474 CHAPTER 25. CURVILINEAR COORDINATES

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.

25.5 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 in Rn,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.

U

D D̂

M N

(x1,x2,x3) (z1,z2,z3)

Thus M (x) = N (z) and so z = N−1 (M (x)) . The point in U will be denoted inrectangular coordinates as y and we have y (x) = y (z) Now by the chain rule,

ei (z) =∂y

∂ zi =∂y

∂x j∂x j

∂ zi=

∂x j

∂ zi e j (x) (25.19)

Define the covariant and contravariant coordinates for the various curvilinear coordinatesin the obvious way. Thus,

v = vi (x)ei (x) = vi (x)ei (x) = v j (z)e

j (z) = v j (z)e j (z) .

Then the following theorem tells how to transform the vectors and coordinates.

Theorem 25.5.1 The following transformation rules hold for pairs of curvilinear coordi-nates.

vi (z) =∂x j

∂ ziv j (x) , vi (z) =

∂ zi

∂x j v j (x) , (25.20)

ei (z) =∂x j

∂ zie j (x) , e

i (z) =∂ zi

∂x j ej (x) , (25.21)