Reading Goldstein's Classical Mechanics as an Undergraduate

[IAmLoco]

We were prescribed Goldstein, Taylor and Marion/Thornton for our first course in analytical mechanics, and I'm about to finish up the course but I feel like I have not gotten a good physical, intuitive grasps of the concepts, so I've been trying to read the texts a bit more.

Taylor and Marion-Thronton have been okay so far, but Goldstein has proved to be a challenge. I understand that Goldstein is for the most part, a graduate level text, while some have used it as an advanced undergraduate text in their institutions. With this in mind, how much of Goldstein should an undergraduate read and try to understand?

I first tried to go through Goldstein's derivation of Lagrange's equations from D'Alembert's Principle(which was not once mentioned in my course, I am actually wondering if things like the virtual displacement/work, D'Alembert's and least action principle are usually discussed in undergraduate analytical mechanics), which was quite tough, so I'm a bit lost on how to proceed.

 

[hutchphd]

Why do you ask? You should learn from the materials you find most conducive. Presumably your course in analytical mechanics covered a representative sampling of a typical undergraduate course. I would recommend that you revisit the coverage of that course until you are comfortable with it. If you find something that interests you particularly, dive right in using everything available.
My first (and only, actually) analytical mechanics course was from Goldstein as a a Junior. It was hard as heck. I am still ambivalent about the book and it has been on my shelves for >45 years.

 

[marcusl]

You will have a good undergrad-level grounding in mechanics if you learn everything in Marion (there was no Thornton when I was in school). Goldstein was used in grad school.

 

[mpresic3]

Goldstein is my favorite physics book. I would say that you can have a strong undergraduate education without any Goldstein. I read the beginning section as an undergraduate (first edition) but did not understand it. I am told the material on holonomic constraints is at best misleading.

Burke's book on Applied Differential Geometry, and other informed members of this forum really pans the treatment of holonomic constraints in Goldstein. I am not comfortable with the delta symbols expressing variation and virual work either.

I like Goldstein, but I feel the first sections are not as well written and the sections on rigid body motion, orbits, and oscillations, and special relativity and Hamilton's treatment are best. I would not at all be concerned if you passed over the first sections in Goldstein, in favor of your intermediate mechanics textbook. I think even at the graduate level, few graduate students are happy with the early sections.

 

[PhDeezNutz]

Speaking about virtual work and non-holonomic constraints.

When I was reading Goldstein I thought “virtual displacements” were completely arbitrary. Apparently they are not.

https://iopscience.iop.org/article/10.1088/0143-0807/27/2/014/meta

and as @mpresic3 points out that Goldstein’s treatment of non-holonomic constraints might be wrong. I am not sure, you would have to ask @wrobel or @vanhees71.

I believe the formula in the latest print of Goldstein has come to be known as “vakonomic”. It has been shown that this formula doesn’t satisfy the zero virtual work principle.Also I still don’t understand canonical transformations and action angle variables.

 

[vanhees71]

Goldstein doesn't treat non-holonomic constraints right when treating them within the action-functional formalism. AFAIK with D'Alembert's principle he does it right, and the correct treatment of non-holonomic constraints within the action-functional formalism (Hamilton's principle) must be equivalent to this. A much better book on classical mechanics is Landau & Lifshitz vol. 1, and there everything is correct.

 

[wrobel]

Let me just recall the following
[Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics) (Anthony Bloch, et al]:

 

[mpresic3]

The general consensus about reading Goldstein as an undergrad is: ?

Just wondering if anyone else found as I did that Goldstein's later chapters seemed to be better written than his first two. Also I know L&L mechanics was recommended, but I found it to be very terse, and did not cover all the material in Jackson.

I think some may agree that although Sommerfeld's first book, Mechanics, is very discursive, it is also very interesting. I think I might recommend that to an advanced undergraduate too. (Any thoughts)

I do like L&L mentioning the exact solution to Euler's free body equations as elliptic functions, (actually theta functions), but this is too advanced even for most graduate students. Whittaker (Analytic treatment of Rigid Bodies) ,and MacMillan, and Ames, and many early treatments develops this more thoroghly, though

 

[wrobel]

I think D. Greenwood's Classical Dynamics is very good anyway

 

[wrobel]

Euler's free body equations as elliptic functions, (actually theta functions), but this is too advanced even for most graduate students

This is just useless. Who needs solutions in elliptic functions when we have a computer on the table? We need qualitative analysis of the equations and qualitative understanding of the motion. This understanding is not based on explicit formulas in wild special functions.

 

[vanhees71]

The general consensus about reading Goldstein as an undergrad is: ?

Just wondering if anyone else found as I did that Goldstein's later chapters seemed to be better written than his first two. Also I know L&L mechanics was recommended, but I found it to be very terse, and did not cover all the material in Jackson.

I think some may agree that although Sommerfeld's first book, Mechanics, is very discursive, it is also very interesting. I think I might recommend that to an advanced undergraduate too. (Any thoughts)

I do like L&L mentioning the exact solution to Euler's free body equations as elliptic functions, (actually theta functions), but this is too advanced even for most graduate students. Whittaker (Analytic treatment of Rigid Bodies) ,and MacMillan, and Ames, and many early treatments develops this more thoroghly, though

Indeed Sommerfeld's lectures are hard to top. For me they are the best theoretical-physics textbooks ever written. I never understood why Goldstein is so much in favor.

 

[vela]

The general consensus about reading Goldstein as an undergrad is: ?

It's overkill. For the typical undergrad, a less advanced text would be better from a pedagogical standpoint, which seems to reflect the experience of the OP. That's not to say an undergrad shouldn't consult Goldstein if interested, but he or she shouldn't be too concerned if they can't fully digest it yet.

 

[mpresic3]

This is just useless. Who needs solutions in elliptic functions when we have a computer on the table? We need qualitative analysis of the equations and qualitative understanding of the motion. This understanding is not based on explicit formulas in wild special functions.

Not always. (A few physicists will always need "wild" special functions) For my work, I had to consider the force-free equations of motion for a rigid body with three unequal moments of inertia. At first, I used a Runge-Kutta to integrate the equations of motion. For the selected spin rates, and the time interval I needed the solutions for, I (and my bosses) were worried that numerical errors in each integration step would degrade the solution when run for a duration. I was able to use the properties of the jacobi elliptic functions and theta functions in the advanced texts, and Abramowitz and Stegun, to use infinite series for these functions, so that integration was non needed, and these solutions could be used as a check on the integration. The computer on the table was used in both cases.

The best check on computer analysis is often a good "toy" technique to be used. The more involved the example, the more it tests the computer analysis. Goldstein and other texts agree after doing a certain amount, the exact full solution would be in his words unrevealing, and moves on to qualitative solutions.

D.F.Lawden suggests in his textbook on elliptic functions that earlier generations of physicists were more conversant with elliptic functions. I am not in the camp that every physics undergrad needs to be educated in this area, the way they are complex variables, (which is critical in the theory of elliptic functions). Perhaps educators understand that physicists need complex variables, just in case there are a few of them that will need them later in studying special functions, including elliptic functions.

 

[mpresic3]

BTW, I like Wrobel's suggestion of Greenwood. Symon, also has a good book on Classical Mechanics.

 

[vanhees71]

Goldstein doesn't treat non-holonomic constraints right when treating them within the action-functional formalism. AFAIK with D'Alembert's principle he does it right, and the correct treatment of non-holonomic constraints within the action-functional formalism (Hamilton's principle) must be equivalent to this. A much better book on classical mechanics is Landau & Lifshitz vol. 1, and there everything is correct.

I must qualify this criticism. It's unjust to blame Goldstein for the blunder concerning nonholonomic constraints within the action principle. It's only the additional authors of the 3rd edition who introduced this nonsense. In the 2nd edition it's correctly treated as constraints on the variations ("virtual displacements") and thus in accordance with d'Alembert's principle and last but not least Newton's equations of motion as it should be.

There's also a nice paper with an experimental test with a two-wheeled cart of standard Newtonian vs. vakonomic dynamics showing that the latter leads to wrong results for a simple example:

https://doi.org/10.1119/1.1701844

Unfortunately there seems to be no free preprint version.

 

[George Jones]

In the 2nd edition it's correctly treated as constraints on the variations ("virtual displacements") and thus in accordance with d'Alembert's principle and last but not least Newton's equations of motion as it should be.

The second edition of Goldstein was used as the text for two undergrad classical mechanics courses that I took.

 

[vanhees71]

Why the heck must people take a textbook, which is considered a classic by many and make it worse?

I once glanced at a "modernized" version of the famous Courant&Hilbert two-volume book. It was a sacrilege!

 

[robphy]

Goldstein (2ed) was used in an advanced undergraduate course I audited [at a nearby university when I was looking to transfer from an engineering program to a physics program somewhere]. It was challenging because I only had an intro course at the time. I did later take a course using Symon's text at the institution I eventually transferred to.

While it was difficult, one lesson I took from it was that
it offered a glimpse of what would be important in my later coursework.
So, when getting my assigned textbook for a course, I would get the next textbook to get a sense of what will be important later. It was, of course, more important to understand my assigned textbook. But I would occasionally skim the advanced book, without letting that advanced book discourage me when I got stuck.

Some undergraduate texts and some engineering dynamics texts do make use of "virtual work".

 

[wrobel]

O yes, it is better to write modern books than refresh outdated ones.
Regarding Courant&Hilbert there are two completely different theories of PDE: PDE before functional analysis and Sobolev spaces and PDE of functional analysis and Sobolev spaces. The Courant&Hilbert textbook belongs to the first kind

 

[andresB]

The general consensus about reading Goldstein as an undergrad is: ?

In my undergrad we took two courses in intermediate mechanics. One on non-analytical mechanics with Marion and the second one on Lagrangian and Hamiltonian formalism with Goldstein. I just can't fathom an undergrad curriculum that doesn't include canonical transformations, Poisson brackets and the Hamilton-Jacobi theory, and Goldstein (2ed) does quite well with those topics.

Now, for D'Alembert's Principle and constraints, I do support Wrobel's recomendation of D. Greenwood's analytical dynamics.

 

[dextercioby]

O yes, it is better to write modern books than refresh outdated ones.
Regarding Courant&Hilbert there are two completely different theories of PDE: PDE before functional analysis and Sobolev spaces and PDE of functional analysis and Sobolev spaces. The Courant&Hilbert textbook belongs to the first kind

For the second, do you know a better treatment than the one by V.S. Vladimirov?

 

[jasonRF]

I just can't fathom an undergrad curriculum that doesn't include canonical transformations, Poisson brackets and the Hamilton-Jacobi theory

At least for the US, I have the impression that those topics are not in every physics departments’ undergraduate curriculum. I suspect that curriculum choice has a large impact on whether Goldstein is assigned for courses.

Where I went the physics department taught two versions of the upper-division mechanics course. I was an EE major and took the easier one that was at the level of Marion, although we covered the topics in a different order (Lagrangians and Hamiltonians were covered earlier) and added some topics like perturbation theory applied to parametric oscillators. It was mostly filled with physics majors who had goals besides physics grad school (there were at least a couple of students preparing for medical school). The harder version covered through Hamilton-Jacobi theory and usually used either Goldstein or Landau and Lifshitz. It also had more advanced math prerequisites.

jason

 

[andresB]

At least for the US, I have the impression that those topics are not in every physics departments’ undergraduate curriculum. I suspect that curriculum choice has a large impact on whether Goldstein is assigned for courses.

Where I went the physics department taught two versions of the upper-division mechanics course. I was an EE major and took the easier one that was at the level of Marion, although we covered the topics in a different order (Lagrangians and Hamiltonians were covered earlier) and added some topics like perturbation theory applied to parametric oscillators. It was mostly filled with physics majors who had goals besides physics grad school (there were at least a couple of students preparing for medical school). The harder version covered through Hamilton-Jacobi theory and usually used either Goldstein or Landau and Lifshitz. It also had more advanced math prerequisites.

jason

Well, I've never understood the US system, that's might be the root of my confusion with this thread.

 

[robphy]

In my undergrad we took two courses in intermediate mechanics.
…
I just can't fathom an undergrad curriculum that doesn't include canonical transformations, Poisson brackets and the Hamilton-Jacobi theory, and Goldstein (2ed) does quite well with those topics.

In my experience as a student and as a professor in the US, a two-course intermediate-mechanics sequence for undergraduates is more likely at a PhD-granting institution, but less likely at a liberal arts college (including the some of the more elite ones).

At liberal arts colleges and at primarily undergraduate-institutions, the large number of core and major credits needed to graduate, the small number of students in these classes, and the small number of available faculty members make such a two-course sequence difficult to offer regularly. Those more advanced mechanics topics will be seen later by students who go on to grad school in physics.

 

[andresB]

I...have no idea what a liberal arts college is.

I do have to admit that the curriculum at my university was a little too ambitious for a brand new program (It was only year and half old when I started it) in a non-phd granting 3rd world country university. They toned it down since.

Though, Goldstein is way more accessible than the book we used in grad school, Jose & Saletan.

 

[Dr.D]

I got a whole lot out of Goldstein. It was enough that when I encountered Greenwood in another course, it appeared pretty elementary.

 

[gmax137]

I...have no idea what a liberal arts college is.

A liberal arts college generally grants Bachelor's degrees (very few if any Masters or PhDs). These schools also require the students to take classes outside their major. So as a physics major at a liberal arts college, I took just over half my classes in physics and math, with others such as:
english
history
psychology & sociology
economics
art history
philosophy
chemistry
classics (greek plays)
etc.

 

[robphy]

A liberal arts college generally grants Bachelor's degrees (very few if any Masters or PhDs). These schools also require the students to take classes outside their major. So as a physics major at a liberal arts college, I took just over half my classes in physics and math, with others such as:
english
history
psychology & sociology
economics
art history
philosophy
chemistry
classics (greek plays)
etc.

It might be enlightening to give a rough schedule of your sequence of courses.
And how many students were in the class.
(In some places, some courses were offered in alternating years.)

 

[gmax137]

It might be enlightening to give a rough schedule of your sequence of courses.
And how many students were in the class.
(In some places, some courses were offered in alternating years.)

Hmm I would have to find my transcript for detail (1978).

We had four classes, in each of two semesters (fall and spring) plus a single concentrated class in January. The first two years I probably took one physics and one math class per semester, then in the 3rd and 4th years it would be three (physics and math).

The physics sequence was mechanics, EM, thermo/modern, quantum, fluids, solid state, more quantum, more classical mech. With a couple math methods classes taught in the physics dept. The fluids and solid state classes were "major electives" I think. Oh, I had a class in electronics and circuits, transistors etc.

Math in the math department was multivariable calculus, linear algebra, real analysis, complex analysis. I guess diff eq was in the math methods classes.

My January "Winter Study" was Fortran programming, lab work with one of the physics professors, and twice in the physics dept machine shop.

The early physics classes had maybe 20 students, dropping down over the years to about 8. That solid state physics class I remember was 4 students.

The other (non major) classes could be large (100+ in chemistry or psychology, these are pre-med classes), or quite small (a dozen in philosophy).

I hope that helps.

EDIT - the student body was about 400 per class, or 1600 total.

 

[andresB]

So as a physics major at a liberal arts college, I took just over half my classes in physics and math, with others such as:
english
history
psychology & sociology
economics
art history
philosophy
chemistry
classics (greek plays)
etc.

...why?

I mean, the US is a powerhouse in science and math, so, the system has to work...somehow. But I just don't get it.

I think the only non-physics courses I took (and they were mandatory) were Spanish (for writing skills), chemistry (I should have paid more attention), and job-related law. I'm counting history of physics as a physics course.

 

[robphy]

Hmm I would have to find my transcript for detail (1978).
...

I have a feeling that your college is one of the elite Little Ivies.

 

[gmax137]

...why?

I can think of several reasons. In no particular order:

Do most 17 or 18 year old college freshmen really know they want to study physics or classical literature? How would they know? Why not allow them to try classes in each (and more) and let them decide later what they want to study in depth?

Must a physics graduate think and talk only about physics? Wouldn't it be nice to know at least something about subjects beyond your major?

Is the purpose of the college/university training or education? The liberal arts approach is trying to teach the students how to think critically, how to write clearly, how to study, how to learn. With that, you can spend the rest of your life studying and learning whatever you choose. The variety of subject matter helps because the study habits are different: for physics you must do problem sets, for math you must do proofs. What would a history or literature "problem set" look like?

I went on to graduate school in engineering. Quite a different experience, much more emphasis on gaining specialized knowledge (almost approaching training). Much more pointed at future employment.

 

[gmax137]

I have a feeling that your college is one of the elite Little Ivies.

I wonder how many Ephs have found their way to physicsforums?

 

[wrobel]

For the second, do you know a better treatment than the one by V.S. Vladimirov?

I like

Michael Renardy Robert C. Rogers: An Introduction to
Partial Differential Equations

L. Evans: Partial Differential Equations

Michael Taylor: PDE

M. Shubin: Lectures in PDE (in Russian, perhaps there exists in English)

O. Oleinik: Lectures in PDE (in Russian, perhaps there exists in English)