Should I Be Concerned About Missing Lagrangian Mechanics in My CM Course?

[jbrussell93]

I'm in my junior year and recently took classical mechanics. We did not cover Lagrangian or Hamiltonian mechanics which was very shocking to me. My instructor said that Lagrangian mechanics involve very little physical intuition and therefore, time would be better spent with Newtonian mechanics. The closest thing we did involved using the "method of constraints" which supposedly leads into Lagrangian mechanics. While I feel that I learned A LOT in this course, I also feel that I missed out on something very important...

I took a math methods course concurrently with mechanics where we, the students, convinced the instructor to at least show us how to solve a problem using Lagrange's equations. He wrote down the equations and said, "That is all you need to know". We solved a single problem and that was it. Aside from the derivations, is this really all there is to it?

My school does not offer a second semester of mechanics for undergrads so I'm considering taking the graduate course next semester, although I don't have much room in my schedule. Of course, I can always learn the material on my own, but I've found that being formally presented the material in lecture and then tested on it helps me retain things. Is it worth taking a second semester? I am planning on graduate school in physics/geophysics.

Any advice is appreciated.

 

[maajdl]

Learn that on your own.
You will enjoy, as this is beautiful.
It might also be useful later.
The Least Action Principle and Lagrangian mechanics is very useful in current theoretical physics.
Read the famous lecture by feynman on the least action principle.

http://www.cs.helsinki.fi/u/ldaniel/mm_cn/FeynmanPrincipleofLeastAction.pdf

also:



as well as -maybe- some courseware you may find out.

And enjoy also this classic reference:

https://archive.org/details/Mechanics_541

 

[Lavabug]

I'm in my junior year and recently took classical mechanics. We did not cover Lagrangian or Hamiltonian mechanics which was very shocking to me. My instructor said that Lagrangian mechanics involve very little physical intuition and therefore, time would be better spent with Newtonian mechanics. The closest thing we did involved using the "method of constraints" which supposedly leads into Lagrangian mechanics. While I feel that I learned A LOT in this course, I also feel that I missed out on something very important...

I took a math methods course concurrently with mechanics where we, the students, convinced the instructor to at least show us how to solve a problem using Lagrange's equations. He wrote down the equations and said, "That is all you need to know". We solved a single problem and that was it. Aside from the derivations, is this really all there is to it?

My school does not offer a second semester of mechanics for undergrads so I'm considering taking the graduate course next semester, although I don't have much room in my schedule. Of course, I can always learn the material on my own, but I've found that being formally presented the material in lecture and then tested on it helps me retain things. Is it worth taking a second semester? I am planning on graduate school in physics/geophysics.

Any advice is appreciated.

Bizarre. A course on this subject really put geometric intuition to good use. I had two analytical mechanics courses in my 2nd year. The first covered point-particle mechanics using Lagrangians among other things (SR and waves), while the 2nd was on rigid body dynamics, small oscillations, and Hamilton-Jacobi theory.

If you're going to grad school, you'll likely do one or two courses just on those topics you're missing, so I would not worry too much.

But if you want to self study on a time and financial budget, get the book on the subject by Calkin and/or Landau (I would pass on buying Goldstein) and study it over the summer. This is overkill for the type of Lagrangian mech problems in the PGRE, but it's good to know nonetheless.

 

[jbrussell93]

I appreciate the videos and the advice. The book we used was Mechanics by Symon.

It sounds like I should definitely do some self study then to at least to familiarize myself with it. Maybe the grad course if I can squeeze it in...

 

[ZapperZ]

I agree with Lavabug's comment that this is bizarre. Considering that QM is based on using a "Hamiltonian" as the starting point, not having an idea of what it is from classical mechanics is a severe problem. This is besides the fact that there are many situations in which using the Lagrangian/Hamiltonian approach is simpler to solve than using the Newtonian method.

I will also suggest that you look up Mary Boas's text "Mathematical Methods in the Physical Science" (which you should own one if you are a physics major anyway), and study the chapter on Calculus of Variation. This introduces you to the concept of Least Action principle that leads to the Lagrangian/Hamiltonian.

Zz.

 

[jbrussell93]

I will also suggest that you look up Mary Boas's text "Mathematical Methods in the Physical Science" (which you should own one if you are a physics major anyway), and study the chapter on Calculus of Variation. This introduces you to the concept of Least Action principle that leads to the Lagrangian/Hamiltonian.

I used Boas in my math methods course so I'll definitely give that chapter a look. Thanks for the suggestion.

Lets say that I am not able to take the graduate mechanics course I mentioned before graduating, and my only exposure to Lagrangian mechanics is through self study. How much will that hurt me in graduate school, besides lacking some familiarity? Do professors generally assume this knowledge? I am worried about falling behind the others.

 

[jesse73]

What is the point of a junior level CM class without lagrangians and hamiltonians. That is the most vital part for having some grounding on how hamiltonians work for QM with classical systems for which you have an intuition for .

 

[jesse73]

A upper level undergrad CM course isn't required for graduating in Physics at MIT. You can get by. However in reality they are really useful.

 

[WannabeNewton]

How much will that hurt me in graduate school, besides lacking some familiarity?

You'll be fine. There is very little theory at the level you'd be interested in. The rest is just practice ad infinitum.

Do professors generally assume this knowledge? I am worried about falling behind the others.

It would probably depend on the class. In my UG QM class you were expected to have already seen Hamiltonian mechanics and Lagrangian mechanics but again there is very little theory at this level. Just do enough practice problems from books like Taylor or Calkin and you'll be just fine. The calculations will mostly amount to clever use of Euclidean plane geometry and coordinate systems. If you want a deep appreciation of the variational formalism and why it's important in classical physics then you would have to work through and understand L&L's beautiful text on classical mechanics (not an easy task by any means!).

 

[homeomorphic]

I can see why the professor wouldn't like Lagrangian/Hamiltonian mechanics. I didn't like it very much either when I first saw it, and it seems that the professor never moved beyond that level of understanding.

This is one area where my extensive background in math seems to serve me very well. It's not so much that I need it explicitly to solve problems. But I do use it to be able to perceive the beauty of Lagrangian/Hamiltonian mechanics. In particular, when dressed in more mathematical clothing, the idea that it involves very little intuition is easily seen to be false.

Some references for this point of view:

Susskind's lectures on classical mechanics:

The Road to Reality, by Roger Penrose.

Mathematical Methods of Classical Mechanics by V. I. Arnold.

Baez's course notes: http://math.ucr.edu/home/baez/classical/.

And if I ever get around to writing my own ideas up, that will filling in the missing pieces not covered by those (all this should be supplemented by doing a ton of problems from more standard course textbooks).

 

[Lavabug]

I agree with Lavabug's comment that this is bizarre. Considering that QM is based on using a "Hamiltonian" as the starting point, not having an idea of what it is from classical mechanics is a severe problem.

On the classical side, any exposure to statistical physics will immediately deal with phase spaces, the theorem of Liouville (which reappears in QM as Ehrenfest's theorem...) and Hamilton's equations (Hamiltonians appear in most of the Gibbs ensembles' partition functions...).

Also, even a basic "poor man's introduction" to General Relativity like I had makes use of the action principle to obtain the geodesic equation of motion.

On the other hand, I've had some private communication with a phd graduate here who confessed he didn't learn Hamilton/Lagrangian mech as an undergrad and just had to work extra hard as a grad, and he turned out rather fine.

In short: you need Lagrangian/Hamiltonian mechanics. Just try to obtain Newton's equations for a point particle moving in a cone, bowl, helix, cylinder or on any arbitrary surface by decomposing your force vectors and you'll see why. In LM all you need to do is properly pick your generalized coordinates with some geometric thinking, crank out your Lagrangian and apply the Euler Lagrange equation. As an added bonus, writing out that curly L is cool and it makes you feel important.

 

[jbrussell93]

I appreciate the input from everyone. I'll definitely check out the links and work through the problems in my mechanics book.

Just out of curiosity, how many semesters of classical mechanics are you all required to take? Is it standard to take 2 semesters? I'm curious because we essentially covered the first half of Mechanics by Symon so ideally a second semester would be required to cover the second half.

 

[Lavabug]

I didn't attend college in the US, but I had 2 as a junior. Calculus-based Newtonian mechanics, oscillations, fluids and thermodynamics were covered in my 1st year annual physics course.

Most US grad school guides I've seen expect undergrads to know mechanics at the level of Taylor (see gradschoolshopper or any grad school's page for prospective students).

 

[dextercioby]

It may be that for a path in engineering, taking a classical mechanics course w/o Lagrangians and Hamiltonians could be the right solution. For engineering, PDEs and numerical methods are much more important. But then on, if you're trying to become a physicist, then taking a second course on mechanics from a more abtract/mathematical perspective is a must.

 

[radium]

I don't know how you can teach that course without lagrangians. That is the key to everything about symmetry and Noether's theorem.
Knowing the Lagrangian formalism of classical mechanics is crucial for quantum field theory. For every field you can start with a classical Lagrangian, find the classical equation of motion with solutions (plane waves solutions for free fields), the symmetries and then go back and quantize it by turning everything into operators. In quantum mechanics, the Lagrangian appears in the action which is all over the place when you get to path integrals, the Aharanov Bohm effect. Gauge invariance and topological properties all come from the symmeties of the Lagrangian.

 

[jasonRF]

jbrussell93,

I doubt this will put you behind in your other classes from this same department - after all, your classmates are all in the same boat. If you are worried about it I recommend spending a few weeks learning it over the summer if you can. Since you are a physics major I suspect you will find it fun to learn, and it will add insight to future classes even if it isn't required prerequisite knowledge. Enjoy!

jason

 

[Jorriss]

Not covering Hamiltonian's mechanics as an undergraduate is nothing to be concerned about, that is not atypical but skipping lagrangian mechanics is more startling. You should study this independently in my opinion.