[問題]about constraints
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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