[問題]about constraints

[njvulfu (njvulfu)]

大家好，

小弟在 classical mechanics ( Goldstein ) 裡，

第47中第二段的一個example中有一個不懂的地方，

原文如下:

As an example , consider a smooth solid hemisphere of radius a place with its

flat side down and fastened to the earth whose gravitational acceleration

is g. Place a small mass M at the top of the hemisphere with an infinitesimal

displacement off center so the mass slides down without friction. Choose

coordinates x,y,z centered on the base of the hemisphere with z vertical and

the x-z plane containing the initial motion of the mass.

    Let θ be the angle from the top of the sphere to the mass. The Lagrangian

       1      .2  .2  .2
is L = ─ M ( x + y + z ) - mgz. The initial conditions allow us to ignore the
       2
                                                    2   2
y coordinate , so the constraint equation is a - ( x + z ) = 0. Expressing the

                    2   2   2       x
problem in term of r = x + z  and  ─ = cosθ, Lagrange's equation are
                                    z
   .2                           2..
Maθ - Mg cosθ + λ = 0 ,and Ma θ+ Mga sinθ = 0. Solve the second equation

                              .2    2g           2g
and then the first to obtain θ = -── cosθ + ── and λ= Mg(3cosθ-2)
                                    a            a

問題來了

         .2
    1.Maθ - Mg cosθ + λ = 0 這一個equation怎麼來的?

       .2    2g            2g
    2.θ = -── cos θ + ── 這怎麼解出來的?
             a             a


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