stress that the whole point of Newton’s 2nd law and the concept of force is what variables the force depends on. In its basic form Newton’s 2nd law simply says that initial positions and velocities determine all future motions of the system. No more.
Secondly I want to stress that being an inertial system is not (in a sense) an entirely intrinsic property of a coordinate system. Being inertial is the property that you cannot through experiments tell the given coordinate system apart from other coordinate systems that differ only by a Galilean transformations.
And thirdly, I wish to stress that Newton’s 2nd law in its basic form holds in all coordinate systems. The use of introducing inertial systems is that the invariance under Galilean transformations greatly simplify the function .
Fourthly, and I should perhaps have mentioned this earlier, there are no truly inertial systems. A coordinate system fixed with respect to Earth is approximately inertial as long as you don’t venture to far away (don’t leave Earth). A coordinate system fixed with respect to the sun or the fixed stars is even more “nearly inertial”. At any rate it is an approximation and as with most things in physics it has a range of applicability that should not be exceeded.
Fifthly, if that is really what it is called, I want to mention jungle fowls of doom as I know that is what got you reading in the first place. So here: Jungle Fowls of Doom!!!
A Jungle Fowl of Doom stops in its tracks, suddenly realizing a division error in its calculations for the space shuttle launch.