How to find the equation of motion using Lagrange's equation?

[Pironman]

Good morning, I'm not a student but I'm curious about physics.
I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces.
From Wikipedia, I found two method:

1. using kinetic energy and generalized forces only:

in this way I must compute the kinetic energy, and the projection of my external forces into the generalized coordinate.

2. using the lagrangian, and generalized forces:

Here I must compute the kinetic and potential energy, and instead of "0" in the right side of the equation I must put my external forces.

So, I think that using the second method I have more calculation to do, because I must compute the potential energy, that I don't need using the first method. An example of the application of these two method are in the wikipedia page of inverted pendulum, where the equations are computed using three method (these two, and Newton method): https://en.wikipedia.org/wiki/Inverted_pendulum#From_Lagrange's_equations

So, where is the advantages of compute the lagrangian? If I have to choose a method, when one is more convenient than the other?Thank you

 

[apostolosdt]

Look again at that Wikipedia article. The two methods are equivalent via D'Alembert's principle (forget Newton). Therefore, which method one chooses depends on what physical quantities one readily has; in other words, it is a matter of convenience.

Now, using the Lagrangian in an Euler-Lagrange equation is more important in beyond-the-Newtonian physics for a number of reasons; perhaps the most important is that the Lagrangian can be formulated to observe the symmetries of the system under study.

On the other hand, there are Newtonian-mechanics-related areas where the Euler-Lagrange approach is the only practical method of attacking a problem. That's because the differentiation is done by generalized, not merely space, coordinates. Mechanical engineering is heavily using that method; look for the so-called theoretical mechanics branch.

Since you are interested In physics at that level, it is vital that you construct a reasonable "roadmap", otherwise you may get lost in the math and physics prerequisites. So, stick with the Lagrangian!

 

[Coelum]

Good morning, I'm not a student but I'm curious about physics.
I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces.
From Wikipedia, I found two method:

1. using kinetic energy and generalized forces only:

in this way I must compute the kinetic energy, and the projection of my external forces into the generalized coordinate.

2. using the lagrangian, and generalized forces:

Here I must compute the kinetic and potential energy, and instead of "0" in the right side of the equation I must put my external forces.

So, I think that using the second method I have more calculation to do, because I must compute the potential energy, that I don't need using the first method. An example of the application of these two method are in the wikipedia page of inverted pendulum, where the equations are computed using three method (these two, and Newton method): https://en.wikipedia.org/wiki/Inverted_pendulum#From_Lagrange's_equations

So, where is the advantages of compute the lagrangian? If I have to choose a method, when one is more convenient than the other?Thank you

Hi Pironman,
in practice the Lagrangian approach is usually more convenient. In most cases, computing the potential energy is relatively easy, while projecting all the forces on the generalized coordinates can be tricky. I have seen the first approach used only to deal with friction, that cannot be modeled using the Lagrangian approach (except a few simple cases).

As a rule of thumb: the more complex the system, the more convenient the Lagrangian approach is. Which is exactly the point of using the Lagrangian. If you think of it, the first approach stops midway: you get the generalized momenta from the kinetic energy, but do not the generalized forces from the potential. If you are not convinced, have a look at your preferred book on Analytical Mechanics and see how many times the two methods are used for moderately complex problems.

To answer your last question: I use the Lagrangian whenever computing directly the generalized forces is more cumbersome than computing the potential and its derivatives. Which happens almost always.

 

[anorlunda]

Perhaps if someone can point the OP to a worked example, it might clarify it for him.

 

[Vanadium 50]

Actually, I was going to suggest the OP look at a text like Marion or Symon. This is a big field, and entire books are written about it (and related topics), A book will answer this, provide examples, provide questions to test understanding, and also do the same for followup questions.

 

[Coelum]

I'd recommend the good old Goldstein/Poole/Safko. Chapter 1 has a nice, clear discussion of the D'Alembert principle and Lagrange equations (section 1.4), along with examples (section 1.6) and a good set of exercises at the end of the chapter. The mathematical prerequisites are minimal.

 

[vanhees71]

Only the 1st and 2nd edition of Goldstein can be recommended! The later editions are spoiled by authors in an attempt to modernize it, but they have introduced serious errors (particularly about non-holonomic constraints).

 

[Pironman]

First of all, thanks to everyone.
Now I understand why use one or another form, so the main topic question is solved.
I will look to the books you advised me.

For a little bit more information, to better understand my problem, I'm mechanical engineer and I'm trying to write a mathematical model of a motorcycle, in my free time, to better understand it's dynamics (it's not my job but I'm really passionate to motorcycle).
I've tried using Newtonian mechanics, but I've failed. So I start with Lagrange, but at university we study only Newtonian mechanics so I must start to study again after 10 years.

 

[apostolosdt]

[...] For a little bit more information, to better understand my problem, I'm mechanical engineer and I'm trying to write a mathematical model of a motorcycle, in my free time, to better understand it's dynamics (it's not my job but I'm really passionate to motorcycle).
I've tried using Newtonian mechanics, but I've failed. So I start with Lagrange, but at university we study only Newtonian mechanics so I must start to study again after 10 years.

I'm sure there are more comprehensive textbooks on theoretical mechanics and more knowledgeable members, but in your case, I would first thumb through Schaum's Series with the same title by Spiegel. (There is also, again in Schaum's series, the Lagrangian Dynamics by Wells.)

 

[andresB]

Look again at that Wikipedia article. The two methods are equivalent via D'Alembert's principle (forget Newton). Therefore, which method one chooses depends on what physical quantities one readily has; in other words, it is a matter of convenience.

This is not true. The equivalence only happens when all the forces can be derived from a potential, the obvious counterexample are dissipative forces.

 

[apostolosdt]

This is not true. The equivalence only happens when all the forces can be derived from a potential, the obvious counterexample are dissipative forces.

I was trying to keep things simple. The least OP is needed are fully detailed answers. I've seen numerous queries from students that elicited monograph-like discussions and must say I'm not in favour of such attitude. What's coming next? Discussion of non-Hermitian Hamiltonians?

Otherwise, your remark is correct; thanks for pointing it out to me.

 

[Coelum]

First of all, thanks to everyone.
Now I understand why use one or another form, so the main topic question is solved.
I will look to the books you advised me.

For a little bit more information, to better understand my problem, I'm mechanical engineer and I'm trying to write a mathematical model of a motorcycle, in my free time, to better understand it's dynamics (it's not my job but I'm really passionate to motorcycle).
I've tried using Newtonian mechanics, but I've failed. So I start with Lagrange, but at university we study only Newtonian mechanics so I must start to study again after 10 years.

I am a Mechanical Engineer, too - though I always worked in IT.
Let me re-state my recommendation: try Goldstein. I agree with vanhees71 that the last editions did not improve over the second one: get it if you can. Solutions to the problems can be easily found online, if you need it. Goldstein will give you a level on insight that Schaum will not.
Otherwise, if you have time, use both - maybe concurrently. Enjoy your trip in Classical Mechanics!

 

[Pironman]

Right now I started with the first volume of the physics series of Lev D Landau, just because I already have it, I bought it some years ago but never readen.

 

[Coelum]

Right now I started with the first volume of the physics series of Lev D Landau, just because I already have it, I bought it some years ago but never readen.

Great book, but not for self-teaching. Quite terse, few exercises. You may want to augment it with Schaum's exercises.