This coming spring term I will have to teach two new classes and I will use this diary to jot down some reflections I’ll have about the material and the about how the students cope with the course. Classical Mechanics is our first theoretical physics course. In the series of theory classes, we take the material that has been introduced in an experimental class before, and take it to a new, more contemporary level. Mechanics is often the first of such. Other similar courses include E&M (usually as E&M1 and 2), Quantum Mechanics (to extend modern, atomic, and nuclear physics), and Stat Mech (to extend thermal physics).

As usual, I will change names even though it is a long shot for anyone to figure out who I am actually talking about. My long-term colleague David retired with short notice at the end of December. Our department was very strained to offer all courses for spring. I was the only realistic option for two classes to take over, and this is one of them. I have never taught it before, so this is a bit of an adventure.

Students are planning to take this as their first serious major class in their second sophomore term. The weakest students will normally not make it here. These thirteen students made it here:

Mel, a grad student in Math; Max, a Mechanical Engineer with a minor in physics (this is usually one of four classes minors have to take beyond the two introductory physics courses 1 and 2), he took a Physics 1 class from me two years ago and defined the top of the class then; Trevor, a double major from Physics Education, apparently, I had him once before - I do not remember him; Caleb, an Astronomy major I have not met in class yet; Eidan, a transfer Physics major who has now completely switched to us (I had him last term in Stat Mech - as a transfer, Eidan takes courses out of order); Reese, a physics major I have not met yet; Gabby, an astronomy major I had in Physics 1 and over summer in Physics 2; Willem, a new transfer astronomy major, his transfer record looks a bit troubled; Corinne, a physics major I had during covid indirectly when I had to take over a colleague’s online class last year; Chris, a Math/Physics double major who I recruited two summers ago when he took my Physics 1; Mason, a Physics Education double major who I had in Physics 2 last term, he is an athlete in the football team and carries an incomplete in Physics 2 with him; Britney, another Physics Education double major, I do not know her yet; Jaclyn, a physics major with a troubling record (failed Physics 1 the first time, took it over summer with me and got a D and took last term my Physics 2, which she finished also with a D).

Lecture one will largely be consumed by talking about the syllabus. The remaining lecture will be used to lay out to the students what the course is about, how it relates to their prior course on Newtonian Mechanics, and how it differs.

I will start that section with an interactive part in which I make the students remember what they have learned and what they have not learned. In that initial assignment I hope students to come up with two memories: Newton's Laws plus the Law of Gravitation, sometimes known as Newton's Fourth Law, and Conservation Laws (for Energy, Momentum and Angular Momentum). I will also make them confront what they did not learn: what the exact nature of the premises of Mechanics are: space, time, and mass. These students will be taking Modern Physics in parallel, a class where I do a similar warmup when I teach it. But this term a Taiwanese colleague is teaching that class and he will likely do a much drier, not conceptual introduction. So I won't rely on what they may or may not learn there and do the parts I need for this course myself.

From there I will go into an overview lecture that starts with what the students are familiar with, but put it into a new light and context, and then augment it with an outlook how we will be changing it. It goes something like this:

(Remember, the students have just recalled the Conservation Laws)

The principle of Conservation of Momentum is a consequence of Newton's Third Law. (in our abbreviated treatment in physics 1 this will not have been on the menu)

Consider a vertical rocket launch. You may think "Apply Newton's Second Law and solve". But that does not work because the mass of the rocket is changing continually as the rocket is being pushed by a force. Newton was aware of this difficulty and saw that it is corrected by equating the force to the rate of change of its momentum. With this in mind, one can calculate the force acting on the rocket in two steps. First, treat the rocket at any moment as though its mass was constant and multiply the mass by its acceleration at that moment. Then treat the velocity as not changing and multiply it by the rate of change of mass at that moment.

The force acting at that instant is the sum of these two products: m dv/dt + v dm/dt, where the terms dv/dt and dm/dt refer to the calculus, that is the derivative of velocity v and mass m with time t. Stated this way, the Second Law indicates that momentum is a candidate for conservation.

Momentum Conservation implies a spatial symmetry of laws of nature because the calculation is the same, regardless where one measures from (where we assign the origin).

Just so, Energy Conservation means a time symmetry of laws of nature (remember that in the StatMech class I drew on that when I introduced entropy), because it is invariant with respect to a shift in time.

This means that if some physics process is some function of space and time (as my students know they always are), then momentum p is related to how the state changes from point to point in space. And energy is related to how the state changes from moment to moment in time. This intrinsic coupling, if you will, plays a very important role in Quantum Mechanics, for example in the Heisenberg Uncertainty Relation, which states "change in momentum times change in position is bounded by a minimum value", the Planck quantum. And similarly, :change in energy times change in time is bounded by the same quantity.

Most of the time we do not think of Newton's Laws as applying to rotational motion, but they do. Moreover, pure translational motion rarely occurs in nature. To better understand the implications, let's first think about pure rotational motion (like a wheel turning on a stand but not moving anywhere). In pure rotation, no straight line remains parallel to itself, except the axis of rotation. Newton's Laws cannot be applied directly, one has to enlarge the formalism:

A force acts on a particle to keep it moving in a circle. Its momentum is not conserved. But its angular momentum is: a new vector of magnitude mvr that points along the axis of revolution. It is related to a variation of force, the torque tau. In linear mechanics we had dp/dt = F, that is the rate of change of momentum p in time t is the Newtonian force F. Here we find that the rate of change of angular momentum L in time is torque: tau = dL/d

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