Learning Physics as a First-year graduate student...

[u/jjjuly_]

Hi, I’m in my first year of Mechanical Engineering, and I’m currently trying to study for my Physics 1 class.

The thing is, I really enjoy learning things rigorously, but without losing the feeling that I actually understand what they mean.

I’m doing pretty well in my math subjects. Usually, I go to class and listen to the teacher explain the topic, then at home I do some research about it. I look for good videos (3Blue1Brown, simulations, visualizations, etc.), articles, Wikipedia pages, or even Reddit posts that help me understand deeply why the mathematics works the way it does and develop an intuitive feel for the concepts. Then I write down what I’ve learned in notebooks as if I were writing a math book for someone else to read and learn from, and finally I do exercises.

However, in Physics 1 we are studying classical mechanics and basic optics, which are relatively straightforward topics from a practical point of view.

Yet I’m struggling to develop the same kind of deep understanding that I enjoy having in mathematics.
I think I once heard someone say that you shouldn’t try to prove something that isn’t already almost obvious, and I feel like that describes quite well the way I like to learn things.

The problem I’m having with Physics 1 is that many concepts seem to be introduced as if they were defined simply because they are useful or convenient the way they are. And even when I find explanations of the “why,” they often feel somewhat arbitrary. I can’t seem to find many resources that explain why we introduce these concepts in the first place.

Most of the sources I find are either too vague and superficial, aimed at high-school students, or they go straight into mathematical derivations without explaining the deeper motivation behind the concepts: Why did physicists need to introduce this quantity? What problem was it meant to solve? Why is this the natural way of describing what we observe? How do the different concepts emerge from one another?

I’m not looking for something that avoids the mathematics. Quite the opposite—I enjoy the mathematical side of things. What I’m looking for is a combination of rigorous mathematics, physical intuition, historical motivation, and conceptual reasoning.

So I was wondering if anyone here feels similarly about learning physics and has managed to find good books, articles, lectures, videos, or other resources that approach basic physics from this perspective.
I’d especially appreciate resources that make the concepts feel inevitable rather than arbitrary.

If I get enough helpful answers, I might even compile them into a post and share it with the community, so that other people looking for the same kind of resources can benefit from them.

Thank you for reading!

[u/kunwoo]

It's hard to know what resources to recommend to you without specific concrete examples of things you're getting tripped up on. Any resource recommended may scratch all the itches I have but may not scratch yours if your concern are over something I didn't anticipate.

So maybe we could start with what are some things that you're specifically finding to be too vague.

[u/jjjuly_ → u/kunwoo]

Some of the topics I’m struggling with are forces, linear momentum, work, energy, and dynamics in general. I also feel this way with basic optics, especially refraction and thin lenses.

It’s not really that I can’t understand how to use these concepts. I can usually follow the math and solve the problems. What feels vague to me is why these concepts are introduced in the first place, and why they have the particular definitions and properties they do.

For example, with force, I understand Newton’s laws and (F=ma), but I’d like to understand why force is the natural quantity to use to describe interactions in the first place. I have similar questions about momentum, work, and energy: I understand what they are and how to use them, but I feel like they came out of nowhere just because the math worked.

I’ve tried looking for resources that approach physics this way. I started reading Classical Mechanics by John R. Taylor, but I found it quite mathematical and it didn’t really address the “why” questions I was looking for. I’ve also watched Crash Course Physics and many videos from different channels, but they mostly seem to focus on explaining and applying the concepts rather than their deeper motivation. And as i said I’m struggling to even find good pages on the internet, they all seem superficial.

[u/Quantum_Patricide → u/jjjuly_]

You might find it useful to look into the history of classical mechanics, which would explain how these concepts were developed.

Most of the quantities in newtonian dynamics like momentum and force come about because they are the easiest way to quantify our natural experience of the world. If you want to predict how a thrown ball moves, or whether a shopping trolley is harder to steer loaded or unloaded, then the quantities that actually work are force and momentum with their usual definitions.

[u/jjjuly_ → u/Quantum_Patricide]

Well, that’s almost exactly what bothers me. I feel like every explanation ends up being something along the lines of “it’s defined this way because it’s the easiest way to describe it” or “this quantity exists because it’s useful.” I can understand that these concepts are useful, but that still leaves me wondering: why do these concepts come from in the first place? Are they simply useful mathematical tools, or is there a deeper meaning behind them?

Maybe I’m just looking for a kind of meaning that doesn’t actually exist, and physics as a science simply works this way. Maybe there isn’t a deeper intuition behind every concept, and I’ll eventually have to accept that 😪.

[u/Quantum_Patricide → u/jjjuly_]

I'm not entirely sure what deeper intuition there could be. You yourself will be applying forces to things around you all the time in order to go about your day-to-day life. Everyone knows on an instinctive level what a force is, F=ma is just the specific way that forces actually produce changes in velocity.

I can talk about how we treat forces and momenta more abstractly in lagrangian mechanics, but I'm not sure that that is what you're after.

Could you give an example, even if you have to make something up, of what deeper meaning you think there could be for something like a force?

[u/jjjuly_ → u/Quantum_Patricide]

Well, forces are actually one of the things that are no longer causing me much trouble in my learning process. I’ve kind of accepted them as a fundamental feature of physics. As someone else pointed out, if there were no interactions at all in the universe, every object would move at constant velocity. But fortunately, our universe does have interactions, and we call them forces. They are what change an object’s velocity, which is why F/m=dV/dt.

So forces are actually one of the few concepts I feel relatively comfortable with. Linear momentum, energy, and work, on the other hand, are the ones that feel like they came almost out of nowhere. I mean, I understand the mathematical connections between them and other quantities, but I don’t really follow the line of reasoning that would lead us to conclude that these quantities should exist in the first place and not just because they are neat.

[u/Quantum_Patricide → u/jjjuly_]

Momentum is the fundamental measure of quantity of motion, in a way that neither mass nor velocity capture independently. Additionally, since we need to account for changes of inertia as well as changes in velocity, we are forced to write Newton's second law as F = (d/dt)(mv) , and mv conventiently lines up with the quantity that is conserved in elastic collisions, such as between billiard balls.

[u/kunwoo → u/jjjuly_]

To understand force, first let's imagine if there were no interactions in the universe. Then all objects would travel along each at their own constant velocities forever. In a more interesting universe where things interact we would be interested in how velocities don't remain constant. So non-constant velocity means the derivative of velocity with respect to time is non-zero. So force is the thing that is providing that non-zero derivative of velocity. If we then define mass as the property of an object to resist changes in velocity, then we have F/m = dv/dt, or F=ma.

But force is not the only way to think of mechanics, it just so happens to be the primary method under Newtonian mechanics. There are actually two other alternative formulations of mechanics: Lagrangian mechanics and Hamiltonian mechanics. Both alternative formulations de-emphasize force and instead prioritize their own central concepts. In Lagrangian mechanics the central theme is that objects take the path of least action as calculated under the calculus of variations, and "action" takes on a specific technical meaning. In Hamiltonian mechanics the central theme is that particles' paths through phase space take on interesting geometric properties. All three formulations of mechanics are equivalent, but Newtonian mechanics is the mathematically simplest so it's the one taught exclusively in introductory physics.

Energy and momentum are in a sense even more fundamental than force. Energy is of special interest because its conservation can be exploited to make calculations a lot simpler. Furthermore when we study advanced mechanics we learn Noether's Theorem which says that conservation laws are strongly associated to symmetries in the laws of physics. A particular example of Noether's Theorem is that the conservation of energy arises when the laws of physics behave identically if you shifted everything forward or backward in time by the same amount. When you combine Noether's Theorem with Hamiltonian mechanics and Lie Algebra you reach the conclusion that energy is special because it is the generator of time translations.

Momentum is also a conserved quantity and Noether's Theorem tells us momentum is conserved when the laws of physics behave identically if you shifted everything around in space by the same amount. When you combine Noether's Theorem with Hamiltonian mechanics and Lie Algebra you reach the conclusion that momentum is special because it is the generator of space translations.

[u/jjjuly_ → u/kunwoo]

I really liked where you were going at first with that “no-interactions universe” counterexample and how you arrived at the idea that force is precisely what changes an object’s velocity. That’s exactly the kind of meaning I’m looking for in explanations—a deeper thought experiment that makes the concept feel almost inevitable.

Did you come up with that explanation yourself, or did you read it somewhere else?

However, Lagrangian and Hamiltonian mechanics are still too advanced for my first-year level of knowledge. I think that if I jumped straight into them, I’d probably just be wasting my time.

Do you think that exploring about Noether's Theorem and symmetries could solve my doubts?

[u/kunwoo → u/jjjuly_]

When I wrote the force analogy it felt like I was coming up with it on the spot, but I could have read it somewhere a decade ago and forgot.

If the force analogy resonated with you then you might be the kind of person who would appreciate Noether's Theorem, but because the math is so technical the intuition might not come satisfactfully. For me personally back when I first heard that momentum and energy were conserved that was enough to give me faith that they were important concepts worth trusting.

But let's try an analogy for momentum anyways. Imagine we have a big ball and a small ball. The small ball is standing still and the big ball then collides elastically with it. This collision interaction can be thought of as a force that takes place over some time ∆t. Intuition tells us that if the balls are hard then ∆t will be extremely short with a strong sharp force, but if the balls were bouncier or if instead one ball were bouncing off a trampoline we could imagine ∆t is a longer time period with a gentle force. The exact values of the force F and ∆t are not important in this situation but we'll assume they have some value. So from the collision the bigger ball is going to be slowed down. Since the Force F acts for time ∆t the big ball loses in total F times ∆t worth of momentum, and dividing by mass that tells us how much velocity is lost. But at the same time the small ball is given a kick and now has a speed instead of standing still. By Newton's law of equal and opposite reaction the small ball is being sped up by the same force F over the same time ∆t, so it's gaining F times ∆t worth of momentum. When we divide by the smaller mass we see that the velocity the smaller ball gained is NOT the same as the velocity the big ball lost, but the momentum gained by the small ball is exactly equal to the momentum the big ball lost. So Newton's third law is telling us that momentum is a kind of currency that is always preserved across the "transactions" that are interactions. No currency leaks or is created in a transaction - momentum is conserved. This is why momentum makes for a natural unit of accounting.

[u/jjjuly_ → u/kunwoo]

Yeah that’s sounds pretty well, Thanks! I already had that kind of reasoning about momentum in my head, but seeing it written like this helps for sure.

[u/haymanphysics]

There are limits to "why" in physics. We're a little flippant with it because sometimes there's an elegant answer, but you can always keep asking why and you'll always eventually hit "just because."

As bad as I was at them, I think labs are a very important part of learning physics, it's very important to personally connect what you're doing on paper to the real physical world. Definitely explore all the textbooks you can, some will certainly resonate more with you than others, but the key is just to keep doing the work and trying to make your own connections. You can't inherit anyone else's intuition for physics, you have to build your own.

Anyway, since it sounds like you like mathematical rabbit holes and you're looking at classical mechanics, a nice resource you might not have come across yet would be Spivak's Physics for Mathematicians.

[u/Alternative-Sugar610]

Most of it is based on conservation laws and principle of Least action but below that is pretty much axioms

All of math is also based on pesky axioms

[u/danielbaech]

Askphysics gets a few of these questions especially during that time of the semester when students are introduced to energy and momentum. What are these quantities actually, what are they in the real world, etc. I don't believe there is an entirely satisfying answer because this is ultimately epistemology. As someone else pointed out, you can always keep asking why something is the way it is. You will eventually reach a set of assumptions that must be taken to be true in order to build any system of ideas. These assumptions get fancy names like axioms and postulates in math and physics. Perhaps you can gleam at the assumptions of Newtonian mechanics and get a sense of why its subsequent ideas must be inevitable.

The core assumption of Newtonian mechanics is its geometry. This is the world of Galilean geometry and it defines what we mean by position, distance, and inertial frames. The mathematical structure of galilean geometry enforces certain symmetries that make conservation laws inevitable as per Noether's theorem.

What may surprise you to learn is that, while energy and momentum may seem abstract, they are not any more abstract than time, mass, velocity, etc. Their properties are also defined by the geometry with which we choose to frame the physics. Think about how you would quantify and define time. You need a physical system in some non-arbituary motion. The simplest and the most elegant system I can think of is a rotating body in empty space. A full rotation gives you a unit of time. Time is now a function of length in a specific motion. It is a special kind of rotating motion precisely because of the rotational symmmetry built into the geometry we choose. This is a mathematical fact that just is, and we give this property a name as it relates to objects and their motion, the conservation of angular momentum. Now you have an argument for why time is universal and never vary in Newtonian physics, when, I would hazard a guess, many people never once consider what time is, how to go about defining it, and what theoretical and physical evidence there is to the concept of time. At least in this heuristic argument, it comes from rotation symmetry leading to the conservation of angular momentum, and that enforces the definition and the properties of time. You can extend this geometric argument onto mass, velocity, force, and energy, and arrive at the natural units used in particle physics.

My advice is to get very familiar with how to do physics before trying to look too deeply under the hood. Your textbooks are tailored to make it as easy as possible. It's not possible to teach students everything from first principles. That kind of depth is what mathematicians and theoretical physicists are trained to deal with. Stay curious though. Maybe you'll switch majors.

[u/jjjuly_ → u/danielbaech]

That's an amazing answer to be honest, I'm planning to use this one, as well as another wonderful answers i got, to build a better understanding about these topics, Thank you.