Consider two neighboring geodesics (great circles) on a sphere of radius a, one the equator and the other a geodesic slightly displaced from the equator (by △θ = b) and parallel to it at ∅ = 0. Let ξ(vector) be the separation vector between the two geodesics, and note that at ∅ = 0, ξ(vector) = b∂/∂θ. Let l be proper distance along the equatorial geodesic, so d/dl = u(vector) is its tangent vector.

(a) Show that l = a∅ along the equatorial geodesic.

(b) Show that the equation of geodesic deviation (25.31) reduces to

(c) Solve Eq. (25.53), subject to the above initial conditions, to obtain

Verify, by drawing a picture, that this is precisely what one would expect for the separation vector between two great circles.

Equation (25.31)

Transcribed Image Text:

d2x0 do² - 0, d2x4 dø² = 0. (25.53)