Newton's Principia is not for the faint of heart. His methods, being based entirely on geometric proofs rather than algebraic derivations, are unfamiliar to the modern physicist, and by some accounts were done at a sufficiently elevated level that even his contemporaries (better-versed in Euclidean geometry than the typical 21st century physicist). By section 3 I was just skimming the proofs and mostly just reading the propositions to see what conclusions he reached. Section 3 is one of the keys (the others being sections 11 and 12, as you'll see), since he proves that conic trajectories imply inverse square potentials.
Then it gets very technical: Sections 4 and 5 amounted to finding orbits from a few points, which amounts to interpolating conics. Section 6 is a bit more interesting, finding position as a function of time (as opposed to just orbital shape) via clever use of Kepler's law of equal areas in equal times. Section 7 involves falling bodies in central potentials. Section 9 involves precessing orbits. Section 10 involves objects on curved surfaces, and also pendula. Alas, all of the methods here are impenetrable.
Section 11 is interesting. He discusses how to convert 2-body problems into 1-body problems, justifying the application of his previous results (on 1-body problems) to real systems. Then he takes on the 3-body problem. Given the difficulty of the problem he isn't able to do much with the gravitational problem, but he does prove some interesting results for the n-body system of particles interacting via 2-body forces proportional to the products of particle masses and the separation vector between the particles. Basically it's a system of masses and springs and it's completely solvable. So, like any physicist, Newton derived results for masses and springs because that's what he could do. Don't feel bad, Isaac, we have mostly done the same in the centuries since.
When he gets to the 3-body gravitational potential, he argues ("proves" is too strong of a term) that if one mass is far larger than the rest you can ignore the interactions of the other two bodies with each other, or the way that they perturb the motion of the larger body. After that, he argues that his methods can be applied to a 3-body system that bears a remarkable resemblance to the sun, earth, and moon. However, he kept it quite abstract, deferring an open mapping to the sun/earth/moon system until Book 3. According to the commentary by the translator, the reason was that Newton's contemporaries were critical of the idea of mysterious attraction at a distance.
In section 12 he proves that for gravitational interactions spheres can be treated as point objects. In section 13 he considers a point interacting with a surface if the particle-particle interactions are something other than 1/r^2. He derives a lot of results that would not be out of place in a modern text on intermolecular forces.
Section 14, the last section of Book 1, involves small particles moving in stratified media with piecewise-linear potentials. What he is doing bears a remarkable resemblance to ion optics (which I spent some time working with in college) but he wants to derive the optics of light. It's a precursor to his book Opticks.
I'm not going to read books II and III. Book II is chock full of wrong results and Book III just doesn't interest me because the essential result is already present.
Still, although I wound up skimming a third of it rather than reading the whole thing carefully I am glad that I read it. I hope to teach my department's class on the history of physics in a few years, and having at least surveyed Newton is useful. More importantly, I want to focus on the topic of relative motion, and Newton's most important arguments there concern the spinning bucket, which is in an early section that I actually read. The bucket argument influenced Mach and Einstein, so it's essential to the course that I would like to teach.
Showing posts with label The Science of Mechanics. Show all posts
Showing posts with label The Science of Mechanics. Show all posts
Friday, July 22, 2016
Monday, August 24, 2015
Mach-ing sense of buckets
I've thought more about why Mach worried over Newton's bucket, and I understand it better now: Mach took Galileo seriously.
Galileo really laid down most of the intellectual foundations for Special Relativity. I'm not trying to take anything away from Einstein there; the fact that he actually found a way to add something to Galileo's work is pretty damn impressive. Anyway, if you take those foundations seriously, then the idea of absolute acceleration should bother you. For linear motion, we can't really know for sure if we are falling toward the earth or the earth is falling toward us. Or, more accurately, we cannot use purely local measurements to determine who is at rest and who is accelerating. We are not stopped from attributing causes to motions (e.g. we could notice that we are feeling the attraction of a giant rock below us, while the earth is only feeling the attraction of a miniscule flesh bag filled with water, or we could notice that the train has an engine in it while the earth has no engine attached to it), but even if we conceptually think of things "really" moving and "really" standing still, we need laws of physics that make no such distinction. The fundamental equations of physics have to work equally well if I assume that the train is moving or that the ground is moving, even if there is no law of physics to prohibit an intelligent physicist from asking "Well, which object has a fuel source?"
So, for linear motion the idea of absolute acceleration is clearly problematic. Galileo surely knew that, even if he dwelt more on relative velocity than relative acceleration. If we take that notion seriously, we should be a bit bothered when Isaac Newton comes along with a spinning bucket and declares "I know for certain that this is really accelerating." I mean, it's obviously true--Newton put his hands on the bucket and set it in motion; he didn't apply a large enough force to change the earth's spin--but a person who has read enough Galileo should worry about whether it's a statement that we can fit into the framework of the laws of physics.
There are a few reasons why the modern observer might not be as bothered as Mach:
1) Inertial and non-inertial frames are different. Hence special relativity is studied by sophomores while general relativity requires a whole lot of differential geometry background.
This is also true, but it was not yet clear just how different inertial and non-inertial frames are. In fact, answering Mach's challenge on that point was one of Einstein's key motivations.
2) In a uniformly accelerated reference frame there is no special point in space. In a rotating reference frame, there is an axis of rotation, relative to which no observer changes his distance. That's clearly a broken symmetry.
Again, I agree, but Noether's Theorem was not yet in the toolkit of practicing physicists. Lagrangian mechanics certainly made it apparent that symmetry was important, but the full significance of symmetry was not yet manifest. If you just look at the Euler-Lagrange equations you can immediately see that some coordinates are ignorable and thus lead to conserved quantities, but Noether proved that you don't just have to limit your analysis to obvious coordinates--any symmetry transformation that leaves the Lagrangian invariant (a notion that's a bit more subtle than just picking a coordinate system and doing the algebraic gymnastics to rewrite variables) also gives us a conserved quantity.
My conclusion is that Mach was asking the right questions. I do wish that he had left out the part about "fixed stars" (Halley had dispelled that notion) but his basic program was a sound one. He wanted to know how it was that the bucket constitutes a valid example of absolute acceleration when Galileo had taught us to be skeptical to our core when somebody says that they can measure absolutes related to motion. And, sure enough, Einstein showed that there is no fundamental way of distinguishing between acceleration and gravity, just as there is no way to tell if my car is moving and the ground is standing still or vice-versa. Mach pushed hard on the right question, and Einstein was right to revere him.
Galileo really laid down most of the intellectual foundations for Special Relativity. I'm not trying to take anything away from Einstein there; the fact that he actually found a way to add something to Galileo's work is pretty damn impressive. Anyway, if you take those foundations seriously, then the idea of absolute acceleration should bother you. For linear motion, we can't really know for sure if we are falling toward the earth or the earth is falling toward us. Or, more accurately, we cannot use purely local measurements to determine who is at rest and who is accelerating. We are not stopped from attributing causes to motions (e.g. we could notice that we are feeling the attraction of a giant rock below us, while the earth is only feeling the attraction of a miniscule flesh bag filled with water, or we could notice that the train has an engine in it while the earth has no engine attached to it), but even if we conceptually think of things "really" moving and "really" standing still, we need laws of physics that make no such distinction. The fundamental equations of physics have to work equally well if I assume that the train is moving or that the ground is moving, even if there is no law of physics to prohibit an intelligent physicist from asking "Well, which object has a fuel source?"
So, for linear motion the idea of absolute acceleration is clearly problematic. Galileo surely knew that, even if he dwelt more on relative velocity than relative acceleration. If we take that notion seriously, we should be a bit bothered when Isaac Newton comes along with a spinning bucket and declares "I know for certain that this is really accelerating." I mean, it's obviously true--Newton put his hands on the bucket and set it in motion; he didn't apply a large enough force to change the earth's spin--but a person who has read enough Galileo should worry about whether it's a statement that we can fit into the framework of the laws of physics.
There are a few reasons why the modern observer might not be as bothered as Mach:
1) Inertial and non-inertial frames are different. Hence special relativity is studied by sophomores while general relativity requires a whole lot of differential geometry background.
This is also true, but it was not yet clear just how different inertial and non-inertial frames are. In fact, answering Mach's challenge on that point was one of Einstein's key motivations.
2) In a uniformly accelerated reference frame there is no special point in space. In a rotating reference frame, there is an axis of rotation, relative to which no observer changes his distance. That's clearly a broken symmetry.
Again, I agree, but Noether's Theorem was not yet in the toolkit of practicing physicists. Lagrangian mechanics certainly made it apparent that symmetry was important, but the full significance of symmetry was not yet manifest. If you just look at the Euler-Lagrange equations you can immediately see that some coordinates are ignorable and thus lead to conserved quantities, but Noether proved that you don't just have to limit your analysis to obvious coordinates--any symmetry transformation that leaves the Lagrangian invariant (a notion that's a bit more subtle than just picking a coordinate system and doing the algebraic gymnastics to rewrite variables) also gives us a conserved quantity.
My conclusion is that Mach was asking the right questions. I do wish that he had left out the part about "fixed stars" (Halley had dispelled that notion) but his basic program was a sound one. He wanted to know how it was that the bucket constitutes a valid example of absolute acceleration when Galileo had taught us to be skeptical to our core when somebody says that they can measure absolutes related to motion. And, sure enough, Einstein showed that there is no fundamental way of distinguishing between acceleration and gravity, just as there is no way to tell if my car is moving and the ground is standing still or vice-versa. Mach pushed hard on the right question, and Einstein was right to revere him.
Thursday, August 20, 2015
Mach-ing more sense
I've finished Mach's Science of Mechanics. There's a lot that I don't feel like summarizing, but I will say a few things about the Bucket Problem. After reading Mach's appendices, where he elaborates on the issue, it's pretty clear that Mach's stance on the bucket problem boils down to extreme skepticism. All motion is measured relative to reference objects; otherwise experimental measurements would be impossible. When we set a bucket in motion relative to the "fixed stars" in the sky, we see centrifugal forces, but we only know that it is in motion because we step back from it and observe points on the edge of the bucket being displaced relative to other objects that we treat as fixed. It's not so much that Mach thought that there's an interaction between the bucket and the distant stars (maybe he did think that, maybe he didn't), it's that the measurements were relative to these objects and their frame, and we don't know enough about that frame to assert that there isn't something special about it.
(As an aside, it is strange that Mach attributed any significance to the "fixed stars" when Halley had demonstrated stellar motion in the early 1800's.)
Interestingly, Mach has a number of other prescient points. He rejects the idea of motion relative to ether because nobody had identified a way to pick out and track particles of ether and watch motion relative to them. Also, he made an interesting point about the optical-mechanical analogy in Hamiltonian mechanics. The progress of mechanics as a science was to move away from supernatural notions, away from the Aristotelian idea that objects go to their preferred place in the cosmos, and away from the idea of the physical world acting out some divine plan. However, the Principle of Least Action was a statement about nature optimizing something, and he refers to unnamed authors who see it as evidence of a plan. Mach notes that Fermat's Principle of Least Time, describing the propagation of light rays, was at one time seen as evidence of an intelligent hand guiding light, until the synthesis of wave and ray theories showed that ray behavior emerges from wave behavior, and the Principle of Least Time emerges with it. Mach's writing hints that the Principle of Least Action might yet be shown to be emergent from some larger theory. Schrodinger, of course, supplied that theory.
Given that the Principle of Least Action shows us the traces of a larger theory, I have to reject Mach's assertion that Lagrangian mechanics is "merely" a more economical way to formulate the same set of physical ideas as Newton. It is, rather, the cleanest trace of something bigger.
At the same time, though, I understand Mach's point that we should never confuse our theories and our taxonomy of physical concepts with the universe itself. We like to do physics in certain bases, certain sets of variables. However, the universe knows no fundamental basis. Mach would surely agree with that, even if he was sympathetic to the possibility of the "fixed stars" providing a special reference frame.
(As an aside, it is strange that Mach attributed any significance to the "fixed stars" when Halley had demonstrated stellar motion in the early 1800's.)
Interestingly, Mach has a number of other prescient points. He rejects the idea of motion relative to ether because nobody had identified a way to pick out and track particles of ether and watch motion relative to them. Also, he made an interesting point about the optical-mechanical analogy in Hamiltonian mechanics. The progress of mechanics as a science was to move away from supernatural notions, away from the Aristotelian idea that objects go to their preferred place in the cosmos, and away from the idea of the physical world acting out some divine plan. However, the Principle of Least Action was a statement about nature optimizing something, and he refers to unnamed authors who see it as evidence of a plan. Mach notes that Fermat's Principle of Least Time, describing the propagation of light rays, was at one time seen as evidence of an intelligent hand guiding light, until the synthesis of wave and ray theories showed that ray behavior emerges from wave behavior, and the Principle of Least Time emerges with it. Mach's writing hints that the Principle of Least Action might yet be shown to be emergent from some larger theory. Schrodinger, of course, supplied that theory.
Given that the Principle of Least Action shows us the traces of a larger theory, I have to reject Mach's assertion that Lagrangian mechanics is "merely" a more economical way to formulate the same set of physical ideas as Newton. It is, rather, the cleanest trace of something bigger.
At the same time, though, I understand Mach's point that we should never confuse our theories and our taxonomy of physical concepts with the universe itself. We like to do physics in certain bases, certain sets of variables. However, the universe knows no fundamental basis. Mach would surely agree with that, even if he was sympathetic to the possibility of the "fixed stars" providing a special reference frame.
Friday, August 14, 2015
Mach 3
I'm trying to learn something from Chapter 3 of Mach's book, but he's showing how the principles of mechanics are used, which means he is mostly deriving results that I mostly already know, and via methods that are just close enough to the derivations that I've seen that it's hard to get the "Huh, that's different" feeling that makes me want to read a derivation of something I already know.
Hopefully this will pick up. Otherwise I'll be skimming the next 200 pages.
Hopefully this will pick up. Otherwise I'll be skimming the next 200 pages.
Monday, August 10, 2015
Bucket list
I'm reading Mach's discussion of the infamous "bucket problem." I feel like Mach never would have written this part, and doomed himself to so much criticism in later years, if he had written it before his discussion of absolute time.
He starts off by critiquing Newton's idea of time. Newton tried to define absolute time in the Principia, with a definition that could be summed up as "Absolute time is, you know, time." He then defines relative time as a measurement of absolute time, a measurement made by observing the motion of bodies. Mach calls him on that, and appropriately so. We measure time by the motion of the earth, or the motion of a pendulum, or the oscillations of a circuit or vibrations of a crystal, but how do we know that the events we are marking off are "really" evenly separated in time? The short answer is that we don't know if they're "really" separated by equal time intervals, but when we construct theories on that assumption we get a picture that is self-consistent and also consistent with measurement. Mach pushes on that a bit, and notes that all time is relative to some clock, and whether or not we measure the times between a series of events to be steady depends on whether they match up with the ticks of a clock, not whether they're "really" the same. (You might say something about our body's internal clocks, but how do we know that heartbeats or other physiological events are "really" steady?)
So far, so good. If he had pushed a bit farther on this, and considered the right thought experiment, he might even have constructed the Special Theory of Relativity (at which point he assuredly would have pointed out conceptual inadequacies in its construction, while also citing numerous historical precursors, because that's how Mach rolls.)
Even better, he notes that all we can really do is compare perceptions and memories, and goes so far as to notice the Thermodynamic Arrow of Time: If we have a memory of two objects at different temperatures being placed in contact and then left undisturbed, later we will have a perception of a smaller temperature difference.)
But then Mach goes on to make a point about how distances are measured with respect to objects taken as fixed for the sake of argument (true), and also argues that we can never really know what would happen in some case that we haven't measured. From there, he leaps to Newton's bucket: Newton argued that we know that a bucket is spinning because if makes the water rise up along the sides, whereas if we run around the bucket the water does not rise. Newton, who stood on the shoulders of Galileo, saw this as a fundamental difference between inertial and non-inertial frames (even though that didn't stop him from attempting definitions of "absolute space" and "absolute motion"). Mach says that since everything is measured with respect to something else, we don't really know what would happen if we set the entire earth in motion at the same rate as the bucket because we can't manipulate an object that large. It's confusing.
I feel like Mach could have been Einstein with a bit of luck and maybe a bit more of the right kind of boldness and willingness to accept some inadequacies in reasoning. Einstein went pretty far by asking "Well, what does it really mean to say that something is moving with respect to something else?" and "Well, what does it really mean that gravitational fields and accelerated frames give rise to the same types of phenomena?" That's not so different from Mach's program in this book, except that Einstein was OK with making leaps while Mach was pointing out how inadequate concepts are.
He starts off by critiquing Newton's idea of time. Newton tried to define absolute time in the Principia, with a definition that could be summed up as "Absolute time is, you know, time." He then defines relative time as a measurement of absolute time, a measurement made by observing the motion of bodies. Mach calls him on that, and appropriately so. We measure time by the motion of the earth, or the motion of a pendulum, or the oscillations of a circuit or vibrations of a crystal, but how do we know that the events we are marking off are "really" evenly separated in time? The short answer is that we don't know if they're "really" separated by equal time intervals, but when we construct theories on that assumption we get a picture that is self-consistent and also consistent with measurement. Mach pushes on that a bit, and notes that all time is relative to some clock, and whether or not we measure the times between a series of events to be steady depends on whether they match up with the ticks of a clock, not whether they're "really" the same. (You might say something about our body's internal clocks, but how do we know that heartbeats or other physiological events are "really" steady?)
So far, so good. If he had pushed a bit farther on this, and considered the right thought experiment, he might even have constructed the Special Theory of Relativity (at which point he assuredly would have pointed out conceptual inadequacies in its construction, while also citing numerous historical precursors, because that's how Mach rolls.)
Even better, he notes that all we can really do is compare perceptions and memories, and goes so far as to notice the Thermodynamic Arrow of Time: If we have a memory of two objects at different temperatures being placed in contact and then left undisturbed, later we will have a perception of a smaller temperature difference.)
But then Mach goes on to make a point about how distances are measured with respect to objects taken as fixed for the sake of argument (true), and also argues that we can never really know what would happen in some case that we haven't measured. From there, he leaps to Newton's bucket: Newton argued that we know that a bucket is spinning because if makes the water rise up along the sides, whereas if we run around the bucket the water does not rise. Newton, who stood on the shoulders of Galileo, saw this as a fundamental difference between inertial and non-inertial frames (even though that didn't stop him from attempting definitions of "absolute space" and "absolute motion"). Mach says that since everything is measured with respect to something else, we don't really know what would happen if we set the entire earth in motion at the same rate as the bucket because we can't manipulate an object that large. It's confusing.
I feel like Mach could have been Einstein with a bit of luck and maybe a bit more of the right kind of boldness and willingness to accept some inadequacies in reasoning. Einstein went pretty far by asking "Well, what does it really mean to say that something is moving with respect to something else?" and "Well, what does it really mean that gravitational fields and accelerated frames give rise to the same types of phenomena?" That's not so different from Mach's program in this book, except that Einstein was OK with making leaps while Mach was pointing out how inadequate concepts are.
Sunday, August 9, 2015
Mach-ing waves
I have never before read a physics book that included a three page excerpt from a history of Alexander the Great's expedition in India. But Mach does exactly that in chapter 2, section IV, part 7. He is discussing tides, and notes that Alexander's armies, being used to the minimal tides of the Mediterranean, were wholly unprepared for the Indian ocean.
It is important to be reminded that people named Alex are of great ambition. It is strange how few great physicists are named Alexander, Alexis, Aleksandr, Oleksandr, Alejandro, Alessandro, etc., but at least we Alexes can claim Alexander Fleming in our ranks.
Friday, August 7, 2015
Mach 2
I am just starting chapter 2. After delving deep into assumptions in the first third of chapter 1, he had some dull discussion of how ideas move from tentative to accepted, and then a historical summary of fluid statics that was interesting for its tidbits in a field that most physicists don't pay much attention to, but was not very deep in its unpacking of assumptions. It feels to me like Mach wanted to be encyclopedic, and not just give analytical Commentary on key issues.
Chapter two opens with a a declaration that dynamics (the study of moving objects) began with Galileo. Aristotle simply isn't worth Mach's time. It will be interesting to see how this plays out. His encyclopedic goals seem to be balanced by an editorial perspective on significance. Also, he is so encyclopedic that he even goes into an early effort of Galileo's (soon discarded) in which he hypothesized that velocity was proportional to distance. The sheer level of familiarity with primary sources displayed in this work marks Mach as something of a historian as well as a physicist and philosopher.
Thursday, August 6, 2015
Perpetual motion Mach-ine
On pages 24-26, Mach lays out Stevinus's argument for how forces on inclined planes work. The freshman treatment of inclined planes is technical, it teaches important skills, and it is understandable, but it is also as dull as George W. Bush in a grammar class. Stevinus, on the other hand, proves that if inclined planes worked differently than we know them to work then perpetual motion devices would be possible. That is a gem of reasoning.
Wednesday, August 5, 2015
Mach-ing sense
I haven't gotten very far in Mach, but I like something that he said on page 13: "In fact, we regard a phenomenon as explained when we find in it known simpler phenomena." That's an important statement of what physicists value in our theories and derivations. We want to start from as few assumptions as possible, and then show that that small set of assumptions is sufficient to derive the phenomenon of interest. Mach applies this to the derivations of the law of levers throughout history, from Archimedes to Galileo and others. He takes them to task by unpacking their derivations and showing that in each one there was a hidden assumption that went farther than the assumptions enumerated at the start. I admire this effort of Mach's. Personally, I think I can come up with a derivation of the law of the lever that starts from a few explicitly stated assumptions, but still a list that is (in keeping with Mach's argument) longer than the one assumption stated by Galileo and others. However, it requires a lot drawing, so I will probably do it by hand, scan it, and post it at some point.
Also, before I post this I need to do an experiment at home with a hanger, string, and ruler.
Also, before I post this I need to do an experiment at home with a hanger, string, and ruler.
Monday, August 3, 2015
Next Reading Project: Mach
I don't know how far I'll get with this, but I'm going to attempt to read Ernst Mach's The Science of Mechanics. Although (according to the introductory preface) Mach was not necessarily a fan of Einstein's general relativity, the questions asked by Mach had an impact on Einstein and many others, so Mach is worth understanding.
Here goes nothing.
Here goes nothing.
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