Classical Kleppner/Kolenkow: Treatment of Kinematics

[QuantumCurt]

For whatever it's worth, I do feel like the first part of K&K that covers vectors and kinematics is the "worst" part of the book. I've been working through this book again recently as a refresher since I'm starting junior level classical mechanics this fall. I think a good amount of prior knowledge is assumed in this chapter. It's also true that many conventional physics textbooks tend to spend far too much time on kinematics, which somewhat convolutes the topic.

 

[robphy]

For whatever it's worth, I do feel like the first part of K&K that covers vectors and kinematics is the "worst" part of the book. I've been working through this book again recently as a refresher since I'm starting junior level classical mechanics this fall. I think a good amount of prior knowledge is assumed in this chapter. It's also true that many conventional physics textbooks tend to spend far too much time on kinematics, which somewhat convolutes the topic.

It seems to me the issue is mathematical preparation.
K&K assume that you had a good course in calculus and analytic geometry.
The conventional physics textbooks assume you didn't.

 

[QuantumCurt]

I've completed calculus I, II, and III plus courses in both differential equations and linear algebra. So mathematical preparation isn't really an issue for me at this point. The conceptual discussions on kinematics in K&K just come across as a bit vague to me. I get that they're trying to demonstrate the simplicity of kinematics, but I don't feel that they convey it well. They take things for granted that should really be specified, and some of the assumptions and methods that are utilized in the example problems aren't really conveyed at all in the text preceding them. I've never really felt this way about any other sections of the book. The kinematics section just feels rushed to me.

 

[verty]

I quote from a post not on this site but on stackexchange.com.

For example, I had lot of difficulties with the book "An Introduction to Mechanics" by Daniel Kleppner, Robert J. Kolenkow, which seemed according to many views to be an easy approach toward Newtonian and relativistic mechanics. The authors in general only and quickly pushes equations in my front without giving any reason for why a certain procedure is correct, and give no explanation on most of the things. I had then one choice: search on the net. But when I do, to search for a term X, I get to wikipedia page X, who give a definition that contains another term Y, where I click to understand the full meaning of term X, but who then contain another term Z, who redirects to... which leaves me with no understanding.
...
I feel like: Mechanics is not well organized. For example, in relativity we first learn about Galilean relativity, then special relativity then general relativity. Everything is in order and it makes of the understanding a lot smoother. (according to my friend) But in classical mechanics I don't know where to start or what to pick.

In Lagrangian and Hamiltonian mechanics book, it is even worse.

Result? I fail to correctly answer some basic questions like: what happens when a cup of water starts to melt? or even more easy physics questions.

So I'm searching for a clear textbook that explains Newtonian mechanics well, then goes to special relativity, then to Lagrangian and Hamiltonian mechanics.

Do you see how this guy's desire to have the book be how he wants prevents him from learning the subject? He likes how relativity makes sense (huh?), is ordered and is well-behaved, but mechanics he finds to be disordered. I guess he sought out a book that was clear, was recommended K&K, searched for reasons and couldn't find them, etc.

Clearly he had a very bad experience. He is still looking for a book on Newtonian mechanics. Why is he still looking for a book, has he not learned it yet? He's the type of person that can't be helped because he wants everything to be his way, how he wants.

I just want to say, don't be like this guy. Don't get so caught up in not liking how a book is written that you fail to see that it is a jewel. This book is a jewel. If one chapter is less good, so what? Does it need a book club meeting about it? The only masochists are people like this guy I quoted who will never find a book they like. The people who take a book like this one, find the questions tough and rise to the challenge, are the ones who aren't masochists because they are saving themselves a ton of trouble by learning the stuff the right way, so that they can solve problems with it. They have faced the problems, they have conquered the problems, they know mechanics. For me, every book should be like this one but at the specific level for the audience. This one is meant for top tier students at schools like MIT who will credit out of calculus. If you're expecting it to be a walk in the park, observing all the physical daisies, it's not going to be like that. It's going to be rigorous, it's going to be tough.

But it's a good book and you need to rise to the challenge. That's all I'm going to say about that.

 

[atyy]

For whatever it's worth, I do feel like the first part of K&K that covers vectors and kinematics is the "worst" part of the book. I've been working through this book again recently as a refresher since I'm starting junior level classical mechanics this fall. I think a good amount of prior knowledge is assumed in this chapter. It's also true that many conventional physics textbooks tend to spend far too much time on kinematics, which somewhat convolutes the topic.

Are you using the blue or red pill?
https://www.amazon.com/dp/0070350485/?tag=pfamazon01-20
https://www.amazon.com/dp/0521198119/?tag=pfamazon01-20

 

[QuantumCurt]

Are you using the blue or red pill?
https://www.amazon.com/dp/0070350485/?tag=pfamazon01-20
https://www.amazon.com/dp/0521198119/?tag=pfamazon01-20

I'm using the red pill.

 

[verty]

About the first chapter, I think the authors assume one is learning multivariable calculus simultaneously with their book. So for example, one would be learning about vectors at the same time as doing the kinematics stuff. The angle between vectors, all that stuff would be covered in the first two lectures of MVC. This may explain why they reduce stuff to math and then just leave the reader to figure it out. And certainly later in the book they use MVC stuff like partial derivatives.

So I do recommend users of this book learn MVC at the same time. Clearly it has steep requirements, I think no one can doubt that.

 

[robphy]

Are you using the blue or red pill?
https://www.amazon.com/dp/0070350485/?tag=pfamazon01-20
https://www.amazon.com/dp/0521198119/?tag=pfamazon01-20

I'm using the red pill.

It seems that the "Preface" and "To the Teacher" sections were greatly shortened in the new [red] version.

I've quoted parts of the first edition, which might explain why kinematics and Ch 1 may appear rushed.

[blue] http://hep.ucsb.edu/courses/ph20/kkfront.pdf

"Our book is written primarily for students who come to the course knowing some
calculus, enough to differentiate and integrate simple functions. It has
also been used successfully in courses requiring only concurrent registration in
calculus. (For a course of this nature, Chapter 1 should be treated as a
resource chapter, deferring the detailed discussion of vector kinematics for a
time. Other suggestions are listed in To The Teacher.)"

In "To the Teacher"...
If the course is intended for students who are concurrently beginning their
study of calculus, we recommend that parts of Chapter 1 be deferred. Chapter 2
can be started after having covered only the first six sections of Chapter 1.
Starting with Example 2.5, the kinematics of rotational motion are needed; at
this point the ideas presented in Section 1.9 should be introduced. Section 1.7,
on the integration of vectors, can be postponed until the class has become
familiar with integrals. Occasional examples and problems involving
integration will have to be omitted until that time. Section 1.8, on the
geometric interpretation of vector differentiation, is essential preparation
for Chapters 6 and 7 but need not be discussed earlier.

[ red ] http://assets.cambridge.org/97805211/98110/frontmatter/9780521198110_frontmatter.pdf

 

[OldEngr63]

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")?

To answer your question based on my own experience, the Hamiltonian formulation is not much help in everyday engineering problems at all. What is a lot of help, particularly in "mixed" type problems is Hamilton's Principle, where by "mixed" I am thinking of something like an electromechanical or electroacoustic or continuum mechanics with some lumped elements included. In all of these cases, Hamilton's Principle can be a great help, but Hamiton's formulation, not so much.

 

[atyy]

To answer your question based on my own experience, the Hamiltonian formulation is not much help in everyday engineering problems at all. What is a lot of help, particularly in "mixed" type problems is Hamilton's Principle, where by "mixed" I am thinking of something like an electromechanical or electroacoustic or continuum mechanics with some lumped elements included. In all of these cases, Hamilton's Principle can be a great help, but Hamiton's formulation, not so much.

Could you have more specific examples in which Hamilton's Principle is useful in everyday engineering? An electroacoustic one would be particularly cool.

 

[Dr. Courtney]

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

Dan Kleppner was my thesis advisor, and he is one of the sharpest physicists I have ever known. The emphasis is in training physicists rather than engineers.

 

[Dr. Courtney]

To answer your question based on my own experience, the Hamiltonian formulation is not much help in everyday engineering problems at all. What is a lot of help, particularly in "mixed" type problems is Hamilton's Principle, where by "mixed" I am thinking of something like an electromechanical or electroacoustic or continuum mechanics with some lumped elements included. In all of these cases, Hamilton's Principle can be a great help, but Hamiton's formulation, not so much.

Could you have more specific examples in which Hamilton's Principle is useful in everyday engineering? An electroacoustic one would be particularly cool.

Hamilton's Principle leads to Hamilton's equations. When calculating the time evolution of a particle (a trajectory or orbit), as a practical matter, it is easier to integrate Hamilton's equations (first derivatives) than Newton's equations (acceleration and velocity with first and second derivatives.

When you actually write the code to do it, it is clear why.

 

[atyy]

Dan Kleppner was my thesis advisor, and he is one of the sharpest physicists I have ever known. The emphasis is in training physicists rather than engineers.

Hi Dr. Courtney! I saw the link to BTG Research on your PF profile and did wonder whether Kleppner was your supervisor (I didn't know whether you were Michael or Amy, since there are two researchers on that site). Anyway, although we have never met, I have actually read quite a bit of your PhD thesis! I was working on a senior thesis with Xiao-Gang Wen on quantum chaos, and Dan Kleppner gave an IAP class on something related (I can't remember), and somehow I ended up chatting with him in his office, and he gave me your thesis and recommended I read it. Anyway, although I am a biologist, I do think the measurements you, and the many others Kleppner did, are very beautiful. My measurements are considerably coarser (I can't even get voltage to within 5 mV) but I hope to get closer someday :)

 

[Dr. Courtney]

Hi Dr. Courtney! I saw the link to BTG Research on your PF profile and did wonder whether Kleppner was your supervisor (I didn't know whether you were Michael or Amy, since there are two researchers on that site). Anyway, although we have never met, I have actually read quite a bit of your PhD thesis! I was working on a senior thesis with Xiao-Gang Wen on quantum chaos, and Dan Kleppner gave an IAP class on something related (I can't remember), and somehow I ended up chatting with him in his office, and he gave me your thesis and recommended I read it. Anyway, although I am a biologist, I do think the measurements you, and the many others Kleppner did, are very beautiful. My measurements are considerably coarser (I can't even get voltage to within 5 mV) but I hope to get closer someday :)

Thank you for the kind words.

All the trajectory calculations in my thesis used integration of Hamilton's equations.

In addition to only having first derivatives (easier to integrate than second derivatives), a second advantage of Hamilton's equations is the relative simplicity of dealing with the scalar potential energy rather than vector forces.

 

[vanhees71]

Of course, the best way to express fundamental physical laws is the action principle in both the Lagrange and the Hamilton formulation. While the former is simpler concerning the formulation of relativistic point-particle and field systems in a manifestly covariant way the latter provides the algebraic formulation needed to switch to quantum theory easily. For classical point-particle systems, when treated numerically the Hamilton formulation is of advantage, because it provides a first-order differential-equation scheme, and you can get accurate results by using algorithms employing the symplectic structure of the phase space.

I don't understand, what's still the issue with kinematics of Newtonian mechanics. If you do "naive" Newtonian mechanics, which you should when starting physics, then it boils down to the definition of the position vector as a function of time to describe the trajectory of a point particle (or of several such vectors when describing many-body systems). Then the velocity is the time derivative of the position vector and acceleration the time derivative o the velocity vector.

The only somewhat more complicated subject is the choice of the comoving dreibein, which provides a coordinate free (intrinsic) characterization of a curve and thus also a trajectory of a point particle. However, this can be easily omitted in the first attempt to learn classical mechanics. The main subject to be learned in Newtonian Mechanics are the techniques to describe motion in terms of ordinary differential equations and their solution.

Of course, analytical mechanics is much more elegant and on a higher level simpler than the naive approach, but you need more advanced mathematics, including variational calculus. In my opinion this course should finish with an introduction of Lie groups and algebras, using the Poisson-bracket formalism, because that makes this rather abstract-looking methods pretty intuitive.

 

[OldEngr63]

When you actually write the code to do it, it is clear why.

This is true (up to a point), but ease of integration is not the only consideration.

For many purposes, we really can't make much use of momenta, but we sure would like to know velocities. This is often a simple conversion, but not in all cases. Those odd cases can be very computer resource intensive.

The real rub about using Hamilton's equations in classical physics come when you try to include things like Coulomb friction in a problem. It just does not fit very easily (nor does linear viscous friction, or v^2 type friction, etc.)

The advantage to using Hamilton's Principle for mixed systems is that the coupling terms are formulated automatically in the process as a result of the integrations by parts.

 

[OldEngr63]

Could you have more specific examples in which Hamilton's Principle is useful in everyday engineering? An electroacoustic one would be particularly cool.

The best book ever written (IMO) on engineering applications of Hamilton's Principle is

Mechanical and Electromechanical Systems
by Crandall, Karnopp, Kurtz, and Pridmore-Brown
McGraw, 1968.

One of the distinctive features of this book is the careful distinction between energy and co-energy that is maintained throughout. This facilitates dealing with nonlinear constitutive relations.

It is filled with engineering applications of all sorts. In particular, on p. 380-385, there is an electroacoustic example.

If you feel up to it, try setting up for yourself a Hamilton's Principle model for an electromechanical sonar transducer. This system has electrical aspects (including piezoelectricity), mechanical aspects (wave propagation down the length of the stack) and acoustics (radiation into the water). It is a challenging problem, but Hamilton's Principle would make it much easier than any other approach.

PS: This book was not a great success sales-wise, I think. It relied on the ability to introduce the Calculus of Variations to undergraduates (it is an undergrad book), and that is a very difficult feat to pull off. I have tried it several times with only limited success.