I know F = ma, but how will a Lagrangian help me?

[Saketh]

My physics career up to this point has been introductory mechanics (an AP Physics: Mechanics course), centered around Newton's second law - $$\vec{F} = m\vec{a}$$. Having finished the course, I sought new methods of solving problems.

I stumbled upon a tome; Goldstein's Classical Mechanics, third edition. After flipping through the first few chapters, I found something that was jaw-dropping and intriguing, but extremely confusing. This was the Lagrangian method of solving mechanics problems, using the identity that $$L \equiv T - V$$. Wanting to learn more, I opened up a textbook online, written by David Morin, a professor at Harvard. However, the approach there confused me as well.

I am fluent in the mathematics behind the Lagrangian method, but the physics of it confuses me. The authors of the textbooks appear to apply constraints as if it were their second nature, while I am struggling to understand how to apply the constraint. Naturally, this is due to my inability to understand the concept, but I cannot seem to find a gentler introduction to the Lagrangian method. David Morin praises the method, saying that it is better than $$\vec{F} = m\vec{a}$$ almost all of the time.

My questions are:

1. Will the Lagrangian method increase the efficiency of solving $$\vec{F} = m\vec{a}$$ mechanics problems? (I realize this is subjective)
2. Where can I find a gentler introduction to the Lagrangian method and the principle of least action, preferably geared to someone coming from an $$\vec{F} = m\vec{a}$$ background?

Thank you for your assistance!

(P.S. I thought this was more appropriate for the Classical Physics forum, since it is neither my homework nor classwork.)

 

[maverick280857]

As you say you are fluent with the mathematics behind Lagrangian formulations (which is basically Variational Calculus) you can try solving simple problems with the Lagrangian method (simple pendulum, block on a wedge when both the block and wedge can move, pulley systems, etc.) to get a feel for it.

As for your first question, it depends on what problem you are solving. If the form of the Lagrangian is too complex to be differentiated when the problem can be done with Newton's laws cold, then you obviously would prefer the second line of attack.

A gentle introduction? Well if you are familiar with variational calculus then almost all introductions should be appear to be gentle You can try http://www.resonancepub.com/lagrangian.htm for a start (it gets complex towards the end, but its good for a start). Google it up as "Lagrangian tutorial" or "Lagrangian dynamics tutorial". By the way, Feynman has dealt with the Principle of Least Action very nicely in his "Lectures on Physics". That's a must read for anyone interested in the area.

If you want, you can discuss some problems on this thread too.

 

[jtbell]

My physics career up to this point has been introductory mechanics (an AP Physics: Mechanics course), centered around Newton's second law - $$\vec{F} = m\vec{a}$$. Having finished the course, I sought new methods of solving problems.

I stumbled upon a tome; Goldstein's Classical Mechanics, third edition.

Oy! As you may or may not be aware, Goldstein is a graduate-school level textbook. I wouldn't expect a high-school student to get much out of it.

College freshman-level physics books don't do Lagrangian mechanics either. You need to look for a sophomore-level book like Fowles & Cassiday. Or look on college and university Web sites for their sophomore-level courses (not the junior/senior level courses which tend to be more advanced).

I haven't taught a course that covers Lagrangians, myself, so I can't think of other books off the top of my head besides Fowles & Cassiday. I remember it only because that's the book I used (in an earlier edition) when I was an undergraduate.

You're already aware that problems can often be solved more simply by using energy-related methods, or comservation of momentum, than starting from scratch with Newton's Laws, right? You can think of Lagrangian mechanics as a more sophisticated way of applying those concepts.

 

[maverick280857]

After all that jtbell said, I think I can safely say that you should read the Feynman Lectures section on the Action Principle, if you are interested. You can skip the more mathematical parts of it if you feel like.

 

[Triss]

I think Morin's book is the most gentle introduction to Lagrangian mechanics that i have seen since it is written for undergraduates and doesn't use a lot of Variational Calculus (the most basic, but not very relevant to you, could be http://arxiv.org/abs/physics/0004029) .

The problem with constraints is that (a certain type constraints) is removed straight away by choosing a decent coordinate system. That is properly why it seems very odd from when you are used to from Newtonian mechanics. That also somewhat answers your first question: Choosing the "right" coordinates removes some of the constraint equations from the problem and makes it easier to solve. You'd have to do a little work do to this in Newtonian mechanics.

 

[Saketh]

Thank you for the replies!

I realize now that my confusion with the Lagrangian stems from my lack of experience with working in the polar coordinate system in physics problems. I will read Feynman's lecture on the Principle of Least Action, while adjusting to the polar coordinate system.

 

[nrqed]

My physics career up to this point has been introductory mechanics (an AP Physics: Mechanics course), centered around Newton's second law - $$\vec{F} = m\vec{a}$$. Having finished the course, I sought new methods of solving problems.

I stumbled upon a tome; Goldstein's Classical Mechanics, third edition. After flipping through the first few chapters, I found something that was jaw-dropping and intriguing, but extremely confusing. This was the Lagrangian method of solving mechanics problems, using the identity that $$L \equiv T - V$$.

That's not the best book to start with gently!
Do you have Feynman's lectures?

I would highly recommend Schaum's outline book on Lagrangian dynamics. I think that it would be good for you to do a lot of calculations to see how it is applied.

I am fluent in the mathematics behind the Lagrangian method, but the physics of it confuses me.

maybe if you would post specific quotes from these books we could try to help you more.

My questions are:

1. Will the Lagrangian method increase the efficiency of solving $$\vec{F} = m\vec{a}$$ mechanics problems? (I realize this is subjective)

1. 
   
   Definitely. But to see this, you would have to do a few more complicated problems than the ones covered in high school (for example a rigid pendulum connected to the extremity of a second rigid pendulum. Or letting a mass slide down an inclined block which itself may slide on a frictionless surface). Again, Schaum's outline is good in that you will see the full solutions. Then if you try to solve those problems with F=m a you will see how harder it would be.
   
   But the power of the Lagrangian/Hamiltonian formulations goes way beyond that.
   One example is the treatment of fields (example: the electromagnetic field). One can define a momentum for an EM field, and a total energy and so on. But one cannot use F=ma (there is no mass around!). The Lagrangian approach is, however, fully applicable and one can derive Mawell's equations from a Lagrangian approach. This can then be used to understand better the symmetries of the system.
   
   Another important motivation: when making the transition to quantum mechanics, F=ma is useless (the acceleration of a quantum particle is not defined except as an average quantity). However, the Lagrangian/Hamiltonian approaches *are* suitable to a generalization to quantum mechanics, which is quite remarkable.
   
   
   My two cents
   
   Patrick

 

[maverick280857]

I think Morin's book is the most gentle introduction to Lagrangian mechanics that i have seen since it is written for undergraduates and doesn't use a lot of Variational Calculus (the most basic, but not very relevant to you, could be http://arxiv.org/abs/physics/0004029) .

Triss, this link was very nice, thanks! When I was in high school (in India, this is Class 12 and we call it higher secondary) I could never find such an introduction to Lagrangian Mechanics and so I had to hunt around and read stuff from several places including a bit of Goldstein. As ngred has stated though, you get the hang of it only when you apply it to more complex problems.

 

[vanesch]

That's not the best book to start with gently!
Do you have Feynman's lectures?

I would highly recommend Schaum's outline book on Lagrangian dynamics. I think that it would be good for you to do a lot of calculations to see how it is applied.

I would second everything that has been said here. Goldstein is a great book (the "bible" on classical mechanics), but not directly the easiest one. The Schaum book is great to get acquainted with practical uses of the Lagrangian method, and Feynman really gives you a great conceptual understanding of it.

Again, Schaum's outline is good in that you will see the full solutions. Then if you try to solve those problems with F=m a you will see how harder it would be.

But the power of the Lagrangian/Hamiltonian formulations goes way beyond that.
One example is the treatment of fields (example: the electromagnetic field). One can define a momentum for an EM field, and a total energy and so on. But one cannot use F=ma (there is no mass around!). The Lagrangian approach is, however, fully applicable and one can derive Mawell's equations from a Lagrangian approach. This can then be used to understand better the symmetries of the system.

Another important motivation: when making the transition to quantum mechanics, F=ma is useless (the acceleration of a quantum particle is not defined except as an average quantity). However, the Lagrangian/Hamiltonian approaches *are* suitable to a generalization to quantum mechanics, which is quite remarkable.

So I agree with all the above as "publicity" for the Lagrangian approach.
However, one caveat: these variational methods are useless in practical *real-world* engineering applications where friction is to be taken into account. From the moment there's friction (and in all engineering applications, this enters into the considerations at one or another moment), the variational techniques fall on their face.
There's no conceptual difficulty with this, because friction is not something which is fundamental (it is just the neglecting of lots of microscopic degrees of freedom to take care of the practical side of the problem). But an engineer doesn't care about that: he has a phenomenological friction law which allows him to estimate the effects of friction, and this can not (or very awkwardly) be incorporated in any Lagrangian technique.

So I'd say: once you've done the double or triple pendulum, with a Lagrangian and an F=m a technique in the ideal, frictionless case (you'll be convinced of the efficiency of the Lagrangian approach), now add the extra item of having some friction couple proportional to the square of the angular velocity at each connection, and try again...

 

[ChrisHarvey]

Try Landau and Lif****z, Course of Theoretical Physics, Volume 1 (Mechanics), chapter 1. My lecturer made a pig's ear out of trying to explain the Langrangian. Thankfully this book was listed in the recommended reading. I borrowed it from the library initially, but finding it such a good book, I then went and bought it. It starts with the principle of least action which I find is a much gentler approach to the derivation.

Your other question: I find the Lagrangian can either hugely simplify a problem, or really complicate things. For 1 DOF systems, I think F=ma is easier. For more complicated 2 DOF systems or higher, use the Lagrangian!

 

[ChrisHarvey]

lol - it doesn't like the word "****". The author is L-I-F-S-H-I-T-Z.