884 APPENDIX C. LAGRANGIAN MECHANICS

+Mgl cosθ =C. (3.8)

We can use the conservation of energy along with the equations of motion to gainunderstanding of the spinning top. From 3.6 we see there is a constant, P such thatI3 cos(θ)φ

′+ I3ψ ′ = P and so

ψ′ =

P− I3 cos(θ)φ′

I3(3.9)

and from 3.5 there is a constant, Q such that I1 sin2 (θ)φ′+ I3 cos2 (θ)φ

′+ I3 cos(θ)ψ ′ =Q. This along with 3.9 implies I1 sin2 (θ)φ

′+Pcos(θ) = Q and so we also have

φ′ =

Q−Pcos(θ)I1 sin2 (θ)

. (3.10)

Therefore, from the conservation of energy,

12

I1

[sin2 (θ)

(φ′)2

+(θ′)2]+

I3

2P2 +Mgl cosθ =C

and using 3.10 to find φ′, and adjusting the constant,

I1(θ′)2

+ I1(Q−Pcos(θ))2

I1 sin2 (θ)+ I3P2 +2Mgl cosθ =C

The expression, f (θ) = I1(Q−Pcos(θ))2

I1 sin2(θ)+ I3P2 + 2Mgl cosθ is concave up and has some

assmptotes. If C happens to equal the minimum value of f then we must have θ′ = 0 and

so the top will circle around the x3 axis with θ a constant. Thus we would observe the anglebetween the axis of the top and the x3 axis would be constant. If C is not the minimum valueof f then we will have θ changing between two values. This is called nutation. Also, from3.10 we see that φ

′ is probably not zero. Thus the line of nodes moves around the x3 axis.Even ψ ′ may change due to 3.9. If ψ ′ were known to be constant, then you could use 3.9to conclude φ

′ = Ccosθ

.