B.5. TRANSFORMATION OF COORDINATES. 347

9. Suppose the pendulum is not assumed to vibrate in a plane. Let it be suspended atthe origin and let φ be the angle between the negative z axis and the positive x axiswhile θ is the angle between the projection of the position vector onto the xy planeand the positive x axis in the usual way. Thus

x = ρ sinφ cosθ ,y = ρ sinφ sinθ ,z =−ρ cosφ

10. If there are many masses, mα ,α = 1, · · · ,R, the kinetic energy is the sum of thekinetic energies of the individual masses. Thus,

T ≡ 12

R

∑α=1

mα |ẏα |2 .

Generalize the above problems to show that, assuming

yα = yα (x,t) ,

ddt

(∂T∂ ẋk

)− ∂T

∂xk =R

∑α=1

F α ·∂yα

∂xk

where F α is the force acting on mα .

11. Discuss the equivalence of these formulae with Newton’s second law, force equalsmass times acceleration. What is gained from the above so called Lagrangian for-malism?

12. The double pendulum has two masses instead of only one.

m1

l1θ

m2

l2φ

Write differential equations for θ and φ to describe the motion of the double pendu-lum.

B.5 Transformation of Coordinates.How do we write ek (x) in terms of the vectors, e j (z) where z is some other type ofcurvilinear coordinates? This is next.

Consider the following picture in which U is an open set inRn,D and D̂ are open sets inRn, and M,N are C2 mappings which are one to one from D and D̂ respectively. The onlyreason for this is to ensure that the mixed partial derivatives are equal. We will supposethat a point in U is identified by the curvilinear coordinates x in D and z in D̂.