Kleppner/Kolenkow: Treatment of Kinematics

[Cosmophile]

While reading through Kleppner & Kolenkow's "An Introduction to Mechanics," I realized something: I am horribly dissatisfied with their treatment of kinematics. Perhaps I am simply spoiled by more run-of-the-mill textbooks, like Tipler and Giancoli, but K&K seem to not give much attention to, say, projectile motion. The first section of the book is "Vectors and Kinematics - A Few Mathematical Preliminaries," and they absolutely do a great job explaining vectors. However, the attention on kinematics seems minimal. Section 1.7 is "Formal Solution of Kinematical Equations," and it is here that they show the formal integration procedure to find velocity from acceleration (and, by extension, position from velocity).

They go on to explain that, under uniform acceleration, the integration procedure yields the equations one who took an algebra-based physics course would be familiar with. Namely, $v(t) = v_0 + at$ and $x(t) = x_0 + v_i t + \frac {1}{2} at^2$. From here we are given one example of motion in a uniform gravitational field, and then we are given an example of nonuniform acceleration (an example, I might add, which is quite neat). After that, move on to describe more in-detail the derivatives of vectors, and we do not hear about kinematical equations again.

Once I reached the practice problems, I found myself zipping through the problems about vectors, but almost at a standstill once I reached the actual "physics" problems.

It is possible that I could simply have not taken enough time on the few pages which discussed kinematical equations (I will be returning to them when I am done here), but I thought I would make a post to see if others had a similar experience.

 

[micromass]

Once I reached the practice problems, I found myself almost at a standstill once I reached the actual "physics" problems.

Yep, this is what doing an actual physics book is like. I hope you'll enjoy it.

 

[atyy]

K&K are great, but you don't have to be able to do their problems to master classical mechanics - you can try the problems from Halliday and Resnick or Young. It's fun to do a couple of K&K problems, but by and large they are far more difficult than one needs unless one is a masochist.

I bet K&K couldn't do their own problems if woken up in the middle of the night

 

[Cosmophile]

K&K are great, but you don't have to be able to do their problems to master classical mechanics - you can try the problems from Halliday and Resnick or Young. It's fun to do a couple of K&K problems, but by and large they are far more difficult than one needs unless one is a masochist.

This is certainly good to hear. I've certainly maintained a level of stress and a slight feeling of inadequacy since looking at these and seeing that others have come up with much nicer solutions.

 

[atyy]

As I said, it's good to start on easy problems, and the K&K problems are largely for masochists, even if one is a physics major. But in the larger scheme of things for what physics majors do, one reason to try at least a few of them is to see how difficult they are when approaching them from the Newtonian formulation of ckassical mechanics. Then when one learns the Lagrangian formulation of classical mechanics, many of the K&K problems become very easy. The Lagrangian formulation is a stepping stone to the Hamiltonian formulation, which is the stepping stone to quantum mechanics (some would say Hamilton-Jacobi is a better stepping stone, but that's nitpicking).

 

[Cosmophile]

As I said, it's good to start on easy problems, and the K&K problems are largely for masochists, even if one is a physics major. But in the larger scheme of things for what physics majors do, one reason to try at least a few of them is to see how difficult they are when approaching them from the Newtonian formulation of ckassical mechanics. Then when one learns the Lagrangian formulation of classical mechanics, many of the K&K problems become very easy. The Lagrangian formulation is a stepping stone to the Hamiltonian formulation, which is the stepping stone to quantum mechanics (some would say Hamilton-Jacobi is a better stepping stone, but that's nitpicking).

Again, that's very good to hear. I've heard that the Lagrangian (and Hamiltonian) formulations make many K&K problems laughable. I suppose I can't help but to worry that, because I am struggling with some of these, I'm just not the caliber I wish I were at, so I appreciate the reassurance that these struggles are not an indicator that I am doomed to mediocrity, haha.

 

[Intrastellar]

As I said, it's good to start on easy problems, and the K&K problems are largely for masochists, even if one is a physics major.

I do not understand this statement. K&K problems are never highly computational, and if you understand the answer, it usually does not take too much space to write it.

@Cosmophile: If you are finding the problems difficult, then that is all the more reason to do them. You will never get better if you do not challenge yourself. Challenging yourself with problems that emphasize the physics is teaching yourself to think like a physicist. The earlier, and more, you can do that, the better.

I am slow. It usually took me a day or more to do the average problems, and more than a week for the really good ones. I hope that this encourages you to spend more time with the problems.

 

[verty]

While reading through Kleppner & Kolenkow's "An Introduction to Mechanics," I realized something: I am horribly dissatisfied with their treatment of kinematics.

You may want to pick a book that you are not dissatisfied with. Being dissatisfied with a book can be an impediment to learning.

 

[atyy]

I do not understand this statement. K&K problems are never highly computational, and if you understand the answer, it usually does not take too much space to write it.

@Cosmophile: If you are finding the problems difficult, then that is all the more reason to do them. You will never get better if you do not challenge yourself. Challenging yourself with problems that emphasize the physics is teaching yourself to think like a physicist. The earlier, and more, you can do that, the better.

I am slow. It usually took me a day or more to do the average problems, and more than a week for the really good ones. I hope that this encourages you to spend more time with the problems.

By masochist, I mean something more like what you say about taking a day to a week to do one problem. My time on a K&K problem was about 1 hour for an average problem and 4 hours for the "really good ones" (which I do count as painful :)

 

[vanhees71]

Well, what else than vector analysis of curves (trajectories) do you want to discuss as kinematics? The important point is to introduce the vector machinery adequately. It's the most important issue at the first two introductory theory lectures (at least in Germany, in the 1st semester we start with "naive mechanics", covering the vector analysis on hand of the quite intuitive examples of Newtonian mechanics in the 2nd we have "analytical mechanics", where (in my opinion) the most important part is to introduce the Hamilton principle of least action in Hamilton's formulation and discuss Lie groups and Lie algebra in terms of phase-space and Poisson brackets as Lie algebra).

The kinematics, as usually understood, is a pretty short application of the vector analysis. It should cover the coordinate free representation of curves in space (tangent, normal and binormal vector, curvature, Fresnel formula). That's it.

 

[robphy]

Kleppner's text is certainly a challenging introductory text, which is used in some places as THE introductory course for physics majors.
(I was a TA for such a course. I think my students already had a simpler course in high school in which the standard projectile problems are treated.
So, Kleppner is intended for the very-well prepared... and so, they are able to discuss topics that are never touched in a typical introductory text. They moved onto Purcell next. As a TA, I did find Kleppner's problems challenging, but I appreciated them... however, I probably would have found Kleppner difficult had it been my introductory text.)

Again, that's very good to hear. I've heard that the Lagrangian (and Hamiltonian) formulations make many K&K problems laughable.

I don't get the point of this comment.* That would apply to simpler intro mechanics texts as well.
(When I was a TA for a course using Serway, in the middle of a exam-grading session, I suggested to another TA that we should slip into the pile of exams one that did the problems using Lagrangian methods.)

(Along the lines of your comment*, as a graduate student, I heard a story of a well-known mathematical relativist flipping through Jackson's Electrodynamics text and wondered why folks seemed to fear it since all of its differential equations were linear.)

 

[QuantumCurt]

K&K is an absolutely wonderful book, but it's not the book for everyone. Some people don't mesh well with the writing style or format of it. A lot less time is spent on details and examples than in many physics texts. In my opinion, K&K isn't necessarily the best book for ones first exposure to classical mechanics. I view it as more of an intermediate level bridge between freshman mechanics and junior mechanics.

As far as Lagrangian and Hamiltonian mechanics are concerned, I wouldn't say that they make the problems in K&K laughable. Lagrangian and Hamiltonian mechanics aren't typically covered until upper division classical mechanics. It's a matter of increasing complexity and more difficult subject matter. I wouldn't necessarily say that the problems in K&K are any more or less difficult than the problems in a mechanics text involving Lagrangian or Hamiltonian mechanics. They're very different approaches to the same types of problems.

 

[verty]

K&K is an absolutely wonderful book, but it's not the book for everyone.

That's exactly right. It's written for entering students that have done a full year of calculus (in the sense that they would credit out of single-variable calculus). If you know calculus and calculus is no more difficult to you than algebra is, you are someone for whom the book was written. Most people do not have the math background and therefore they find it difficult.

Cosmophile for example says he is "horribly dissatisfied" by how kinematics is explained. But actually if you look at it from this point of view, that they reduce it to math, it is explained. You'll know if this book is for you. If it isn't and you are dissatisfied by it, choose another book.

PS. Sorry, I didn't mean to be as harsh as it sounded, I just wanted to express that it isn't the book for everyone but it is a great book and one should judge it based on what the authors were trying to do, which is to write a book for entering students who are mathematically advanced.

 

[atyy]

I don't get the point of this comment.* That would apply to simpler intro mechanics texts as well.
(When I was a TA for a course using Serway, in the middle of a exam-grading session, I suggested to another TA that we should slip into the pile of exams one that did the problems using Lagrangian methods.)

The OP was probably basing it off my comment in post #5. I meant that in say Halliday and Resnick, I can basically do any problem in 5 to 10 minutes, but in K&K I can take up to 4-5 hours. However, if I have the Lagrangian formalism, then the Halliday and Resnick and K&K problems usually take 5-20 minutes, but nothing becomes so much faster like 1 minute that it is worth learning that Lagrangian formalism to do a Halliday and Resnick problem (except to learn the concept).

I have a technical question. When I made the comment on Lagrangian mechanics, I was mainly thinking that it is easier to "blindly" treat constraints in that formalism than in the Newtonian approach which requires some "intelligence". However, I don't usually think of the Hamiltonian formalism as good for anything in classical mechanics, except that it exists and is very beautiful, and a stepping stone to quantum mechanics. Is this wrong - is the Hamiltonian formalism also "practical" in classical mechanics (please do not answer with ADM, which is the only place I know where it's "practical")?

 

[robphy]

Without quantum mechanics, I think Hamiltonian mechanics would be less emphasized in undergraduate physics.

However, they are good for:
symmetries and conserved quantities, nonlinear dynamical systems (phase space) and statistical mechanics, numerical methods (symplectic integrators), perturbation theory [in celestial mechanics], optics,...

 

[verty]

That's exactly right. It's written for entering students that have done a full year of calculus (in the sense that they would credit out of single-variable calculus). If you know calculus and calculus is no more difficult to you than algebra is, you are someone for whom the book was written. Most people do not have the math background and therefore they find it difficult.

Cosmophile for example says he is "horribly dissatisfied" by how kinematics is explained. But actually if you look at it from this point of view, that they reduce it to math, it is explained. You'll know if this book is for you. If it isn't and you are dissatisfied by it, choose another book.

PS. Sorry, I didn't mean to be as harsh as it sounded, I just wanted to express that it isn't the book for everyone but it is a great book and one should judge it based on what the authors were trying to do, which is to write a book for entering students who are mathematically advanced.

I should probably change my previous post further but that would be editing way beyond when one should edit. If you imagine that a Russian wrote it, it'll come across in the right way. The Russian society is much more competitive especially in sports, and people can be ice cold to each other when it comes to getting ahead. That is the kind of steel-hearted mentality you want to have with this book. I want to be the best, I want to use the best book, I won't let anything stop me.

One can see for example that this is a concept that Stallone thought about because his Rocky movies were based around that, the idea that one gets little if one doesn't fight. So that is the interpretation for my previous post. Cosmophile has entered the arena and got punched in the face by K&K and he doesn't seem to be bleeding just yet. He should keep at it because this book is great.

 

[atyy]

@verty, I object! K&K is a warm and fuzzy book. Nothing wrong with reading the text and doing problems from elsewhere.

 

[verty]

@verty, I object! K&K is a warm and fuzzy book. Nothing wrong with reading the text and doing problems from elsewhere.

Well Cosmophile certainly didn't seem to like the actual text too much. If he is an example of a person who would want to do that, probably it wouldn't be better than just choosing another book. But I think it just surprised him that it seemed to be a bit, um, gladiatorial. It's like, okay now defeat the problems.

I can see why someone would find that to be a bit adverse and not in the usual style of a textbook. But as I said, one should think in the Russian way, I need to do this because this is the book. And that's all I have to say about that.

 

[Cosmophile]

I should clarify: I am not upset with the mathematical nature of the book (in fact, I'm thrilled by it). I was one such person who experienced a very spoon-fed version of physics in high school (and again in my community college), which spent a lot of time on projectile motion. All of the "big 5" equations of kinematics were explicitly put forth. I suppose one could say I was "absolutely taken by surprise," instead of "horribly dissatisfied." Clearly, the coverage of Kinematics is sufficient (if it were not, I doubt the book would be so highly regarded!). It simply was not at all what I was expecting. I'm taking the time to push through it, though, and I can tell that it is paying off. My purpose in posting this was to see if others had felt a similar shock when they reached the problems section. It seems I'm not too alone.

I should probably change my previous post further but that would be editing way beyond when one should edit. If you imagine that a Russian wrote it, it'll come across in the right way. The Russian society is much more competitive especially in sports, and people can be ice cold to each other when it comes to getting ahead. That is the kind of steel-hearted mentality you want to have with this book. I want to be the best, I want to use the best book, I won't let anything stop me.

One can see for example that this is a concept that Stallone thought about because his Rocky movies were based around that, the idea that one gets little if one doesn't fight. So that is the interpretation for my previous post. Cosmophile has entered the arena and got punched in the face by K&K and he doesn't seem to be bleeding just yet. He should keep at it because this book is great.

10/10 comment. For the record, I took no offense to your comment (nor anyone else's). I picked this book for a reason, and that reason is that it seems to be geared for those who wish to be the best. Now, to get back to the problems. (Round 2: FIGHT!)

 

[atyy]

I beg to differ again. K&K is one of my favourite books. Read Halliday and Resnick and Young and do the problems from there, then take a look at K&K and it will seem very sweet and gentle, like ...

 

[WannabeNewton]

K&K are great, but you don't have to be able to do their problems to master classical mechanics - you can try the problems from Halliday and Resnick or Young. It's fun to do a couple of K&K problems, but by and large they are far more difficult than one needs unless one is a masochist.

K&K problems are actually ideal for someone who wants to be a good physicist. They teach very important techniques that are used time and again in more advanced physics classes. The masochistic problems are really in books like Morin and Irodov.

 

[atyy]

K&K problems are actually ideal for someone who wants to be a good physicist. They teach very important techniques that are used time and again in more advanced physics classes. The masochistic problems are really in books like Morin and Irodov.

But aren't you into numerical GR?

 

[WannabeNewton]

But aren't you into numerical GR?

I do theoretical GR actually. And the techniques I've learned from doing problems in K&K have certainly come in handy in numerous physics classes. Plus the problems are really fun to do; they're worth doing just for the satisfaction you get from solving them.

 

[verty]

I beg to differ again. K&K is one of my favourite books. Read Halliday and Resnick and Young and do the problems from there, then take a look at K&K and it will seem very sweet and gentle, like ...

Are you saying the questions only seem difficult but once you lose your fear of them, they are easy?

 

[atyy]

I do theoretical GR actually. And the techniques I've learned from doing problems in K&K have certainly come in handy in numerous physics classes. Plus the problems are really fun to do; they're worth doing just for the satisfaction you get from solving them.

Ah that's interesting. I was going to say anyone who does numerical GR is a masochist. Maybe GR theorists can qualify too.

 

[atyy]

Are you saying the questions only seem difficult but once you lose your fear of them, they are easy?

I'm saying that K&K is a great piece of literature, and so people will have different interpretations of it. I don't think I like being punched in the nose, but I like K&K very much.

 

[verty]

People are going to have to make their own decision, that is the bottom line.

 

[atyy]

I do theoretical GR actually. And the techniques I've learned from doing problems in K&K have certainly come in handy in numerous physics classes. Plus the problems are really fun to do; they're worth doing just for the satisfaction you get from solving them.

I guess I remembered you were reading about point particles and small bodies in GR, and somehow I classified that under "numerical GR", but actually the work is all done by theorists. Is that at least still part of what you are interested in? I think that would count as masochistic by most people's standards.

On the other hand, I remember some talk where the guy said something like these other guys did all these things in 10 years, and we used mathematica and got lots more in a few months (very broad paraphrase, I can't remember who said this).

(Anyway, yes, I agree the K&K problems are fun, at least they were for me, since I didn't do them with any grades on the line.)

 

[WannabeNewton]

Is that at least still part of what you are interested in?

Indeed; I'm still working on stuff in that area.

 

[vanhees71]

Well, so different people can think about analytical mechanics. For me analytical mechanics was a revelation. It got me rid of the clumpsy Newtonian formalism of forces, and you start thinking in terms of symmetries. I remember when we were questioned in the beginner's lab when we did the Atwood machine to derive its theory. I wrote down the Lagrangian, and the whole thing was done in 5 minutes. The tutor was baffled and said that this was the record in finishing the theory part quickest. Even simple problems get even more simpler with analytical rather than naive mechanics. As physical theories they are, of course, absolutely equivalent, and if you can solve a problem better using Newton's fomralism (or better said Euler's, because I guess nobody among us would be able to do a problem in the purely geometrical way as Newton wrote in his Principia ;-)), then use it. It's as good as the analytical approach.

If the purpose to do mechanics is, however, to prepare for quantum theory, there's no way out. You must learn Hamilton's canonical formalism, because otherwise you wan't have a chance to grasp the fundamental ideas behind quantum theory.