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
d @L @L d ¡ 2_¢
dt _ = @µ =) dt m(‘ + x) µ = ¡mg(‘ + x)sinµ
=) 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,
¡(‘ + 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
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.