Trouble with classical mechanics self-learning (How to avoid going down the Physics rabbit hole?)

Question

I'm a retired police officer trying to learn classical mechanics on my own. I have gone through many links on the Internet including the classical mechanics quick reference textbooks from Physics Stack Exchange. But, I always have the same problem just as anyone trying to learn classical mechanics on his/her own has had the experience of "going down the Classical Mechanics Rabbit Hole".

It turns out that only classical mechanics is the most difficult part of physics to learn on ones own. I had a friend who confirmed this by comparing how difficult it is to learn classical mechanics (including Lagrangian and Hamiltonian formulation) on his own with electrodynamics and general relativity. (Who are much much more difficult that all the field of CM)

For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn what a field is, but it's the same story all over again: all the definitions you find refer to something called a group. Off to learn about what a group is. Ad infinitum. That's what I'm calling here "to go down the Math Rabbit Hole."

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.

Another thing is that when I go here on Physics Stack Exchange, and when I see answers like:

• https://physics.stackexchange.com/a/14752/
• https://physics.stackexchange.com/a/67705/
• https://physics.stackexchange.com/a/71093/
• https://physics.stackexchange.com/a/64976/
• and many many others... (like an answer by David Z for a question that kinda looks like: 'Would a heavier object fall faster because they attract earth stronger', I have no idea where he found the equations he wrote down. Also in many applied physics questions and answers by Lubos Motl. And in some questions: like: 'Why Newton's third law apply to all inertial frame?' I have no idea about that even if I already learn a lot from Daniel's book.)

I don't know where those guys got all that stuff. 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.

My dream for the next years of my life is to understand mechanics: Newtonian, SR, Lagrangian and Hamiltonian. And to start writing a web page about explanations of different phenomena like John Baez this week on mathematical physics. And maybe to do research on problems in classical physics which would make of me the most happy man in the world.

Regards. Thanks for your understanding and time. My situation is similar to this guy

My background: I'm very old, so I forgot almost all the math/physics I've got in school, however, I've taken courses on Algebra, trigonometry and single variable calculus using KhanAcademy and some MIT videos. I've taken an MIT test on CalcI (just downloading the test online and verifying the solutions) and I scored 90%.

Answer 1

You just need an introductory book. Kleppner and Kolenkow is for ambitious college students who've already had a year-long high school physics course. And of course you don't understand when people are talking about mechanics on this site. I don't understand random passages of Chinese. If I want to learn Chinese, I'll have to start from the basics.

Physics majors generally don't learn Lagrangian and Hamiltonian mechanics until their third year of college, meaning they've taken quite a lot of physics classes before they get to it.

Most of the recommendations people are giving here are not at the right level. It's easy to forget how hard it was to learn early on. Taylor's book, for example, is good, but too advanced for you. Things like Spivak's Calculus on Manifolds are ludicrous to suggest to someone who's only just gone through some basic Khan-academy style calculus. You don't need to know about affine geometry or symplectic endomorfisms (whatever they are) in order to learn basic mechanics.

The class of books to start with have titles like "Conceptual Physics". I haven't read those books, so I don't know which one to recommend. I'd suggest looking at Amazon reviews of books called "Conceptual Physics" and picking one of those. These books are chatty and can be read fairly quickly. Once you read that, try the first half of Lewis Epstein's Thinking Physics to see how much you understood. You will find that there was a lot that seemed like it made sense before, but suddenly became confusing or tricky when you try to solve Epstein's problems. That's okay; it means you're learning.

There's a test of basic conceptual understanding called the "Force Concept Inventory". After you learn conceptual physics you can download that test. If you get around 25/30 on it or so, you're ready to tackle a more advanced textbook at the introductory college level. These things are called "Physics for Scientists and Engineers" or "University Physics" and such. These books are long and boring, but you probably need to read one and do lots of the practice problems. It will take a much longer time than the conceptual physics book and you'll have to go slower.

There are probably a thousand introductions to special relativity out there. I have no idea which one is best for you. The book "Spacetime Physics" by Taylor and Wheeler is a common one, but lots of people I know didn't like it. I'm not sure what a better book is. I'd just pick whichever one on Amazon is rated the highest.

After you read that, you can try a book specifically devoted to college-level mechanics, such as Kleppner and Kolenkow, David Morin, or John Taylor. These include brief descriptions of Lagrangian and Hamiltonian mechanics and the mathematics of relativity. You might also need to learn some more math to understand these books. There are books called "Mathematics for Physics Students" and similar things. The one by James Nearing is free and pretty good.

Those books will get you a solid understanding of classical mechanics. There is still a lot more you can learn after that, but I think it would take more than a year to get through to the next level of graduate-level material.

A couple other things you might like are the first volume of The Feynman Lectures and the lectures by Leonard Susskind. Most people can't learn from the Feynman Lectures as a starting point, so I'd read them at the end of the year, not the beginning. The lectures by Susskind are on youtube. It's about a 20-hour course on Lagrangian and Hamiltonian mechanics that covers the essential ideas. He's also released them as a book called "The Theoretical Minimum". You might read this after the "University Physics" level book, or as a break in the middle of it. It's short and fairly straightforward, but also very abstract.

A couple of unconventional recommendations are to at some point in the year read Five Easy Lessons by Knight, which is about teaching introductory physics. It gives good perspectives on the difficulties and hangups students have when learning. Maybe also Two New Sciences by Galileo. He wrote it to convey his own conceptual understanding of basic physics to common people, back at a time when we only had bits and pieces of the math we need. It's an interesting historical viewpoint that could also help things click for you.

Learning physics is hard. I tried to describe what I know about it here.

Answer 2

A book recommendation is sometimes a hard thing to do in retrospect. (There are several books that I hated when I was using them in class but have since nearly worn out.) For this reason I will recommend the three physics texts which I feel are best suited to study in general and perhaps you can do some research and decide for yourself.

First, the most recent edition of Serway and Jewett is pretty close to a standard as far as introductory texts go. It is aimed at college freshmen. There are a lot of examples and problems and the material is pitched low and slow across the plate. However, it will not cover Lagrangian or Hamiltonian Mechanics and it takes only a cursory glance at relativity.

Second, John Taylor's Classical Mechanics is widely used and very well-liked. I also like it, and whenever I get confused about something I pick it up to see what he has to say. He covers everything (in greater or lesser detail) including the Lagrangian and Hamiltonian formalisms and relativity. Strictly speaking, you don't need much mathematical background here, but it helps. (If I were to recommend only one book, this would be it. I highly recommend it.)

Third, Goldstein is a graduate level text, and as such, few punches are pulled. He covers all of the bases (at times in excruciating mathematical detail). I suspect you will find this text pitched too high and too fast for your liking.

You can probably borrow a copy of any of these if you have an acquaintance in the physics department of a local college. That would give you an opportunity to see if you like any of them, which is the most important thing: if you are going to spend a few years on this, you need to have a book that you can enjoy reading. In fact, you might find it useful to start with Serway-Jewett and then move on to Taylor and finally graduate to Goldstein.

Now that I've made my book recommendations I'll give you three more small tips:

(1) I suggest brushing-up on your single-variable calculus. A lot. You'll need it, especially if what you are interested in is the "why" rather than the "how." In fact, if you get comfortable with one-variable, you should glance at multi-variate calculus as well. It's not much different so don't be worried, but you should know the difference between partial derivatives and total derivatives and be able to handle multiple integrals (particularly in polar coordinates).

(2) You'll need other math, too. Trigonometry is very important, but the headliner here is vectors. "Little arrows with lengths and directions" is basically all you need in simple Newtonian mechanics, but if you want to get down and dirty with relativity, you'll need linear algebra (i.e. row vectors, column vectors, and matrices). Orthogonal transformations will become your new best friends, and you will eventually have a brief and passionate romance with symplectic matrices, and this is only the tip of the iceberg. However, this only applies if you want to get your hands dirty with the real mathematical machinery. The concepts are not difficult, but the math can be overwhelming and very confusing.

You will probably want text books specifically for linear algebra and calculus, too. There are many good calculus texts which deal with vector calculus (i.e. dot products, cross products, vector-valued functions, line integrals, etc.), but they won't deal with linear transformations (matrices) in detail. You'll have to decide for yourself how deep into this rabbit hole you'll go. Which brings me too my final piece of advice...

(3) Remember that Classical mechanics took hundreds of years to get to where it is today. To get the type of firm grasp that you may be seeking will take great effort because it will constitute mastering the life's work of many highly intelligent individuals. Take it slow and expect to struggle. And sometimes you just need to accept things as axiomatic, which is to say that they are true because they work. Newton's Laws are of that kind: they cannot be proved, only corroborated by experiments. They must be taken as they are, as axioms of a theory. Eventually, if you follow the rabbit-hole long enough, all roads lead to axioms.

I hope this helps. Best of luck to you!

Answer 3

I'll just note a couple of things when self-studying.

In order to avoid "the rabbit hole" you should have a look at several books, choose one and stick to it. Reading other books and Wikipedia is useful because you'll have different views, but it will reduce your concentration (hyperlinks are specially dangerous).

You should also try to understand every derivation. For example, a good technique is to have a piece of paper and "fill in the gaps" in a theorem proof. If you feel that you don't understand the math behind, it's time to study from a math book.

In particular, I think that John Taylor's book is a good introduction.

Answer 4

but who then contain another term Z, who redirects to ... which leaves me with no understanding.

Don't try to take it all at once. Mechanics is a vast subject and it takes a $lot$ of time to "get it". This is not so much due to complexity of the basic theory, as due to variety of applications that often require some smart combination of the basic principles interpreted in the right way and often also some mathematical resilience. The best way is to start with simple elementary things and understand them through solving problems.

I learned mechanics step by step. Take few good high-school physics textbooks on mechanics, and try to do the problems, even invent ones - think about what you learn. 80% of university mechanics is in the high school textbook, only with simplified math and language. If you can solve the problems and understand majority of it, then you should try some university-level textbooks.

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

It is much better to seek for those topics in several books. More people know more, and often author A is good on mechanics, but not so good on relativity and vice versa.

For basic mechanics, my favourite were:

• Feynman's lectures on physics, vol. 1; http://www.feynmanlectures.caltech.edu/I_toc.html

This is great book in all chapters. Mechanics is exposed very nicely with great deal of originality and personal approach of Feynman.

• textbook by Gettys, Keller, Skove http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=keller%20gettys%20skove

standard American style textbook, the best of its kind I've seen.

• Berkeley course on physics, part I. Mechanics

http://www.amazon.com/Mechanics-Berkeley-Physics-Course-Vol/dp/B000OFGY3S/ref=sr_1_cc_2?s=aps&ie=UTF8&qid=1388691985&sr=1-2-catcorr&keywords=berkeley+course+on+physics+mechanics

more advanced stuff, and great.

You should be able to find these in physics university library.

Then for relativity, there are many good and even some superb books. I learned a lot from a great intro paperback book by a Czech autor, with blue cover, I think it was Karel Bartuška, Deset kapitol ze speciální teorie relativity (only in Czech, should be in public libraries). The chapter on Special relativity in Serway: Physics for Scientists & Engineers

http://www.amazon.com/Physics-Scientists-Engineers-Chapters-23-46/dp/1439048398/ref=sr_1_12?s=books&ie=UTF8&qid=1388693282&sr=1-12&keywords=serway+physics

is a nice-and-short intro. Also more advanced textbook by Katz: Introduction to the Special theory of relativity

http://www.amazon.com/introduction-Special-Theory-Relativity/dp/B0000CMHIO/ref=sr_1_1?s=books&ie=UTF8&qid=1388692825&sr=1-1&keywords=katz+relativity

seems very nice, and Einstein's elementary book Relativity: Special and General theory

https://archive.org/details/relativityspeci00einsgoog

is very readable.

For Lagrangian and Hamiltonian mechanics, you will find some good texts on the Internet, for example by David Tong: Classical Dynamics: http://www.damtp.cam.ac.uk/user/tong/teaching.html

And, of course, the greatest but formidable book on this subject is

Landau & Lifshitz: Mechanics, Course of theoretical physics

http://www.amazon.com/Mechanics-Third-Course-Theoretical-Physics/dp/0750628960/ref=sr_1_1?s=books&ie=UTF8&qid=1388692793&sr=1-1&keywords=landau+mechanics

In few dozens of pages, there it is all. It is a hard book, but very much worth reading. Will be digestible after you've waden through Tong or some other beginner-friendly source:-)

Answer 5

First, I want to say clearly that I'm a second-year Physics student, so I have only a little experience about the matter. However, maybe I can give some advice.

1. As you have realized, the lack of good mathematical background can be a real distraction and can also make you waste a lot of time in trying to understand things that are properly explained in some mathematical terms that you haven't seen before. So my first advice is: learn basic mathematics. Some of the things that you need to know well from the first analysis course (this is not intended as a list):

1. The notions of limit and continuity.
2. The notion of derivative. Taylor expansion.
3. The notion of the Riemann integral over real intervals and its link with the derivative.
4. Differential equations.

About points 1 and 2, my opinion is that single-variable calculus is too limiting. Since the extension of the concepts of continuity and differentiation to functions from $\mathbb R ^n$ to $\mathbb R ^m$ is really not a big jump, I suggest to study these concepts in this little more generality. It'll make easier to understand concepts such, for example, “potential energy”.

Integration over subsets of $\mathbb R ^n$ is more delicate and you really won't need a lot of it to learn basic newtonian mechanics, so I think you can postpone it to a second moment (however you will need it to understand propositions like the Liouville's theorem, for example).

There isn't a lot to say about the importance of differential equations. In particular: $$\ddot x=-\omega ^2x.$$ There's a lot of good analysis books, an example being Rudin's PMA (try to give a look). However, this is a really though book. Maybe a softer approach is given in Spivak's Calculus, I haven't read it though. But he's an excellent writer, in my opinion.

Some concepts from linear algebra that you need to know are:

1. Vector spaces (and affine spaces).
2. Linear applications.
3. The vector space structure of $\mathbb R ^n$. Linear systems, matrices.
4. Affine geometry.
5. Scalar products, vector spaces with a metric given by a SP. Isometries.
6. Eigenvalues. Symmetric endomorfisms. The spectral theorem.

The abstraction of “vector space”, together with the study of the natural applications beetween these sets, i.e. linear ones, is something of huge importance. The concept of affine space, that rests on the former, is important too, since any natural description of the space-time is in terms of an affine space (and not a linear space). About point 6, it's really important to understand at least the statement of the Spectral theorem. A first example of its power will arise from the study of rigid bodies.

Apart from their importance in classical mechanics, I suggest to study hard all this stuff simply beacause it's all really beautiful.

A book that deserves mention, even if treats more advanced subjects, is Spivak's “Calculus On Manifolds”. I think that this book is really precious to a physician that needs to learn a lot of advanced mathematics in a relatively small time.

2. Now we come to physics. I don't want to list topics, because there's not much to advice about it. I'll mention some of the books which I studied and I'm currently studying from.

1. Feynman lectures, book 1. This is a MUST. Someone says that it's better to read him later. I don't agree, I think that this lessons are enlightening at all level. From the first time I read Feynman, I found him brilliant and his style very engaging.
2. Taylor - Classical Mechanics. This is a good introduction to Newtonian mechanics: a nice and easy to follow approach to the basic theorems, with full explanations in all its derivations and remarks where some simplification is being made. It contains some interesting problems and also a brief introduction to Lagrangian Mechanics.
3. V.I. Arnold - Mathematical Methods of Classical Mechanics. I'm actually reading this. It's an incredibly fascinating book, and also a laborious one. I suggest to go through this once you've mastered the Newtonian formulation. I'm aiding myself through this book with “Calculus on Manifolds“, which was one of the reasons to mention it.

This answer grew up a little more than I was expecting, but I think I've said all I had to say. If you have questions, feel free to ask. I hope this can help.