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Lagrangian Method.pdf

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University of Waterloo
PHYS 112
Emiko Yoshida

6.1. THE EULER-LAGRANGE EQUATIONS VI-3 There are two variables here, x and µ. As mentioned above, the nice thing about the La- grangian method is that we can just use eq. (6.3) twice, once with x and once with µ. So the two Euler-Lagrange equations are d ‡ @L · @L = =) mx˜ = m(‘ + x)µ + mg cosµ ¡ kx; (6.12) dt @_ @x and ‡ · d @L @L d ¡ 2_¢ dt _ = @µ =) dt m(‘ + x) µ = ¡mg(‘ + x)sinµ @µ =) m(‘ + x) µ + 2m(‘ + x)_µ = ¡mg(‘ + x)sinµ: =) m(‘ + x)µ + 2mx_µ = ¡mg sinµ: (6.13) Eq. (6.12) is simply the radial F = ma equation, complete with the centripetal acceleration, 2 ¡(‘ + x)µ . And the flrst line of eq. (6.13) is the statement that the torque equals the rate of change of the angular momentum (this is one of the subjects of Chapter 8). Alternatively, if you want to work in a rotating reference frame, then eq. (6.12) is the radial F = ma equation, complete with the centrifugal force, m(‘ + x)µ . And the third line of eq. (6.13) is the tangential F = ma equation, complete with the Coriolis force, ¡2m_µ. But never mind 2 about this now. We’ll deal with rotating frames in Chapter 10. Remark: After writing down the E-L equations, it is always best to double-check them by trying to identify them as F = ma and/or ¿ = dL=dt equations (once we learn about that). Sometimes, however, this identiflcation isn’t obvious. And for the times when everything is clear (that is, when you look at the E-L equations and say, \Oh, of course!"), it is usually clear only after you’ve derived the equations. In general, the safest method for solving a problem is to use the Lagrangian method and then double-check things with F = ma and/or ¿ = dL=dt if you can. | At this point it seems to be personal preference, and all academic, whether you use the Lagrangian method or the F = ma method. The two methods produce the same equations. However, in problems involving more than one variable, it usually turns out to be much easier to write down T and V , as opposed to writing down all the forces. This is because T and V are nice and simple scalars. The forces, on the other hand, are vectors, and it is easy to get confused if they point in various directions. The Lagrangian method has the advantage that once you’ve written down L · T ¡V , you don’t have to think anymore. All you have to do is blindly take some derivatives. 3 When jumping from high in a tree, Just write down del L by del z. Take del L by z dot, Then t-dot what you’ve got, And equate the results (but quickly!) But ease of computation aside, is there any fundamental difierence between the two meth- ods? Is there any deep reasoning behind eq. (6.3)? Indeed, there is... 2Throughout this chapter, I’ll occasionally point out torques, angular momenta, centrifugal forces, and other such things when they pop up in equations of motion, even though we haven’t covered them yet. I flgure it can’t hurt to bring your attention to them. But rest assured, a familiarity with these topics is by no means necessary for an understanding of what we’ll be doing in this chapter, so just ignore the references if you want. One of the great things about the Lagrangian method is that even if you’ve never heard of the terms \torque," \centrifugal," \Coriolis," or even \F = ma" itself, you can still get the correct equations by 3imply writing down the kinetic and potential energies, and then taking a few derivatives. Well, you eventually have to solve the resulting equations of motion, but you have to do that with the F = ma method, too.
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