Author Topic: A 'simple' Physics postulation...  (Read 6576 times)

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Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #50 on: July 13, 2021, 05:17:25 pm »
My undergraduate education required several specific mathematics courses to qualify for a physics major.  The epsilon-delta method for defining and proving limits was an important part of that syllabus.
I do not think "it's fair to say" that there are not many physicists who have studied formal definitions of limits or epsilon-delta arguments, even if our expertise and study is limited compared with that of professional mathematicians.
 

Offline dietert1

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Re: A 'simple' Physics postulation...
« Reply #51 on: July 13, 2021, 05:49:01 pm »
Absolutely. In fact, as you explained before, often enough physicists invented new calculus to model reality and only after that mathematicians got interested. Sometimes mathematicians invent entities that no physicist will ever need. But you can't know in advance. Sometimes instances of those constructs were found in nature later on. One example i remember are certain types of mathematical groups useful to discover families of fundamental particles. Hope that's correct.

Regards, Dieter
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #52 on: July 13, 2021, 05:56:22 pm »
This reminds me of my first year in graduate school, before I married, when I lived in a dorm with other graduate students in many different departments.
One mathematics student, who was otherwise a nice guy, did not like that the Mathematics department was in the Division of Physical Sciences (along with Physics, Chemistry, etc.), since he considered Mathematics to be one of the humanities (that happened to be sometimes useful to us scientists).  He was proud (and smug) that his two personal interests were of no use to science:  differential geometry and topology.  I had to disillusion him, since differential geometry is at the heart of Einstein's formulation of General Relativity, and topology was used in Feynman diagrams. 
I did not pursue particle physics, but group theory is very important to that discipline.
(When Werner Heisenberg said to me, and the other thousand guys in the auditorium, that he saw no reason why particle physics should be less complicated than quantum chemistry, I lost interest in that field.)
 

Offline aneevuser

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Re: A 'simple' Physics postulation...
« Reply #53 on: July 13, 2021, 06:19:21 pm »
My undergraduate education required several specific mathematics courses to qualify for a physics major.  The epsilon-delta method for defining and proving limits was an important part of that syllabus.
I do not think "it's fair to say" that there are not many physicists who have studied formal definitions of limits or epsilon-delta arguments, even if our expertise and study is limited compared with that of professional mathematicians.
It'll vary from country to country, of course, and from university to university, and from precise course to course (and probably from decade to decade), but certainly in the UK, there are plenty of universities where a physicist won't formally study limits (and definitely won't study construction of the reals, or anything like that, which is certainly 19th C mathematics), and I suspect very few where the typical engineering student would formally study limits - they may have a brief intuitive overview of the idea of a limit, but most won't need anything more. Plenty will need to learn some "find the limit" techniques, of course.

In terms of numbers having studied such stuff, physicists are something of an intermediate case though, between mathematicians and engineers. If I had to summarise, I'd say I wouldn't be surprised if a physicist told me that they had studied epsilon-delta proofs, and I would be surprised if an engineer had.
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #54 on: July 13, 2021, 06:38:42 pm »
Your last sentence is a more reasonable statement than your earlier statement "I think it's fair to say. You won't find many 'users of mathematics' (physicists, engineers, etc) who have studied formal definitions of limits, ...  But I don't think it matters too much, most of the time; you don't need to be able to make an epsilon-delta argument"
My personal experience is that we used epsilon-delta arguments in freshman calculus class, probably in the first academic quarter, to prove limits as part of understanding the nature of the differential calculus, etc.  (Perhaps times have changed since 1967.)  We did have one math-department course on "applied mathematics for scientists", but the other required courses were taken by science and math majors alike.  My undergraduate college did not have an engineering school, but some physics students went on to engineering school after graduation.  The epsilon-delta argument is not very difficult.
 

Offline dietert1

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Re: A 'simple' Physics postulation...
« Reply #55 on: July 13, 2021, 07:01:59 pm »
It's long time ago, but i also think i learned that at school, during the last years - what anglo-americans call college.
An example i remember from mathematics lectures during the first years at university was about functions that are in general continuous except in an infinite number of points. Useless nonsense in retrospective, but somewhere i may still have notes..

Regards, Dieter
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #56 on: July 13, 2021, 07:29:26 pm »
My memory from 50 years ago is hazy, but that sounds like the generalization of Riemann integral (the one we all use) to the Lebesgue integral, which I have not needed since.
 

Offline Nominal Animal

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Re: A 'simple' Physics postulation...
« Reply #57 on: July 14, 2021, 03:38:13 am »
My personal experience is that we used epsilon-delta arguments in freshman calculus class, probably in the first academic quarter, to prove limits as part of understanding the nature of the differential calculus, etc.
My own from 1994 in Finland was exactly the same.  Not sure since.



A key point here to remember is that math is the language used to describe physics.  We discover physics, don't invent it.  Whatever expressions we use to describe physics, the first thing we always must to do is to examine whether it makes sense or just happens to match the data in the limited region we used to construct the expression in.

Going below say nanosecond scale without at the same time considering the "object" a collection of atoms and the associated phenomena (including Brownian motion) is picking a subset of the physically applicable rules and knowingly rejecting the rest, and yields garbage physics, even if mathematically correct.

Here, to repeat myself, the only sensible approach (i.e. that makes physical sense given the context of the question) is to consider the trajectory (height component) a second-degree polynomial (given the limitations of the question), which is obviously continuous and differentiable, and therefore has an extremum where the derivative is zero.

Even if we define "stop" to mean "a period of time when the object does not move", the answer does not really change.  You see, near the extremum, velocities are low, so if we pick two points in time symmetrically before and after the extremum, the difference in height is minimal.  Mathematically, the trajectory of the object does change, but considering the kind of approximations we are already making and the associated limits to precision, we can tell those two points in time as separate before we can tell a change in height.

As an example, if you throw a baseball up just right, so its apex is at shoulder height, your friend can hit it with a bat at just the apex, and make it fly perfectly horizontally.  Well, okay, the ball-bat angle with respect to ground can compensate for small vertical velocities.. but you could make a machine that does the same with a flat vertical hitting surface (a pool cue with a plate at the tip).  So, the only sensible interpretation of the original question is that there indeed is such a moment, because we can act (or create a machine that acts) during that moment, poking the object with a bat or cue right then.  The precision at which we can do so, is "just" an engineering problem.
 

Offline magic

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Re: A 'simple' Physics postulation...
« Reply #58 on: July 14, 2021, 01:44:37 pm »
Here's an interesting idea:

If we consider a function whose value is 1 on [0,⅓]∪[⅔,1] and zero elsewhere, any Riemann integral thereof with a mesh finer than 0.1·⅓ will be within 20% of ⅔.

If we then set the function to zero on the middle third of each of those intervals, any Riemann integral with a mesh finer than 0.1·(⅓)² will be within 20% of (⅔)².

We clearly have a limit in zero here. Therefore function which is 1 on the Cantor set and 0 elsewhere is Riemann integrable and the integral is zero. The Cantor set being, of course, an infinite, uncountable and continuum-cardinality abonimation :scared:

This appears to complete the proof that you don't need to understand all that :bullshit: to be a physicist for your whole life :-DD
« Last Edit: July 14, 2021, 01:51:47 pm by magic »
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #59 on: July 14, 2021, 01:48:55 pm »
This is why the "other people" invented the Lebesgue integral for those who find it necessary.
By the way, if you sample a real-world waveform inappropriately, you may find these high-falutin' concepts useful in understanding the result.
 

Offline magic

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Re: A 'simple' Physics postulation...
« Reply #60 on: July 14, 2021, 02:06:42 pm »
I don't even remember what Lebesgue integrals were useful for that Riemann couldn't solve. Supposedly, a function is Riemann integrable iff it is bounded and discontinuous at most on a set of Lebesgue measure zero; that seems to cover all reasonable functions and many unreasonable ones as well.

Now, the Lebesgue measure itself I remember as being an interesting and elegant concept, but the details of it are quite gone from my memory.
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #61 on: July 14, 2021, 03:38:19 pm »
Yeah, that is how my education worked as well.  All I remember about Lebesgue integrals is that one could reduce the Lebesgue integral to a Riemann integral that can be solved normally.
Here's how I (now retired) look back on my formal education.  (I had excellent teachers and professors, two of whom later won Nobel prizes due to no fault of mine.)
Over the years, I encountered several things that I thought were "cool stuff", worth remembering.  Here are a few examples:
1.  Formal theorem proofs, from definitions and axioms and other theorems, in modern treatments of Euclidean geometry.
2.  The epsilon-delta method for proving a limit (which I didn't realize would kick up such a ruckus in this thread).
3.  The method in Purcell's electricity and magnetism textbook  https://archive.org/details/ElectricityAndMagnetismPurcell3rdEdition  demonstrating that the magnetic field from a current results from special-relativity effects on the charge carriers in motion.
4.  Variational principles, such as Fermat's principle of least time in optics and Lagrangian mechanics for systems with interesting degrees of freedom.
I do not claim expertise in any of these areas, nor can I remember every jot and tittle of the proofs, but memory allows me to know what to look up in textbooks or online, should I encounter a relevant problem.
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #62 on: July 14, 2021, 05:33:24 pm »
Let us go back to the original legitimate question, from July 9, that invoked the ancient problem of the “continuum”:
“If you were able to throw an object 'exactly' vertical, such that it came back to you due to gravity, did the object ever 'Stop' at its apex?”,
where Glenn Sprigg is concerned about the change from positive to negative velocity at the apex, and explicitly is not interested in astronomical scales or non-terrestrial situations. 

Here, I try to answer that question in three sections:  Physics, Engineering, and Quibbling.  I start with the velocity directly, since that is more direct than starting with the altitude.  Applying this question to pop-up foul balls in baseball could lead us to a “Unified Infield Theory”.  (For those who may not be familiar with baseball, that was a joke.  Everything else below is serious.)

Physics discussion: 

Consider the (idealized) one-dimensional vertical motion of an object in an inertial reference frame with a uniform gravitation field, where the velocity’s vertical component v = v0 > 0 at t = 0, and we define the vertical position y (altitude) at t = 0 to be y = 0 .  We do not specify how the object attained that velocity at that position and time, only that, for t > 0, there is no force on the object except gravity.  The object goes up (v > 0) to an apex, then falls down (v < 0).  The gravitational acceleration (downwards) is -g < 0, where the constant g is positive.

Newtonian physics tells us that, in this model, gravity constantly decelerates the velocity:

v(t) = dy/dt = v0g t    where v = 0 at a time t = tAv0 / g , and goes negative thereafter.

Simple calculus tells us that

y(t) = v0 t – (gt2 /2)  , where the constant of integration gives  y(t) = 0 at t = 0 ,

and that, at time t = tAv0 / g ,  the apex altitude is

yA = y(v0 / g) = (1/2) (v02/g) ,

which is a well-known result, but not needed to discuss the velocity.  The altitude y is a parabola (quadratic function of time). 

We need discuss only the velocity.  In the first equation, the velocity is continuous and monotonic, and a simple linear function of time.  Looking at the equation for velocity vs. time, we see that it goes continuously through zero. 

One could object that the variable does not remain at any value, including zero, for a finite amount of time.  However, we define the average velocity at any time t0, over a finite symmetrical time interval 2s, as the mean value from t = (t0s) to (t0 + s). 

Since the velocity is a linear function of time, the average velocity is independent of the interval length 2s, so long as the interval stays within the time range before the object hits the ground (proof left to the reader).  Therefore, the limit of the average velocity as s goes to zero is a trivial computation (since the average velocity is independent of s), equal to v (t) as in the equation above, including the value 0 at the apex (t = tA), where the velocity is positive before the apex and negative thereafter, with an average value of 0 (independent of the interval length).

Engineering discussion:

When higher accuracy is required, engineers modify the above ideal case by adding other details to the calculation, including other factors for the specific problem.  Among others, these include:
--Aerodynamics and air resistance.  Boeing and General Motors have good software packages for computing these effects.  Baseballs have interesting aerodynamic features due to their spin and external stitching.  Wrigley Field is famous for winds off Lake Michigan.
--Non-inertial reference frame (earth’s rotation underfoot).  Baseball is often played at roughly 40 deg N latitude, and this effect can be added, along with deviations from a true vertical initial velocity.

Quibbles:

Here, I group phenomena that are physically real, but whose effects on the basic question are unmeasureable, especially with respect to uncertainties in the engineering corrections above.  Examples include:
--Planck distance.  Trivial compared with the 74 mm diameter of a baseball: any effect is not measurable in this problem.
--Non-uniform gravitational field.  If the pop-up attains 50 m altitude, the variation in g, with the Earth’s radius of roughly 6400 km, is approximately 16 ppm.
--Special relativity.  This is trivial compared with the typical baseball velocity of 45 m/s.
--Heisenberg’s Uncertainty Principle:  left as an exercise for the reader.
 
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Offline CatalinaWOW

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Re: A 'simple' Physics postulation...
« Reply #63 on: July 15, 2021, 04:36:06 am »
I think the way to put this question in perspective is to build on T3sl4co1l's observation that the velocity is never constant.  Velocity can only be defined as a derivative or as an average over an interval. 

So in the sense of the OP's thought that the ball never stops, it never has any velocity at all.  The most elusive thing there ever was, moving without velocity.

If you look at things the wrong way you get very weird answers.
 

Offline vad

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Re: A 'simple' Physics postulation...
« Reply #64 on: July 15, 2021, 10:32:31 am »
Velocity can only be defined as a derivative or as an average over an interval.
What if we postulate the velocity, and define distance as integral of v dt? E.g. like SI defines meter.
 

Offline CatalinaWOW

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Re: A 'simple' Physics postulation...
« Reply #65 on: July 15, 2021, 11:45:00 am »
Velocity can only be defined as a derivative or as an average over an interval.
What if we postulate the velocity, and define distance as integral of v dt? E.g. like SI defines meter.

Sure.  An integral is an average.  This can also be misinterpreted if you think of it incorrectly.  (There exist velocity histories for which the integral is zero over some interval.  The object is in the same place at the ends of the interval, hence there was no motion.  This is analogous to the original posters "conundrum".)
 

Offline TimFox

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Re: A 'simple' Physics postulation...
« Reply #66 on: July 15, 2021, 01:39:28 pm »
In my long post above, I pointed out that for this particular case, where the velocity is a linear function of time, if you calculate the average velocity over a symmetrical interval around any time, the average is independent of the length of the interval.
 

Offline GlennSpriggTopic starter

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Re: A 'simple' Physics postulation...
« Reply #67 on: July 16, 2021, 11:23:40 am »
O.P. here....   Oops!!, I think I opened Pandora's Box !!!   :scared:

Diagonal of 1x1 square = Root-2. Ok.
Diagonal of 1x1x1 cube = Root-3 !!!  Beautiful !!
 
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