Does Physics get more abstract with the more advanced topics?

[Matt2411]

In high school I absolutely hated Physics. The class was just all about memorizing formulas and stuff, and I thought I wasn't learning anything useful for real life.

Yet now that I'm older I have re-discovered my passion for science and I'd like to give the subject another try. The Internet especially is a wonderful resource for me, as I can look for the explanations of the formulas (something my textbook doesn't always have).

So far I've only delved into the basics (and I'm already having some problems lol). What really gets me nuts though is, for example, the fact that I can't understand how Newton came up with the Law of Universal Gravitation (mathematically speaking) or how Kepler discover his Third Law of Planetary Motion. I don't know if I'm too dumb to realize that by myself or if the approach of the textbooks is just stupid. I'm starting to feel the same I did in high-school; I'm somehow not getting the full picture.

What I wanted to ask anyway was if the subject got more and more abstract and mathy in advanced topics. Is Introductory Physics a bit dogmatic at first (and you later find out more about why some formulas work, why they came up with this law..) or is it just a taste of what is to come? If it's the latter (as I'm supposing) then it's a shame. Because I'm really interested in science. But it's like there's so much information to process that I just can't make sense of it all.

 

[adjacent]

At grade 8,I didn't know why kinematics formulas worked or how it was derived.But with time ,more physics and maths,I can finally understand how they work and why they are giving the correct results.
So I found out that at the start,I couldn't get the full picture but after getting familiar with it,I can understand it fully.

 

[oddjobmj]

As an undergraduate I can say I have a good idea but probably not a complete picture. With that said:

With regard to the topics you are discussing there are a few points to make.

1) Your textbooks aren't written with the intent to give you a complete picture of every concept you use. The book would never end. They usually derive a formula from past ideas, give a little context, and explain how it is used. You're right that they don't usually explain to you how every formula came about.

2) Each class you take in physics seems to fill in gaps left in previous courses but also open up new gaps to be filled later. As you progress in your study of physics you will realize that those formulas you were confused by all the sudden make sense on a fundamental level. The ideas you learned some time before that seemed abstract become concrete and clear. Again, at the same time, you start to learn more complex theories that boggle the mind. Of course, on the fringe of the study of physics is some exceptionally complex theoretical physics that no one yet fully understands which is very abstract.

3) If you want to know how certain formulas came about there is a wealth of history available for probably any common formula in an introductory level physics book. Planetary motion, Newton's laws, mathematics, etc. These ideas are studied, their histories available, and often even their author's derivations and work leading up to that as well as their final published work is available to peruse. An example of this is this book where a well known physicist took a few fundamental/historical works regarding/relating to physics and added his commentary/explanation on top of the published works of those scientists.

To consolidate a bit here physics isn't all abstract. There are good concrete explanations for basically everything but the cutting edge. It takes a lot of work to get to the point where you really understand even some seemingly simple concepts. If you proceed in your studies you will run into complicated concepts that you won't immediately understand but that is the nature of any field, really.

Lastly, I want to point out that the name of the linked book is very relevant here. It is a metaphor popularized by Newton when he said "If I have seen further it is by standing on the shoulders of giants." He didn't derive his formulas/laws from scratch. He called upon his teachings and understanding passed on to him from generations of scientists and philosophers before him. The formulas in your book come about not by some easily explainable recipe but by hundreds of years of work leading up to that point by countless contributors to the field and even after all that work it still took the final author/scientist years, or even their lifetime, to really hash out what it is they wanted to say in a concise form.

 

[WannabeNewton]

What I wanted to ask anyway was if the subject got more and more abstract and mathy in advanced topics.

It will get more abstract and "mathy" but that's not necessarily a good thing. Classical electrodynamics is basically the best of both worlds in terms of physics and math.

 

[oddjobmj]

It will get more abstract and "mathy" but that's not necessarily a good thing.

I don't know about this. You definitely need a lot of math but only because that's the language we use to talk about these concepts and really put values to them.

To say something is mathy, to me, means that it is purely abstract, if not almost arbitrarily defined, because -that's the way we decided to define it-. Ideas such as whether or not the set of all sets contains itself. There is no physical reason why it would or wouldn't. The only reason we can say it doesn't is because we defined a series of concepts mathematically that lead up to that conclusion. It is not tied down to physical reality but rather, well, some abstraction of rules.

Physics is -math heavy- and some math is abstract BUT these concepts are part of physics because they are tied to reality. They are physical laws and phenomena that we can interact with and thereby, usually, gain some perspective. The speed of light isn't something we decided was a nice round number but rather something you can measure, for example.

 

[WannabeNewton]

That's all nice in theory but in practice if you depend too much on abstraction and mathematical formulas you will find yourself struggling to solve actual physical problems. A deep conceptual understanding is much more important than mathematical fluency in the tools of a physical theory not to mention much harder to acquire.

 

[oddjobmj]

That's basically what I just said. We use math to talk about these concepts but they are tied to reality which is the more important part.

 

[homeomorphic]

You might be interested in this.

http://www.math.uga.edu/~shifrin/Spivak_physics.pdf

It's a draft copy. He finished the book and you can now buy the whole thing, but the rest of it gets pretty intense mathematically. He has some discussion of where the basic laws of Newtonian mechanics "come from".

Most students would be screaming if you tried to explain a lot of this sort of thing to them, so it is dumbed-down. That is the real reason, more than just trying to keep the books short enough.

I never really liked the idea that you have to fill in the gaps later, because I learn to remember, not to forget like most people do. And what makes me remember? Understanding. Not only that, but the whole point of learning physics and math is understanding to my mind because that is the only thing that makes it interesting. Everything else about it is completely boring to me, although it can happen that I have some understanding that is built on top of something that I don't understand.

I did like physics when I first took it because even though I didn't understand things like why you would come up with something like Newton's laws, you could still relate the math to real life and physical intuition. I had never seen anything like it before, so I still found it pretty enlightening, despite the shortcomings. Not only that, but, ironically, even with the incomplete understanding I was getting, it really helped me to catch on to the idea of trying to understand. I got a lot better at math as a result because I applied the same idea to math as well as physics and started thinking for myself more and figuring out why things worked.

I don't know if this is "right" or "wrong", but I do know that the way things are taught makes it very hard to learn my way and understand things thoroughly. Often, it would be 100 times easier to learn this way if people taught with people like me in mind, so I think the problem is mostly that stuff is taught with people who are okay with an incomplete understanding in mind.

But maybe sometimes, it is intrinsically a good idea to take people's word for things. It can save time. It's good to have a little flexibility and be able to use other people's ideas like driving a car, without understanding how it works. Because sometimes, the situation may call for that, and other times, it may call for understanding it for yourself. I have a small amount of flexibility this sort of flexibility in taking things on trust, but I think one of the reasons it was hard for me to contribute anything to research was that I am more or less unwilling to build on other people's work if I don't understand it thoroughly myself. On the other hand, when I was working on my thesis, I tried to compromise a bit on this issue, and I think it actually hurt me and made my problem harder to solve because I was trying to cut corners in understanding in order to get the stupid thing done fast enough.

 

[SteamKing]

So far I've only delved into the basics (and I'm already having some problems lol). What really gets me nuts though is, for example, the fact that I can't understand how Newton came up with the Law of Universal Gravitation (mathematically speaking) or how Kepler discover his Third Law of Planetary Motion. I don't know if I'm too dumb to realize that by myself or if the approach of the textbooks is just stupid. I'm starting to feel the same I did in high-school; I'm somehow not getting the full picture.

In Kepler's case, it took a lot of hard work to come up with his laws of planetary motion, even though Kepler possessed many years of celestial observations made by his former mentor, Tycho Brahe. Reducing and analyzing Brahe's observations by hand, it reportedly took Kepler about ten years to formulate the first two laws of planetary motion. A further ten years' work was required before Kepler came up with the third law of planetary motion.

In Newton's case, he reportedly had derived the fact that planetary orbits have the shape of ellipses, as Kepler had shown, using the hypothesis that the planets are attracted by the sun with a force whose magnitude varies inversely with the square of the distance separating the two bodies. Not feeling that his work justified publication at that time, Newton set his calculations aside (and reportedly lost them), forgetting about them until he later was posed a question about the shape of planetary orbits in a letter written to him by Edmund Halley. Newton replied to Halley that he had shown mathematically that a planet subject to an inverse square law force would travel in an ellipse and would send him the calculations after he had reconstructed them. Eventually, this work would grow into Newton's Principia Mathematica, his best known work, but by no means the only influential scientific publication authored by Newton.

Newton's mind was so restless that he would quickly grasp and often solve a particular problem, perhaps make a few notes, and then quickly move on to another topic, without taking the time to write a fully developed paper on the subject unless there was a great motivation to do so on Newton's part.

Although Newton did prodigious scientific work apparently effortlessly, he acknowledged that he owed a scientific debt to those who had come before with his comment about seeing so far because he stood on the shoulders of giants.

In learning a subject like physics today, or mathematics, or chemistry, we often want to be able to discover the great principles which form these disciplines like they were discovered by a Kepler or a Newton, but often it is not realized how much work was invested (as Kepler did) or what a singular intellect was able to see immediately (as Newton did) the solution to a problem which had eluded others.

 

[homeomorphic]

In learning a subject like physics today, or mathematics, or chemistry, we often want to be able to discover the great principles which form these disciplines like they were discovered by a Kepler or a Newton, but often it is not realized how much work was invested (as Kepler did) or what a singular intellect was able to see immediately (as Newton did) the solution to a problem which had eluded others.

That's true, but a lot of their work was probably dead-ends, missteps, and just being stuck, which wouldn't add any insight. An alternate, simpler motivation from the historical one is often possible, as well. We don't need to keep re-inventing the wheel, but it's simply not necessary to present unmotivated or unsubstantiated concepts. If you don't convince me that Newton's laws work, why should I believe them? Answer: I shouldn't.

So, yeah, this doesn't mean you should go read Newton's Principia. But it does mean Spivak's book is better than standard physics textbooks, and not excessively long, so yes, there is a better way to do it.

 

[mal4mac]

So far I've only delved into the basics (and I'm already having some problems lol). What really gets me nuts though is, for example, the fact that I can't understand how Newton came up with the Law of Universal Gravitation (mathematically speaking)...

It's the simplest law that fits the observations known at that time (i.e., outside the quantum and relativistic realms.) Isn't it obvious how he might have come up with it? Maybe it's too obvious :)

Note that is *axiomatic* (hence the "Law" in the name!) - so there is no mathematical derivation of it. It is quite remarkable that force was *seen to be* proportional to mM/r^2, in all cases from apples to planets.

You should just accept this as a fundamental law (certainly at this stage) Some bright spark might derive it from more fundamental laws, but you might need five years of physics courses to understand anything like that (and, in any case, you'll still be stuck with accepting some other law(s) as axiomatic.)

So physics *is* dogmatic, at base; there is always some formulae you will have to accept as basic, either (at best) because it agrees with our observations & experiments (however limited...) or (at worst - string theory) because it's the latest fad...

 

[PhotonicBoom]

From my experience as an undergrad student, physics do get more abstract with more advanced topics. Me and my flatmate both study physics and both of us have completely different interests.

I on one hand like abstract concepts and I'm more of a QM kind of guy, while my friend doesn't like stuff he can't visualise (He's extremely smart and an A student in all subjects, so no, its not that he doesn't get the subjects, he just doesn't enjoy them), and is following a different path.

For example he chose Climate and Energy Physics while I chose GR and Chaos Theory etc.

Keep in mind though that even the less abstract subjects like Classical Mechanics get more and more abstract the more you study them (e.g Lagrangians and Hamiltonians, 6N Dimensional Phase Spaces etc).

My point is, don't get demotivated. Newton and Kepler while truly amazing geniuses, didn't come up with stuff on the spot. It took them years of observation, trial and error etc. There is nothing you can't get imho, as long as you put passion and dedicate time to it.

 

[homeomorphic]

It's the simplest law that fits the observations known at that time (i.e., outside the quantum and relativistic realms.) Isn't it obvious how he might have come up with it? Maybe it's too obvious :)

Yes, and why the hell does it fit the observations? That's just a tautology. Restating the problem with no new info. Actually, it turns out Newton did some hardcore geometry and showed that the inverse square law yielded Kepler's laws. Even better, ONLY inverse square laws and linear force laws produce elliptic orbits in central force motion (assuming decay is a power function of distance from the center). Something like this should at least be mentioned to a bright student. And there are pieces of it that really aren't that hard to understand with a little effort.

Note that is *axiomatic* (hence the "Law" in the name!) - so there is no mathematical derivation of it.

No mathematical derivation in standard textbooks. But, as I said, Newton's geometry is the reason he introduced it. He didn't just say, "hey, wouldn't it be cool if gravity were described with an inverse square law," propose it, and then, lucky guess, it turns out to fit the observations.

That picture of the discovery is quite misleading. Also, this doesn't quite remove the axiomatic quality, but never the less makes it more motivated/meaningful: there's another thing that's special about inverse square laws. The surface area of a sphere. It varies as a square of the radius. Inverse square is precisely what is need to cancel it out. Physically, this means the same flux is going through a sphere of any size. So, somehow the inverse square law is a law that spreads the force out evenly over all spheres.

It is quite remarkable that force was *seen to be* proportional to mM/r^2, in all cases from apples to planets.

It's much less remarkable than you seem to suggest, since you left out what Newton actually did. Wasn't just a wild guess. It's still all the more remarkable, if you see what he did. That was one of his greatest achievements. But it wasn't magic. Genius, yes. Magic, no.

You should just accept this as a fundamental law (certainly at this stage) Some bright spark might derive it from more fundamental laws, but you might need five years of physics courses to understand anything like that (and, in any case, you'll still be stuck with accepting some other law(s) as axiomatic.)

First of all, the more advanced physics courses are not really necessary, except that they'd give you more practice, and the only one that could possibly be relevant is more classical mechanics. But if you want to understand what Newton did, it's not intermediate classical mechanics. It's geometry. You don't have to take more physics classes, you just have to read about it. Unless you want to go the general relativity route or something. Accepting other, more basic laws as axiomatic may be a real gain because they might be much more intuitive.

So physics *is* dogmatic, at base; there is always some formulae you will have to accept as basic, either (at best) because it agrees with our observations & experiments (however limited...) or (at worst - string theory) because it's the latest fad...

Doesn't have to be that way. People don't just come up with random theories and discover the laws of physics by blind trial and error. They make their best educated guesses, based on sophisticated reasoning.

 

[Matt2411]

You might be interested in this.

http://www.math.uga.edu/~shifrin/Spivak_physics.pdf

It's a draft copy. He finished the book and you can now buy the whole thing, but the rest of it gets pretty intense mathematically. He has some discussion of where the basic laws of Newtonian mechanics "come from".

Most students would be screaming if you tried to explain a lot of this sort of thing to them, so it is dumbed-down. That is the real reason, more than just trying to keep the books short enough.

I never really liked the idea that you have to fill in the gaps later, because I learn to remember, not to forget like most people do. And what makes me remember? Understanding. Not only that, but the whole point of learning physics and math is understanding to my mind because that is the only thing that makes it interesting. Everything else about it is completely boring to me, although it can happen that I have some understanding that is built on top of something that I don't understand.

I did like physics when I first took it because even though I didn't understand things like why you would come up with something like Newton's laws, you could still relate the math to real life and physical intuition. I had never seen anything like it before, so I still found it pretty enlightening, despite the shortcomings. Not only that, but, ironically, even with the incomplete understanding I was getting, it really helped me to catch on to the idea of trying to understand. I got a lot better at math as a result because I applied the same idea to math as well as physics and started thinking for myself more and figuring out why things worked.

I don't know if this is "right" or "wrong", but I do know that the way things are taught makes it very hard to learn my way and understand things thoroughly. Often, it would be 100 times easier to learn this way if people taught with people like me in mind, so I think the problem is mostly that stuff is taught with people who are okay with an incomplete understanding in mind.

But maybe sometimes, it is intrinsically a good idea to take people's word for things. It can save time. It's good to have a little flexibility and be able to use other people's ideas like driving a car, without understanding how it works. Because sometimes, the situation may call for that, and other times, it may call for understanding it for yourself. I have a small amount of flexibility this sort of flexibility in taking things on trust, but I think one of the reasons it was hard for me to contribute anything to research was that I am more or less unwilling to build on other people's work if I don't understand it thoroughly myself. On the other hand, when I was working on my thesis, I tried to compromise a bit on this issue, and I think it actually hurt me and made my problem harder to solve because I was trying to cut corners in understanding in order to get the stupid thing done fast enough.

Thanks, I feel so understood now! I absolutely agree with you, there's no point in learning "by heart" for two reasons: 1- It's boring (and tiresome) to study something you don't understand 2- Learning by memory does not work... unless you remind yourself every now and then.

And it seems like our educational system does not consider all these things. Or maybe it's what you said, most people aren't interested in learning the why of things, and it takes a long time to explain it. I applied the same principle to maths (just like you) so I searched online for a justification of any unexplained theorem we had to learn (such as the law of cosine; sine; etc.). Indeed, I was "wasting" time to understand a proof which ultimately derived in a simple formula but I NEEDED to do it. I hate trusting blindly what my textbooks say. Isn't the whole point of science to use your mind and not to trust everything you're told? It sometimes feels like nobody cares whether students know why they're doing what they're doing. As long as they're useful for the labor market, I guess

Responding to your other post, you make a point that most of the run-up to a big scientific discovery involved making many wrong hypotheses. It could all be summed up in "After much research...", so as to later explain the correct theory/law and the evidence that support it. But again, I guess that until businesses, universities and students start caring about this people like me are turned off by the subject (now I'm into the more understandable but much less respectable social sciences).

 

[Matt2411]

That's all nice in theory but in practice if you depend too much on abstraction and mathematical formulas you will find yourself struggling to solve actual physical problems. A deep conceptual understanding is much more important than mathematical fluency in the tools of a physical theory not to mention much harder to acquire.

Exactly. Sometimes I see so much math in my book and get so involved in the maths that I forget what the purpose of the said formula actually is and why it proves the conclusions later stated.

 

[WannabeNewton]

Exactly. Sometimes I see so much math in my book and get so involved in the maths that I forget what the purpose of the said formula actually is and why it proves the conclusions later stated.

I'm studying from a couple of quantum field theory texts right now and I can tell you I feel the exact same way so you aren't alone.

 

[Matt2411]

As an undergraduate I can say I have a good idea but probably not a complete picture. With that said:

With regard to the topics you are discussing there are a few points to make.

1) Your textbooks aren't written with the intent to give you a complete picture of every concept you use. The book would never end. They usually derive a formula from past ideas, give a little context, and explain how it is used. You're right that they don't usually explain to you how every formula came about.

2) Each class you take in physics seems to fill in gaps left in previous courses but also open up new gaps to be filled later. As you progress in your study of physics you will realize that those formulas you were confused by all the sudden make sense on a fundamental level. The ideas you learned some time before that seemed abstract become concrete and clear. Again, at the same time, you start to learn more complex theories that boggle the mind. Of course, on the fringe of the study of physics is some exceptionally complex theoretical physics that no one yet fully understands which is very abstract.

3) If you want to know how certain formulas came about there is a wealth of history available for probably any common formula in an introductory level physics book. Planetary motion, Newton's laws, mathematics, etc. These ideas are studied, their histories available, and often even their author's derivations and work leading up to that as well as their final published work is available to peruse. An example of this is this book where a well known physicist took a few fundamental/historical works regarding/relating to physics and added his commentary/explanation on top of the published works of those scientists.

To consolidate a bit here physics isn't all abstract. There are good concrete explanations for basically everything but the cutting edge. It takes a lot of work to get to the point where you really understand even some seemingly simple concepts. If you proceed in your studies you will run into complicated concepts that you won't immediately understand but that is the nature of any field, really.

Lastly, I want to point out that the name of the linked book is very relevant here. It is a metaphor popularized by Newton when he said "If I have seen further it is by standing on the shoulders of giants." He didn't derive his formulas/laws from scratch. He called upon his teachings and understanding passed on to him from generations of scientists and philosophers before him. The formulas in your book come about not by some easily explainable recipe but by hundreds of years of work leading up to that point by countless contributors to the field and even after all that work it still took the final author/scientist years, or even their lifetime, to really hash out what it is they wanted to say in a concise form.

Thanks you so much for this. I really want to read that book now.

Indeed, one cannot help but wonder at the effort all these people put into to create a coherent paradigm of the subject (in this case of motion).

I know that all the abstraction has to make sense (if not, how was it all accepted by the scientific community?), yet it's kind of a turn-off. I feel like I'm driving through a misty road: I can't see anything in front of me, but I trust that it is the correct way. However, that means it is not much fun to drive in the first place.

 

[Matt2411]

I'm studying from a couple of quantum field theory texts right now and I can tell you I feel the exact same way so you aren't alone.

Heh, then we're all together in this :)

I sometimes wish I had been born in the seventeenth century (or even a century or two later) because although back then we didn't know half of what we know today, it was so much easier and it took less effort to be a well-informed layman in all the sciences.

In our present time, no matter how much I try, it seems impossible for me to find out and understand all that's going on in all the sciences.

 

[mal4mac]

Yes, and why the hell does it fit the observations? That's just a tautology.

It is not a tautology. How can observations be tautological?

Restating the problem with no new info. Actually, it turns out Newton did some hardcore geometry and showed that the inverse square law yielded Kepler's laws. Even better, ONLY inverse square laws and linear force laws produce elliptic orbits in central force motion (assuming decay is a power function of distance from the center). Something like this should at least be mentioned to a bright student. And there are pieces of it that really aren't that hard to understand with a little effort.

That's good supporting evidence, but without observations it's not going to win the prize.

It's much less remarkable than you seem to suggest, since you left out what Newton actually did. Wasn't just a wild guess. It's still all the more remarkable, if you see what he did. That was one of his greatest achievements. But it wasn't magic. Genius, yes. Magic, no.

It seems remarkable to me that such a simple law fits most of the facts - whether you guessed it, or derived it from geometry. I didn't say it was a wild guess.

They make their best educated guesses, based on sophisticated reasoning.

Like that guy who discovered the ring structure of benzene. A dream about snakes biting their tails? Very sophisticated, very geometrical , very rational... :) You can't lay down the law about how great discoveries are made.. sometimes it's a guess based on sophisticated reasoning, sometimes it's a lucky guess, sometimes it's a dream...

 

[homeomorphic]

It is not a tautology. How can observations be tautological?

Because we know that already.

That's good supporting evidence, but without observations it's not going to win the prize.

Of course not. The point you are missing is that these things provide a hypothesis to test. Now, there may not be an absolute rule that you can never test a hypothesis that doesn't have some kind of theoretic backing, but it is generally true that it's not the best use of your time. Furthermore, it gives us a reason to believe in the law of gravitation. It makes it more psychologically plausible. It adds to our understanding. Understanding is important, not just results. In fact, the experimental verification is something I am happy to leave to other people in most cases, although it can be interesting, too. What I am really after, mostly, is theoretical understanding. And that's not an arbitrary goal. It's a goal I have because it enhances the learning process, makes things meaningful, and ties them together.

It seems remarkable to me that such a simple law fits most of the facts - whether you guessed it, or derived it from geometry. I didn't say it was a wild guess.

Fair enough, but you did leave out any mention of the reasoning used, so it was misleading.

Like that guy who discovered the ring structure of benzene. A dream about snakes biting their tails? Very sophisticated, very geometrical , very rational... :) You can't lay down the law about how great discoveries are made.. sometimes it's a guess based on sophisticated reasoning, sometimes it's a lucky guess, sometimes it's a dream...

Right there, you are talking about the SOURCE of the idea. Most of the time, after you get your creative idea, you are going to have to have some extra reasoning to justify why it is a good hypothesis. As I said, random guesses are generally not the best use of your time and resources, although there is no absolute rule against testing them.

 

[AlephZero]

I sometimes wish I had been born in the seventeenth century (or even a century or two later) because although back then we didn't know half of what we know today, it was so much easier and it took less effort to be a well-informed layman in all the sciences.

??
In the 17th century, most of the population barely knew how to read, and even if they could read, many of them only owned one book - and that book was about religion, not science.

It may have taken "less effort" for the very few who were wealthy enough that they didn't have to work, and lived in a large city or near a major university. But not for everybody else.

 

[jbrussell93]

Actually, it turns out Newton did some hardcore geometry and showed that the inverse square law yielded Kepler's laws. Even better, ONLY inverse square laws and linear force laws produce elliptic orbits in central force motion (assuming decay is a power function of distance from the center). Something like this should at least be mentioned to a bright student. And there are pieces of it that really aren't that hard to understand with a little effort.

Actually, we did this derivation in my mechanics course. I believe there was also a homework problem or two... It was not covered in our book though.

 

[homeomorphic]

Actually, we did this derivation in my mechanics course. I believe there was also a homework problem or two... It was not covered in our book though.

Well, good for your mechanics course. It seems to have been better than the norm, although I would have to see for myself the quality of the actual derivation before I could really sign off on it. But, even if the derivation was not the most enlightening, which I can't judge, it's good that it was covered.

 

[jbrussell93]

Well, good for your mechanics course. It seems to have been better than the norm, although I would have to see for myself the quality of the actual derivation before I could really sign off on it.

Actually, it's funny you say this. My mechanics course did not even cover Lagrangian mechanics (as mentioned in a previous post) but we definitely went into depth on several topics I assume other "typical" mechanics courses do not such as dynamical systems... Not to digress from the OP's post.

In response to the original question, I definitely have noticed that upper division coursework has gotten much more "mathy" and interesting ("mathy" doesn't necessarily always correlate with interesting).

You are usually not presented with equations without a derivation of where it comes from. Of course, there are exceptions for example Newton's laws and Coulomb's law.

 

[homeomorphic]

Actually, in the intermediate classical mechanics class I took, we did derive Kepler's laws from the inverse square law, I think (though the derivation was not an enlightening one, because too many people are afraid of geometry, these days, preferring to just move lots of symbols around), but I don't think the converse result was mentioned, which, to me, is a lot more convincing. It implies inverse square is really the only option. Not, just "we calculated the orbit, under the inverse square force law, and it works out". No, there's only one reasonable choice, at least assuming a power law (admittedly, I don't know how to justify the power law assumption, all I know is that it ought to be a decreasing function of distance, so power law certainly fits the bill).