Kleppner/Kolenkow: Treatment of Kinematics

[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.