What mindset is required for Classical Mechanics

[chris_0101]

I hope this is in the right sub forum, but my question is simple. What type of mindset is required to complete problems in a 2nd year classical mechanics course.

Comparing a typical classical mechanics problem to a 1st year physics problem, they are both completely different. I find that a classical mechanics problem requires a person to be imaginative and clever while the other only requires algebra and plugging in formulas in order to solve a problem.

It is this imaginative and clever characteristic that I lack and there are times when I am given a problem, I simply do not know how to begin that problem. I am asking if anybody can suggest ways to escape a 1st year physics mindset and enter a mindset that will allow me excel in this course.

Thanks for your input.

 

[Pengwuino]

It's all about energy! The hardest part, from what I recall in taking that course, was setting up the problem. The actual calculus and linear algebra really are just tools that you become proficient at on the side, but the problem setups are the real difficulty.

If there is a mindset that you must get into, it's about realizing that all of classical mechanics is about energy and there are so few sources of energy. You basically have energies in springs, gravitational potential energy, and kinetic energy (restricting ourselves to classical mechanics).

I saw an interesting presentation on teaching physics that gave a glimpse at why teaching lower-division 1st year physics is so poorly done that applies here. Students in the 1st year courses, and this includes physics majors, engineers, scientists, and non-scientists, see a wide range of possible physics problems. They see spring problems, roller coaster problems, cannon problems, circular motion problems, DC circuit problems, heat engine problems, etc etc. Physicists see these same problems and see conservation of energy problems, conservation of charge problems, and a few other types. The point is, the physicist sees far fewer types of problems than a 1st year student does and this is the kind of mindset you must get into. You must see all the stuff you did in the 1st year and see them as different applications to conservation of energy/charge/momentum. This kind of mindset allows you to become more comfortable in higher level mechanics in terms of setting up problems and looking at problems.

 

[A.T.]

...requires a person to be imaginative and clever...

You answered you own question right there!

But seriously, just deal with enough of those problems, and you will learn how to approach them. It usually boils down to energy conservation, momentum conservation and vector geometry.

 

[AlephZero]

"Imaginative and clever" isn't the right way to look at it, unless you are taking the defeatest attitude that "I'm not smart enough to do this stuff".

Mechanics is probably the first course where you have to jump the barrier of understanding some general principles (Newton's laws, conservation of energy and momentum, equilibrium, etc) and apply them to a whole range of different situations , including ones that you have never seen before.

That's the difference between "doing science or engineering" and "learning how to pass tests in school".

 

[Ken G]

It is this imaginative and clever characteristic that I lack and there are times when I am given a problem, I simply do not know how to begin that problem.

Oftentimes the reason that students don't know how to begin is simply that they don't know where to begin-- at the beginning! In other words, students want to be able to leap right to the final equation that will solve the problem (like you say, in first year, you can often just pull out one formula with the appropriate variables, and poof, you're done). That's a bit like crossing a river by trying to jump over it in a single bound. But that's not beginning at the beginning of the problem, it's beginning at the end. At the beginning, you need to ask, what am I given? What laws seem relevant to this situation? (Be they conservation laws, or F=ma, or constraints like which forces need to balance, etc.). You generally start by writing down what you know, even before you see how you are going to use it, but those are like stepping stones for getting across the river-- one step at a time. You often don't know if you will get across, but it's what you have to go on, so you see where it leads. If it falls short, you are probably forgetting some important relevant principle or constraint that you need to recognize.

What might help is to realize that you have a white sheet of paper, which has no idea what is given in the problem, or what laws of physics apply. You have to tell your paper those things, and then start manipulating them on your paper. Your paper has no knowledge, but terrific memory-- it recalls everything you write on it, and you can use that memory in your manipulations. But if there's something crucial about the problem that you have not written on that blind and dumb page, then no amount of manipulations on the page will make that key factor spring forth. Recognizing those key factors are what it means to begin at the beginning, and not get frustrated if you find it is simply not possible to leap to the final expression without laying out those stepping stones.

 

[fluidistic]

Isn't the 2nd year CM course the OP is talking about a course dealing with the Lagrangian, Hamiltonian, small oscillations, canonical transformations, etc? Rather than "just" Newton's laws.
In which case I do not think that using conservation of energy/momentum is enough at all.

 

[Ken G]

Most second-year courses do a lot of things. They probably include Lagrangians, but also more traditional Newtonian problems, perhaps involving fictitious forces. The OPer is right that the problems require more thought and more putting together the pieces, rather than just writing down the answer (conservation of energy can often be used to just write down the answer, so is usually not enough in second-year problems, but it can often be one of the pieces, depending on the problem). If it is a Lagrangian problem, then what's tricky is identifying the generalized coordinates in ways that automatically include the constraints on the system. The rest is often quite cook-booky.

 

[fluidistic]

Most second-year courses do a lot of things. They probably include Lagrangians, but also more traditional Newtonian problems, perhaps involving fictitious forces. The OPer is right that the problems require more thought and more putting together the pieces, rather than just writing down the answer (conservation of energy can often be used to just write down the answer, so is usually not enough in second-year problems, but it can often be one of the pieces, depending on the problem). If it is a Lagrangian problem, then what's tricky is identifying the generalized coordinates in ways that automatically include the constraints on the system. The rest is often quite cook-booky.

I see. At my university (in Argentina) CM is a 3rd year course but since we take "only" 3 courses per semester it would correspond to a 2nd year course in the US. The recommended books are Goldstein's and Landau & Lifgarbagez's books (our programme consists almost exclusively of the name of the chapters in the latter book). We have 7 sets of assignments, number 0 (called "Before the beginning") has only Newtonian mechanics problems. The final exam counts for 100% of our mark for that course. I've never seen any Newtonian mechanics problem in past exams.
For this course at my university, knowing how include the constraint equation(s) into the potential energy part of the Lagrangian isn't enough, for a Lagrangian problem. Indeed, they can ask us to solve the problem using Lagrange multipliers and thus to write the "modified" Lagrangian. So it's not only a matter on how to transcript the problem into mathematical expression and then solve it; it's also about solving the problem in all possible ways.
Also, they ask us what quantities are conserved. We cannot say "angular momentum" without any algebra that back up this claim. We have to show that a coordinate is cyclic in the Lagrangian and therefore the generalized momentum associated to this coordinate is a constant of motion.
So unlike the first year course, you can't just assume that the angular momentum is conserved and use this fact to solve a problem. This may sound ridiculous but this is the way it is.
P.S.:1)I know a member in this forum from Spain and he described his course as very similar to mine, with the same books recommended, etc. So I don't think my university is a "special case".
2)I didn't take the final exam yet but I can take it on the 7th of March if I feel ready. I still have a mountain to study. I must reach Poisson's bracket and Hamilton-Jacobi reformulation of CM. :/ Also have to study Euler's angles and canonical transformations.
3)To answer the OP: I've noticed that this 2nd year course takes a very, very long of my (and friends) time. If you don't know how to even start a problem, try to find a similar problem in as many books on CM as you can and on the Internet. If you're stuck you can post here or ask friends/professors to help you. Do not "suffer" silently.

 

[Ken G]

Yes, that does sound like a pretty complete version of that level of mechanics. It would be a very good basis for graduate-school-level CM. What bothers me about it is that it seems so centered on formal approaches to solving problems, but that's actually pretty far from what actual physics research is like. It's not that obvious what all that work is really buying for you, other than preparing you for more of the same in graduate school and on qualifying exams. Those are very important, of course, and you do build useful concepts (like where conservation laws come from), but it still sounds to me that it is very canned, being so based around solving various classes of rather stale problems. It doesn't really sound much like either physics research, or like the concepts of physics that drive my interest in it. It brings to my mind images of late nights hammering away at yet another variation on the basic theme of canned problem solving, scouring books and classmates for anyone who has seen a similar problem solved. That was never the reason I got into physics, is it really a necessary component of the education of a physicist, or just some kind of rite of passage?