Page 173 - Intro to Tensor Calculus

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and in the special case where n 1 =cos β, n 2 =sin β,n 3 = − tan α is expressible in the form

tan α −1 tan α
cos(φ − β)= or φ − β =cos . (4)
tan θ tan θ
The above equation defines an explicit relationship between the surface coordinates which defines a great
circle on the sphere. The arc length squared relation satisfied by the surface coordinates together with the
equation obtained by differentiating equation (4) with respect to arc length s gives the relations

dφ tan α dθ
2
sin θ = q (5)
ds tan α ds
2
1 − 2
tan θ
2
2
2
2
2
ds = ρ dθ + ρ sin θdφ 2 (6)
The above equations (1)-(6) are needed to consider the following problem.
(a) Show that the differential equations defining the geodesics on the surface of a sphere (equations (1.5.51))
are
2
d θ dφ 2
− sin θ cos θ =0 (7)
ds 2 ds
2
d φ dθ dφ
+2 cot θ =0 (8)
ds 2 ds ds
2
(b) Multiply equation (8) by sin θ and integrate to obtain
dφ
2
sin θ = c 1 (9)
ds
where c 1 is a constant of integration.

(c) Multiply equation (7) by dθ and use the result of equation (9) to show that an integration produces
ds
2 2
dθ −c 1 2
= 2 + c 2 (10)
ds sin θ
2
where c is a constant of integration.
2
(d) Use the equations (5)(6) to show that c 2 =1/ρ and c 1 = sin α .
ρ
(e) Show that equations (9) and (10) imply that

2
dφ tan α sec θ
= 2 q
dθ tan θ tan α
2
1 − 2
tan θ
and making the substitution u = tan α this equation can be integrated to obtain the equation (4). We
tan θ
can now expand the equation (4) and express the results in terms of x, y, z to obtain the equation (3).
This produces a plane which intersects the sphere in a great circle. Consequently, the geodesics on a
sphere are great circles.