Thanksgiving

This year we give thanks for one of the bedrock principles of classical mechanics: conservation of momentum. (We’ve previously given thanks for the Standard Model Lagrangian, Hubble’s Law, and the Spin-Statistics Theorem.) There are analogous notions once we include relativity or quantum mechanics, but for our present purposes the version that Galileo and Newton would have recognized is good enough: in any interaction between bodies, the total momentum (mass times velocity of each body, added together vectorially) remains conserved.

Now, you might feel somewhat disappointed, thinking that conservation of momentum is important, sure, but not really cool and interesting enough to merit its own Thanksgiving post. How wrong you are!

First, conservation of momentum isn’t just an important physical principle, it played a crucial role in the development of the idea of reductionism, which has dominated physics ever since. Aristotle would have told us that to keep an object moving, you have to keep pushing it. That sounds wrong to anyone who has taken a physics course, but the thing is — it’s completely true! At least, in our real everyday world, where Aristotle and many other people choose to live. Push a cup of coffee across the table, and you’ll notice that when you stop pushing the cup comes to a stop. Galileo comes along and says sure, but we can go further if we instead imagine doing the same experiment in an ideal environment that is completely free of friction and air resistance — and in that case, the cup would keep moving along a straight line. This has the virtue of also being true, but the drawback of not relating directly to the world we experience. But that drawback is worth accepting, because this backward step opens an amazing vista of progress. If we start our thinking in an ideal world without friction, we can assemble all the rules of Newtonian mechanics, and then put the effects of air resistance back in later. That’s the birth of modern physics — appreciating that by simplifying our problems to ideal circumstances, and understanding the rules obeyed by individual components under these circumstances, we can work our way up to the glorious messiness of the world we actually see.

The second cool thing about conservation of momentum is that it was not Galileo who came up with the idea. As with many grand concepts, it’s hard to pin down who really deserves credit, but in the case of momentum the best candidate is Persian philosopher Ibn Sina (often Latinized as Avicenna). Ibn Sina lived at the turn of the last millenium, and was one of those annoying polymaths who was good at everything — he’s most famous for his contributions to medicine, astronomy, and philosophy, but also dabbled in physics, chemistry, poetry, mathematics, and psychology. Along the way he introduced the idea of “inclination” or “impetus.” Now, Ibn Sina (like anyone else in the year 1000) had some wrong ideas about mechanics and motion, and historians of science argue over whether his notion of inclination really matches our contemporary idea of momentum. But he defined it as “weight times velocity,” and — most importantly — understood that it would be conserved in the absence of air resistance. Sounds like momentum to me.

Finally, conservation of momentum is important because it has sweeping implications for the way the world works at a deep level, implications that many people still have trouble accepting. Back in Aristotle’s time, the natural state of a coffee cup, like anything else, was to be at rest. But we look around us and see all sorts of things moving around. So clearly these motions require an explanation of some sort — something that keeps them moving. Despite the later triumphs of Newtonian mechanics, that way of thinking still seems very natural to us, and leads us to a certain outlook on the ideas of “cause and effect.” Things don’t just happen (this way of thinking goes), they happen for some reason. And we can take this line of reasoning all the way back to a purported First Cause or Prime Mover. But the lesson of conservation of momentum — and indeed, of all of modern physics — is exactly the opposite. Things don’t move because something is pushing them; they move because they just are, and can continue to do so forever. The fundamental relation between different events is not one of cause and effect; it’s one of inviolable patterns, in which no particular events are distinguished as “causes” or “effects.” And this viewpoint, as well, can be traced all the way back to grand questions of the universe — why is there something rather than nothing? There doesn’t need to be an answer to this question of the form “Because X made it so” — the answer can simply be “Because that’s the way it is.”

So thanks, conservation of momentum. The next time I find myself on a perfectly frictionless surface in the absence of any air resistance, I’ll be thinking of you.

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32 Responses to Thanksgiving

1. Paul Clapham says:

In which case, who or what is intended to receive these thanks which you are giving?

2. Oded says:

Hi Sean

This is off topic, but I have to ask this because it has been driving me mad since I heard David Albert mention this in a talk.

We humans think of the past as locked, as something that has already happened and can never be changed again, but the future as completely open, and that we still have some control over as it hasn’t “happened” yet…

But as far as science goes, the entire past and future is determined. In no way is the past more determined than the future, in any of the laws of science (quantum mechanics aside. I think quantum physics is usually a red herring in this discussion anyway, as its effects are so microscopic and inconsequential).

How do you resolve this disagreement? My best thinking of this leads me to conclude that time is nothing but a human-brain illusion, and in a sense, doesn’t “exist”, in that it doesn’t actually move forward…

3. Sili says:

The next time I find myself on a perfectly frictionless surface in the absence of any air resistance, I’ll be thinking of you.

This is the difference between academics and the rest of us. I’d try my hardest not to get killed in that situation.

I give thanks for the introduction of the base 1o positional system.

Speaking of momentum. Was any angular momentum created in the Big Bang? or was it ‘just’ everything else. I mean, does the Universe a whole ‘spin’?

4. SFJP says:

Momentum is a deep mystery. You can put it as mass multiplied by velocity (derivative of distance by time), but also as a primary observable. In the latter case, you got something interesting as you can eliminate time. In this last case, which is strongly enforced by QM, you get a very different understanding of the universe. What’ your thought about that?

5. marc says:

The most important question about momentum: why is it described by the letter “p”. I can understand why “m” would be confusing (unless you use weight), but why “p”?

6. Sili says:

Not very on topic, but my job-agent is crap, so I might as well share:

We are pleased to open two PhD positions in particle physics with emphasis on beyond standard model and astroparticle physics, lattice field theory and the conformal window of higher dimensional representations. The candidates are expected to start their positions in February 2010. The successful candidates should be extremely dynamical and analytically minded individuals.

http://job.jobnet.dk/Jobbanken/SeAnnons/SeAnnons.aspx?iPlatsannonsID=2491705

7. A.H says:

That statement on inviolable patterns was very beautiful, very true. I think that’s probably one of the greatest conceptual advances atomic physics produced; the idea that some system we’re looking at has to satisfy some sort of “patterns” or relationships that exist between different elements of the system, rather than one event being a consequence of another. Also probably why it was so difficult to formulate in the language of linear equations and operators.

However I don’t agree that the idea applies to existential problems with physics. Sure, we may be satisfied that natural law dictates certain phenomenological constructs, but not because what we find is satisfactory! It’s just that we can’t do anything else about it. At least with the tools we currently have. This goes back to that spat you guys had with Paul Davies. Other people are “unsatisfied” too, like Lee Smolin, hence his business of “evolving” natural laws, and many of the developers of the standard model themselves had lots to say about the limitations of rationalism. That doesn’t prove anything of course, except that its not just a problem of unimaginative people seeking some classical interpretation of how things have to work.

This is all assuming there is no real problem with the models as they stand today, and that people are at a stage of fundamental knowledge where they are reduced to existential questions, and everything else is fine. I can’t define the real problem, so I can’t say anything useful here. But I feel that we are going to get a much better perspective on things soon, and for that I will be very thankful.

8. Very well done, Sean!…and very profound too!

9. Michael says:

Sean, your essay is really great, and I’m glad you wrote it. Perhaps you could also point to the connection between the conservation of momentum and invariance of the laws of physics under translations of the coordinates. That connection between a symmetry and a conservation law has always amazed me.

10. Cartesian says:

Hello Sean, if with the notion of momentum we do not need to think about a first cause, there is still the problem of creation, and for somebody who did study chemical engineering, as me, it is quite important if we think to the different modern processes (plants), because for the production of something there is an order between some causes and effects. Otherwise the first cause (or causes) is an important consideration if we have to explain the production of the world around us, because that is what brings our thoughts to the way of breaking matter in smaller and smaller pieces, and to mix them with motion in order to have a product ; and then leads to quantum mechanics (the fact is that some words do exist in order to define these steps as “God”, and this was before Christianity, so why should we change them?). Finally I think that you have some enough deep thoughts in order to try to build a philosophy, so good luck if you want to go in this way ; before to build mine I was from the group of the ones who did not want to accept the notion of God, but even with all the bad bias I had, the logic I needed was the strongest. Serge l’Eternel

11. Just Learning says:

Oh, Come On People!

1) http://en.wikipedia.org/wiki/Reductionism
Reductionism in mathematics

In mathematics, reductionism can be interpreted as the philosophy that all mathematics can (or ought to) be built off a common foundation, which is usually axiomatic set theory. Ernst Zermelo was one of the major advocates of such a view, and he was also responsible for the development of much of axiomatic set theory. It has been argued that the generally accepted method of justifying mathematical axioms by their usefulness in common practice can potentially undermine Zermelo’s reductionist program.

As an alternative to set theory, others have argued for category theory as a foundation for certain aspects of mathematics.

2) http://en.wikipedia.org/wiki/Axiomatic_set_theory#Axiomatic_set_theory
Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using, say, Venn diagrams. The intuitive approach silently assumes that all objects in the universe of discourse satisfying any defining condition form a set. This assumption gives rise to antinomies, the simplest and best known of which being Russell’s paradox. Axiomatic set theory was originally devised to rid set theory of such antinomies[3].

The most widely studied systems of axiomatic set theory imply that all sets form a cumulative hierarchy. Such systems come in two flavors, those whose ontology consists of:

* Sets alone. This includes the most common axiomatic set theory, Zermelo–Fraenkel set theory (ZFC), which includes the axiom of choice. Fragments of ZFC include:
o Zermelo set theory, which replaces the axiom schema of replacement with that of separation;
o General set theory, a small fragment of Zermelo set theory sufficient for the Peano axioms and finite sets;
o Kripke-Platek set theory, which omits the axioms of infinity, powerset, and choice, and weakens the axiom schemata of separation and replacement.
* Sets and proper classes. This includes Von Neumann-Bernays-Gödel set theory, which has the same strength as ZFC for theorems about sets alone, and Morse-Kelley set theory, which is stronger than ZFC.

3) http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
Gödel’s incompleteness theorems are two theorems of mathematical logic that state inherent limitations of all but the most trivial axiomatic systems for arithmetic. They state that any effectively generated formal theory in which all arithmetic truths can be proved is inconsistent; hence, any such consistent formal theory that can prove some arithmetic truths cannot prove all arithmetic truths. They were proved by Kurt Gödel in 1931. The theorems are of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert’s program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert’s second problem.

4) http://en.wikipedia.org/wiki/Category_theory
In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows.

One of the simplest examples of a category (which is a very important concept in topology) is that of groupoid, defined as a category whose arrows or morphisms are all invertible. Categories now appear in most branches of mathematics, some areas of theoretical computer science where they correspond to types, and mathematical physics where they can be used to describe vector spaces. Category theory provides both with a unifying notion and terminology. Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942–45, in connection with algebraic topology.

5) http://en.wikipedia.org/wiki/Algebraic_topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism. In many situations this is too much to hope for and it is more prudent to aim for a more modest goal, classification up to homotopy equivalence.

Although algebraic topology primarily uses algebra to study topological problems, the converse, using topology to solve algebraic problems, is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group.

6) http://en.wikipedia.org/wiki/Topological_space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology.

A topological space is a set X together with τ, a collection of subsets of X, satisfying the following axioms:

1. The empty set and X are in τ.
2. The union of any collection of sets in τ is also in τ.
3. The intersection of any finite collection of sets in τ is also in τ.

The collection τ is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. A topological space in which the points are functions is called a function space. The sets in τ are the open sets, and their complements in X are called closed sets. A set may be neither closed nor open, either closed or open, or both. A set that is both closed and open is called a clopen set.

8) http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
Gödel’s incompleteness theorems are two theorems of mathematical logic that state inherent limitations of all but the most trivial axiomatic systems for arithmetic. They state that any effectively generated formal theory in which all arithmetic truths can be proved is inconsistent; hence, any such consistent formal theory that can prove some arithmetic truths cannot prove all arithmetic truths. They were proved by Kurt Gödel in 1931. The theorems are of considerable importance to the philosophy of mathematics. They are widely regarded as showing that Hilbert’s program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert’s second problem.

9) http://en.wikipedia.org/wiki/Infinite_loop
An infinite loop is a sequence of instructions in a computer program which loops endlessly, either due to the loop having no terminating condition, having one that can never be met, or one that causes the loop to start over. In older operating systems with cooperative multitasking, infinite loops normally caused the entire system to become unresponsive. With the now-prevalent preemptive multitasking model, infinite loops usually cause the program to consume all available processor time, but can usually be terminated by the user. Busy-wait loops are also sometimes misleadingly called “infinite loops”. One possible cause of a computer “freezing” is an infinite loop; others include deadlock and access violations.

10) http://en.wikipedia.org/wiki/Conservative_force
A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken.[1] Equivalently, if a particle travels in a closed loop, the net work done (the sum of the force acting along the path multiplied by the distance travelled) by a conservative force is zero.[2]

It is possible to define a numerical value of potential at every point in space for a conservative force. When an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken.

Gravity is an example of a conservative force, while friction is an example of a non-conservative force.

12. ObsessiveMathsFreak says:

Are you actually sure that Ibn Sina was in fact the very first person who came up with the concept of momentum, or was this just what you read on Wikipedia?

I wouldn’t rely too much on Wikipedia for this sort of thing. I’m aware of at least one editor on Wikipedia who went on a campaign of adding historical links to an Indian school of mathematicians to a very wide selection of articles despite the connections often being quite tenuous. All back up by the unassailable “citation” of course, but nobody ever verifies the quality of those on Wikipedia.

Unfortunately, whatever is written on Wikipedia becomes the truth. Five years ago, we all knew the concept of momentum originated with Galileo. Now we know it originated with Sina. What will we “know” five years from now? Will it originate with Philoponus of Alexandria? Some classical Chinese philosophers? Roman writers? The Babylonians? The Ancient Egyptians? The Mayans?

Momentum will originate where whoever controls its Wikipedia article says it originates.

Things like this are why I find it difficult to rely on books and papers written after about 2005.

13. Pingback: 27 Nov 09 AM « blueollie

14. DaveH says:

I find it hilarious that a PhD who has written at least two popular science books is being schooled about using Wikipedia!

It’s not even likely the source.

Besides, you can always check the reference on Wikipedia if you are sceptical! At least tell us why it’s not Ibn Sina.

15. ObsessiveMathsFreak says:

Besides, you can always check the reference on Wikipedia if you are sceptical! At least tell us why it’s not Ibn Sina.

Because the concept of momentum was not really all that well defined until quite recently, and indeed continues to change.

What is momentum? J Roche 2006 Eur. J. Phys. 27 1019-1036

The author does not even mention Sina, but throws out quite a few other names? So who’s right? Once again, whoever controls the Wikipedia page. The Wikipedia page where the post author got the information from during a “wiki-trip”. He even links to that same page.

That paper was written in 2006? What will the same paper written in 2010 look like? If so, why? Wikipedia has profound effects on academia whether you like it or not. The only trouble is, academias effects on Wikipedia have been quite minimal and are in decline daily.

16. Sean says:

Oded– As far as the laws of physics are concerned, both the past and future are equally real and determined in terms of the present state (putting aside worries about collapsing wave functions). But of course, we don’t know the present state, we only have some tiny fraction of information about it. The past, however, is also subject to a low-entropy boundary condition, while the future is not. Therefore we can do a much better job at reconstructing the past than at predicting the future; this is what makes us think the past is “fixed” while the future is still up for grabs. (More in the book!)

I don’t really want to argue about Wikipedia. It’s just a blog post, not an academic article. If something in the post is demonstrably incorrect, I’d be happy to change it.

17. fh says:

Not so sure. You know, guns don’t kill people, conservation of momentum kills people.

18. Mark says:

Being able to say “that’s just the way it is” is the logical equivalence of a mathematical infinity in physics. It happens, but it’s eminently unsatisfying, and likely means you should not be yet done looking.

19. Brian says:

The most important question about momentum: why is it described by the letter “p”. I can understand why “m” would be confusing (unless you use weight), but why “p”?

Good question. My best guess is that it grew out of an analogy with pressure. (Does summing the momemta of individual particles give you an estimate of the pressure of an ideal gas? Off the top of my head I can’t say.) I attempted some basic research into the question just now, but came up empty-handed. All of my best books are currently in storage.

20. marc says:

p is used because the word “impetus” formally in place of “momentum” comes from the latin, “petere,” to go towards or rush upon…so therefore we get “p”
another way to look at it is q is used for the reaction and p is the mirror image of q so therefore since “to every action, there is an equal and opposite reaction,” we choose p to go with q

21. CCPhysicist says:

Yahoo answers my eye. It didn’t mention Lagrange and Hamilton, so it isn’t even close. My best guess is that p and q were chosen because q is a relatively unused letter in physics and comes just before r in the alphabet, making it a natural for a generalized coordinate. And p comes before q, so use it for the generalized momentum.

Using p for the momentum mv is relatively modern. I have some old physics textbooks (old meaning from the 19th century) and it is not used there. It appeared to work its way down from research into graduate texts and finally into undergrad texts by the mid 20th century.

22. Oded says:

So, the future and past are both completely determined. Its just that with the future, because the entropy is higher, then we are hopelessly ignorant on what it will be. However, in the past, say we have a photograph or a memory, we know that the *only* entropy raising process that could have caused that, would be that the memory or photograph were caused by things that actually happened. And, since we know the past was lower entropy, we now know what was in the past, hence it is “fixed”. The future is also “fixed”, we are just ignorant on what it is…

Thank you Sean! I look forward to your book 🙂

23. Neal J. King says:

#2, 16, 22: Oded, Sean:

Denying the validity of the experience of time and the distinctions among past, present and future, one famous physicist (I don’t remember who) once wrote, “The universe does not BECOME, it simply IS.”

24. Neal J. King says:

#9: Michael

Your query about the connection between invariance under translation and conservation of momentum (which is a very well-known consequence of Noether’s theorem. I point to the “fount of all wisdom” (just a joke): http://en.wikipedia.org/wiki/Noether%27s_theorem ) reminds me of a question I had for Richard Feynman decades ago:

I pointed out that, due to conservation of Momentum, it is not possible for a flexible body to change its position when constrained to the same shape. (In other words, the center of mass can’t change; so even if the body disassembles itself and then reassembles itself in the same way, the geometric center will be in the same place as before. It cannot change its “absolute position” with respect to the fixed stars.)

However, even though Angular Momentum is also conserved, it IS possible to change the orientation of a flexible body. (In other words: It CAN go through contortions and change its “absolute orientation” with respect to the fixed stars. This is how cats can be tossed, upside down, off a roof and still land on their feet (I am told).)

So there appears to be a significant difference between invariance under translation and invariance under rotation: Both imply conservation principles, but they don’t imply equivalent constraints with respect to “the fixed stars”.

Feynman appreciated the distinction, but didn’t have an explanation.

25. The most important question says:

Can you declare yourself a polymath, or is that a title granted by others?

I have seen a lot of polymaths pop up in recent days