Classical Mechanics Primer

There are at least two senses in which the term “Classical Physics” is used. First, classical may be used to mean “all physics that is not quantum physics”. The second sense is more similar to what most people usually have in mind, and basically refers to all physics which is not “modern”—that is, physics before General Relativity and Quantum Mechanics.  Physicists always mean to use the first sense of the term, and laypeople usually switch from one to the other, sometimes without noticing. From this point on, I’ll be using the physicist’s sense unless noted otherwise.

Now, the expression “not quantum” needs some unpacking. As an example, consider a toaster with a knob that you can turn at your leisure over some period of time, t. If you were to plot the energy flowing through the toaster’s resistance as you turn the knob, the graph that you will get might look something like this:

In Classical Physics, there is no limit to the number of times that you could zoom into any section of this plot, and every single time you would see a smooth, continuous line like the figure above. If you moved along the curve, you could always ride the line like a smooth ramp to any value of E that you liked, and in doing so you would pass through all points in between.

This is not the case in Quantum Mechanics, where many physical quantities are said to be quantized.  In the toaster example above, the plot will look the same initially, but after zooming in a finite number of times, eventually you will reach a plot that looks more like this:

This is what makes Quantum Mechanics so weird. And so, Classical Physics includes all the branches of physics where this effect is not taken into account, regardless of when said branch developed. This includes theories such as Classical Mechanics, Electromagnetism, Thermodynamics, Fluid Mechanics, and General Relativity, among others. Usually, classical theories are an excellent approximation for all kinds of real world phenomena, even though the real world is actually quantized.

Newton

 

Newton’s Three Laws of Motion are as follows, though you may find some different wording depending on who you ask:

1. Objects move at constant velocity as long as there is no net force acting upon them. Velocity is not the same as speed, as it also includes direction and is therefore a vector quantity (usually notated in bold). Also, zero is a perfectly valid value for a constant velocity.

2. If there is a net force acting upon an object this will cause it to accelerate, according to the relation \(\mathbf{F} = m\boldsymbol{a}\).

3. Every action produces a reaction of equal magnitude and opposite direction. So if a certain object A pushes or pulls on B, B will answer with a push or pull of its own. This is usually notated like this: $$\mathbf{F_{AB}} = -\mathbf{F_{BA}}$$.

The whole of Classical Mechanics emerges from these three laws, though by far the one used the most in calculations is the second one. At the high school level, this usually means doing lots of trigonometry for setups such as the following, which may distress some readers:

A typical problem in mechanics is to figure out the equation of motion of a body, given some initial conditions. We begin with some information about the object’s starting position, forces and constraints it is subjected to, and work our way up to the equation that will tell us how the object will actually move thereafter. In the case of Newton’s Second Law, all solutions are essentially variations on solving the differential equation $$ \mathbf{F} = m\mathbf{a} = m\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} t}= m\frac{\mathrm{d} }{\mathrm{d} t}\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} t}=m\mathbf{\ddot{x}}, $$ which would yield x as a function of t, or x=x( t ).

Unfortunately, most real world problems are not as clear cut as blocks on inclined planes, and Newton’s Laws are utterly impractical in their original formulation for solving anything but the most simple physical systems. Fortunately, there exist a couple of alternative approaches that manage to elegantly generalize Newton’s Laws so that they’re applicable to nearly any classical system one can think of, at the cost of introducing some abstraction. These approaches are absolutely equivalent to Newton’s Laws, and indeed incorporate them at their core, but their mathematical appearance is quite different.

Lagrangian Mechanics

 

If we think of space in three dimensions, then the position of a particle at any given moment can be described by a position vector from the origin, with individual components in the x, y, and z directions: \(\mathbf{r}=(x,y,z)\). The path that the particle will take over time will once again depend on the forces, constraints, and initial conditions it is subjected to. So, what we’re looking for is the way that \(\mathbf{r}\) will evolve over time, \(\mathbf{r}= \mathbf{r}(t)\).

However, working with cartesian coordinates (x,y,z) doesn’t help very much when, in general, physical systems aren’t easily describable by them. So instead, we introduce some abstraction and think of generalized coordinates, each of which we call \(q_{i}\). Each one of these coordinates could be cartesian \((x,y,z)\), cylindrical \((\rho ,\theta ,z)\), spherical \((\rho ,\theta ,\phi)\), or even other ways of placing a particle in space, such as a distance traveled along a constrained path, or an angle defined by the movement of a particle, or anything that suits us. Each generalized coordinate can also change in time, yielding a generalized velocity, and these are denoted using the dot notation for derivatives as \(\dot{q}_{i}\). By a simple extension, we can speak of other generalized quantities, such as generalized momentum \(p_{i} = \frac{\partial L}{\partial \dot{q}_{i}}\). This particular quantity will come in handy in later on.

A Lagrangian is defined as the difference of the kinetic and potential energies of the system in question, and it will have the following form: $$ L = \frac{1}{2}m\mathbf{\dot{r}}^{2}-U(\mathbf{r}).$$ The function U corresponds to the potential energy, and in general it will depend on the particle’s position. The first term is the familiar kinetic energy term \(\frac{1}{2}m\mathbf{v}^{2}\) from kindergarten, with the velocity expressed in terms of r.

Notice that the Lagrangian depends on the generalized coordinates and velocities, since each r is defined in those terms. L being a vector equation, we can separate the different \(q_{i}\)’s and their velocities into what are known as the Euler-Lagrange equations, which can be conveniently abbreviated as $$ \frac{\partial L}{\partial q_{i}} = \frac{\mathrm{d} }{\mathrm{d} t}\frac{\partial L}{\partial \dot{q_{i}}}. $$
So the rough strategy for solving problems in the Lagrangian formulation of Classical Mechanics is this: write L explicitly in terms of your generalized coordinates and velocities, making sure to include the specific (mathematical) description of U as well if it’s available. Then, plug the resulting expression for L into the Euler-Lagrange equation for as many coordinates as you have, perform the corresponding derivatives, and finally integrate whatever is left to solve each equation (i.e., get rid of whatever time derivatives are left over after you plug things in). If you’re successful, you will end up with as many equations of motion as you had generalized coordinates, and those equations of motion combine to give you the overall motion of the particle.

There are many nuances and neat things to go into with regards to the Lagrangian approach, but I’ll leave them aside for now. As a summary, notice that you end up with a second-order differential equation for each coordinate involved, and also notice that the setup for solving a problem begins with considerations of energy rather than forces. One important consequence of this is that any forces of constraint that may act on the object are naturally incorporated into the equations, without the necessity of force diagrams.

Hamiltonian Mechanics

 

A second approach to generalizing Newton’s laws is that of Hamiltonian Mechanics, which is a similar abstraction to that of Lagrange. First, we remember the concept of a generalized momentum, and also consider that for many systems the Lagrangian doesn’t depend explicitly on time $$\frac{\partial L}{\partial t} = 0.$$ When this is the case, we can define another quantity, the Hamiltonian, as $$H = \sum_{i=1}^{n}p_{i}\dot{q}_{i} – L.$$In addition, if the relationship between generalized and cartesian coordinates doesn’t depend on time either, the Hamiltonian is equal to the sum of the kinetic and potential energies of the system: $$H = T + U .$$In the same way that we used L in the Lagrangian formulation to derive our equations of motion, the Hamiltonian formulation relies on operating on H. In this case, the Hamiltonian is plugged into the equations $$\dot{q}_{i} = \frac{\partial H}{\partial p_{i}}\:  ,\:  \dot{p}_{i} = -\frac{\partial H}{\partial q_{i}}$$ to yield the equations of motion. This may seem like more work than the Lagrangian approach, but notice that we are left with sets of first-degree differential equations instead of second-degree ones. Even though we have twice as many equations as the Lagrangian approach, they are usually trivial to solve, and they include all the advantages of the Euler equations as far as dealing with constraints.

Hamiltonian mechanics in particular is important for later use in Quantum Mechanics and, as was the case with the Lagrangian formulation, there’s many details and asides that I left out for the sake of brevity. Both formulations are usually overkill for problems about blocks sliding down incline planes, but they become indispensable when working with more complicated systems. As I re-study mechanics for my PhD, I will try to post more on the details of each of these approaches, as well as practical examples of their use.

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This post was the result of summarizing some of my notes on the subject, but obviously you would do well to look up these topics online. The Wikipedia entries are quite good, and then there are several recommendable textbooks. First, the Wiki links:

Newtonian Mechanics
Lagrangian Mechanics
Hamiltonian Mechanics

Books that I’m familiar with include the following:

1) Taylor, John R., Classical Mechanics. Great for undergraduate level students and laymen with calculus background. Informal, conversational style.
2) Goldstein, H. Classical Mechanics. Standard graduate text for students who want to get serious and be grown-ups. Very long-winded at times, but solid.
3) Landau & Lifshitz, Mechanics. Standard graduate text for masochists and people who like their textbooks lean and mean.

Cross-post on atheism

In other, unrelated content, I wrote a short piece on atheism over at Medium:


God is not a good theory

Books, Books, Books!

As I mentioned earlier, I had ordered a few books in anticipation of my PhD studies starting next January.  Two of the books in the order arrived today, and I'm quite giddy:

Tow out of four are here.

The Relativity book, as you can see, is Sean Carroll's Spacetime and Geometry, which I hear is awesome and will be putting to the test shortly. And the other book is... wait! You ask: didn't I mention I ordered three books on Relativity? Well yes I did, but I also ordered And yet..., which is a selection of essays by one of my favorite authors of all time, Christopher Hitchens (the anniversary of his death was just this past December 15^th).  I have lots and lots of stuff by him, and this latest addition will be the cherry on top of my collection:

Not too bad, huh?

There are a few missing from this anthology. Right now I can think of his book on Palestine co-written with Edward Said, Blaming the Victims; then there's the book on the arguments for the Irak War, A Long Short War; and finally another essay collection, much older than the rest, called For The Sake of Argument (hint: my birthday is in September, in case Christmas is too soon for you).  The man was as eloquent, lean and mean as a writer could get, and he's certainly my top choice for best essayist in the English language, sneaking in before Orwell.  So far, my favorite tome of them all is Love, Poverty, and War.  The writing is absolutely beautiful, and the depth and scope of topics covered is mesmerizing.

But wait, there's more! My wonderful sister, who's also quite a bibliophile, got me this little gem for Christmas:

Awesome!

So it looks like I will be very productive over the Holidays, or at least be very entertained.  Hopefully, I will be up to date on tensors and ready to hit the ground running next semester.  I also have a few academic papers on Causal Dynamical Triangulation that I want to go over, which would be great, since that's what my work will be about.

Speaking of next semester, the folks at admissions rejected my paperwork again... the amount of documentation one has to provide is of a Kafkian scale.  I was livid last Friday when, after the guy had told me on Wednesday that all I needed was one more paper, and having procured it, he looked at the documents once more and said that I still needed even more documentation, and that I had to go three doors down the aisle (again) to begin another transaction.  They're on break for the next two weeks, and I have only two more days, the 7^th and 8^th of January, to go over there again and finally finish the paperwork for good.  After those dates I'm in trouble, because the admissions results have to be published in the university gazette on the 12^th. Just writing about it is putting me in a bad mood, so I'll end here for now.

Third World Problems

And so, as I feared, I am not yet finished with the paperwork I have to get done.  Yesterday, a guy at the admissions office told me to my face that I was missing only one document, and that I could come back the next day from 9 to 3 and turn it in.  He said everything else was in order and all I needed was this one document (to be fair, I screwed up big time, since this was a basic document that I had simply left on my desktop ready to be printed, but I never did).  So I got there today and what I found when I got there was the doors closed and the following notice:

Sadly, this is not surprising.

Rough translation (the grammar isn't very good in the original): Notice: On Thursday 17 operations will be from 12:00 to 15:00.  Tomorrow, normal hours from 9:00 to 15:00.  Sorry for the inconvenience. -Admissions.

No explanations.  The staff were just AWOL and left a note.  When I got there, it was 11:50 and there were about 10 other people already there waiting.  I had a doctor appointment at 13:00, and a 45 minute bus ride to get there, but I decided to hang on in case they showed up.  Other people, who must have been there even earlier, left, and were now coming back, started to arrive.  Soon, there were a couple dozen of us waiting.  I waited until 12:10 and then left.

Fellow students who were "inconvenienced".

This is not just an instance of what happens at CUCEI on a regular basis.  This is what most of Mexico is like, especially in public institutions.  For example, often this happens at the IMSS, our Third World attempt at a National Health Service.  Only there, it's doctors who don't show up, or maybe they do but they don't have equipment or medicine.  And people run into those situations often with pressing medical conditions, and they are told to go home and come back later (nearly every Mexican who can afford private health care ignores the IMSS altogether, even though they pay for it with their taxes.  The number of Mexicans who can 'opt out' in this way is very small, and gets smaller every year, but that's a whole other essay.)  Things have been slowly getting better over the past decade or two, but we're still decades behind the great country that we could be.  And so, tomorrow I head over there again, and hopefully I can get the bureaucracy off my mind for the holidays.

Getting ready for the holidays

So I'm almost done with the paperwork I have to do to complete my admission process.  I'll have to go to the CUCEI campus once again tomorrow and, if all goes well, that will be all and I will be able to focus on more important things over the Christmas break.  What bothers me most about the paperwork isn't so much the amount of stuff that I have to get, but the unnecesaryness of the whole thing to begin with.  It is indeed a Kafkian circumstance that CUCEI requires from me all these documents that basically amount to the same information, makes me pay for some of them, and has me going around in circles from one office to the next.  And, all the while, I'm obtaining the documents from... CUCEI! They're asking me to prove to them that I was a student under them.

And so, in this digital age where all the information they need is already in some database that they own, it would be a matter of the admissions people logging on to some system, searching my name, and verifying that yes, indeed I got a degree in physics and it's certified by the folks three doors to their left.  If they ever wanted the actual physical documents, then they could just ask those fellows for my name, and go over there and get them themselves.  But no, it is I who has to request those documents (and pay for them!) and get them from one bureaucrat to the other one a dozen steps away. Most of Mexico works this way, I'm afraid.

In other matters, I've started a wonderful little book I ordered a long time ago in anticipation of a moment like now: A Student's Guide to Vectors and Tensors, by Dan Fleisch (I also own another great little book by him on Maxwell's equations).  The first three chapters are a review me about vectors, but then things get novel when tensors are introduced.  It's in these later chapters that I'll slow down considerably and do the excercises (he also has a supporting website, with all kinds of cool stuff to complement the text).

All of this is in anticipation of three other books that are more on-topic for my study.  These could roughly be classified as being in a beginner, intermediate, and advanced level.  I ordered them a coulple of weeks ago, and it's possible they'll arrive sometime before New Year's eve.  They are the Relativity textbooks by Schutz, Carroll (whose website is featured in my blogroll), and Wald. There are many others, but I'm on a budget and I went for the most bang for my buck.

The semester formally begins on January 18, and I hope to have made some headway as far as learning some of the contents of the Fleisch text at least.  Then things get interesting as I begin to work with my advisor and finally plunge head-first into the most beautiful theory ever devised that didn't end up in the trash bin immediately.

Paperwork-cracy

I really love the atmosphere at my campus, CUCEI, which is the science and engineering branch of the University of Guadalajara. The people feel authentic and committed to their studies. This may seem like an odd thing to say, but here in Mexico education is greatly segregated by class. Poor people use public education almost exclusively, and rich people use private education only. My family is in what's left of the middle class, and I went to a private university for my engineering degree but then switched to the public system for music and then physics graduate school. There are some nice people in private education, but most are actually shallow and fake; they're only there to kill time while they wait for their parents to hand them the family business or, in the case of many young women, to look for someone to get married to.

(Now that I think about it, I have to back up a bit for the sake of readers outside of Mexico. Public education in Mexico is virtually free all the way through undergraduate degrees; graduate level studies have a mostly symbolic hike in fees that's covered by many grants and scholarships and so it ends up being almost free anyway. How much is “almost free”? Try a tuition of $300 USD for a Master's in Physics per semester. The application fee for my PhD was about $60 USD. By contrast, a semester in a private university is at least $7000 USD in tuition. Remember, average per capita income in Mexico is less than $20,000 per year.)

It was when I made the switch to the public system, first in music and then in physics, that I sometimes felt like I was the one who was shallow and fake. Everybody was there because they actually wanted to, and they were good, hard working people. I'm in touch with almost nobody from my private education years—especially classmates from high school, who I mostly despised then, and still do. Some friends from engineering linger in my social media, but it's mostly my public school friends that I cherish the most and stay in touch with.

One point against public education, however, is the awful amount of paperwork that one has to do for everything. Among the documents I have to produce to complete my PhD application process, I have to submit a copy of each of the following:

• Master's Degree
• Master's Thesis Defense Certificate
• Master's Studies Certificate

So I have to spend time (and some money) procuring three documents that amount to the same thing: that I was a competent physicist at the Master's level. These papers have virtually the same information printed on each of them. One wonders how it could be that, in the digital age, I have to produce hard copies of each when the university itself owns a few. After my Master's, I didn't bother to do the paperwork so they could hand me my actual degree, mostly out of procrastination, all I got was the Thesis Defense Certificate. The procedure takes a few months and, because I have to procure these documents by January 18, I now have to get another document requesting an extension.  Private institutions are way ahead in this regard, and I can remember how my passage through engineering was a breeze as far as administrative paperwork went.

Besides this awful tramitocracia ("paperwork-cracy"), I want to devote the few weeks I have left as a relatively free man to review some tensor analysis, but also to just relax, read a book or two, and watch videos on YouTube for the Holiday season. If the tensor analysis goes well, I may have time to do a piece on the basics of tensors, so that we can actually get started on real physics.

In the beginning

My mind was void, and without form. Or at least, that must have been the impression I made on Dr. K., the department head for my local university’s Physics PhD program. I had heard that he was a difficult person to get along with, but I hadn't actually talked to him at any moment during my Master's degree a couple of years ago. Indeed, back then I was actually advised not to take his Quantum Mechanics course, on the grounds that I would suffer greatly and learn nothing because of his disdain for engineers and astronomers. So at that time I decided to wait a semester and take the class with another professor, Dr. G. That was perhaps the best decision I made during my two-year Master's program, second only to the election of my Master's thesis director, Dr. R.

But now I was stuck having to confront Dr. K., having finally made up my mind that I would attempt to get into the PhD program. Since finishing my Master's more than a year earlier, I had my sights set on making up for my weaknesses by studying relentlessly before even trying to enter the PhD program, in order that I could work on what really interested me, which was General Relativity. Another professor, Dr. N., had already pretty much accepted me as his pupil, despite my best efforts to prove my incompetence to him (more on this later). But now I had to take the formal step of approaching Dr. K. and officially declaring my intention to enter the PhD candidate pool to him.

“So what did you work on for your Master's?” he asked. His Russian accent was noticeable, but much more subtle than Dr. N.’s.

“Astronomy, with Dr. R.”, I said.

He immediately responded, “Yes, but what actual physics did you work on?”

I wasn't expecting this. You see, astronomy is a branch of physics, and I had worked on spectroscopy. That seemed like physics to me, and so I told him.

“Spectroscopy? Well, then. Tell me about the physics of spectroscopy.”

I rambled and stuttered through the principles at work, mostly butchering the explanation of atomic energy levels, emission of photons in the mid-infrared portion of the spectrum, how they corresponded mostly to molecular hydrogen, how they were then captured by the spectrograph on a space telescope, and how I used a Boltzmann distribution to deduce the energy and therefore the temperature of the hydrogen. . . I trailed off, and his expression somewhat begrudgingly acknowledged there was some physics in what I was trying to say.

“So what do you know about Relativity? Have you worked on it before?”

I mumbled something, but before I could go on he interrupted and asked, “Do you know what a Killing vector is?”

I was blank. “Uh, no,” I said.

“And what is the Schwarzschild metric?”

I knew that one, but couldn't find the words quickly enough. There was an interruption. A student entered the office and discussed some result or calculation with Dr. K. for a few minutes. After he left, he returned to me:

“Look, this is my off-the cuff impression. I'm not judging you, and I don't know you, but what you want to do is very risky. Relativity is really difficult. If you haven't worked with it before, you're not going to get to a high level and produce meaningful results in 3 years. You're setting yourself up for failure,” he said, “You're an engineer, right?”

Dr. N. must have told him something about me, but I had no idea how much had been said. I confirmed I had a degree in chemical engineering, from many years ago. Then I explained—–attempted to explain–—that I had already made my best effort to tell Dr. N. that I didn't feel ready, but that he insisted that I sign up for the entrance exam and interview anyway. I also said Dr. N. had given me some papers to read, and had basically accepted me as his student already, even though the exact topic of my research wasn't yet defined specifically. I mentioned I had good programming skills, and that I could study and learn on my own. He shook his head slightly at this, somewhat dismissively.

“The work Dr. N. does is very advanced mathematically. I don't think you're ready if you come from astronomy. He's been my friend for many years, but we haven't spoken about your case in detail. Are you sure he said he's accepted you?”

I nodded. We got into a brief conversation about how I got to Dr. N. in the first place, how I was interested in Relativity but open to working on something else, and how Dr. N. had advised me to work on what I really wanted rather than what would be easiest or more realistic. I described how I had gone through all the points on the entrance exam study guide one by one with Dr. N., and how I had failed miserably, but Dr. N. told me not to worry about it and just make sure I understood the basic principles at work—–I had about a month to study up. He had said that the panel of professors took many factors into account during the examination, and being a robot that remembered all the equations wasn’t going to win me many points if I didn’t have certain other traits that they looked for in a candidate. Dr. K. seemed very conflicted at this point, keeping his gaze down, and it was mostly me who did the talking. He mentioned the possibility of looking up another astronomy professor and working on a PhD with him instead.

“Look, I encourage you to take the exam. I really do. But you must reconsider your field of study. On the day of the exam, you must have your decision made. Really, take the exam,” he said, “But reconsider. If Dr. N. has already agreed to work with you, I won't get in the way. But honestly I think it would be a dangerous mistake.”

So that's what went around my head at that moment and the rest of the day (and night). Not the excitement of finally reaching a milestone I aimed at since childhood, nor even the stuff that I was supposed to study up on over the next month, but rather the idea that all along I'd been setting myself up for failure by making mistake after mistake after mistake, only to plow along anyway and potentially be miserable for the next three years, if I even made it that far.

*   *  *

The previous scene took place over a month ago.  I was very bothered by it, but it only lasted a couple of days.  Slowly, the feeling that I could succeed became stronger and I didn't need to consider it any further.  On the day of the exam, this past Monday, I reaffirmed my intention to work with Dr. N. on GR as originally planned.  Dr. K. was still somewhat worried, but he managed to make a comment along the lines of "Well, I think we've had successful students who came from engineering before," which is as much of an encouragement as one could reasonably expect from him. At the time of this post, I am roughly a month away from the formal beginning of my studies. In case you were wondering, I'll be working on a description of spacetime by means of dynamical triangulation.  Don't ask me for any details on that just yet, though.