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rather than s, in Submodule 10.3. | dy/ds| is always 1
On a surface in three-dimensional space, a geodesic ó is a curve for which the vector d2y/ds2 is perpendicular to the surface at the point y(s). For now,
let’s assume that all curves are parametrized by arc length, so ó' means dy/ds.
If any curve y (not necessarily a geodesic) on the surface is parametrized at constant speed, we are guaranteed that y" is perpendicular to y', but not necessarily to the surface. To see this, observe that y' ¦ Y = 1, where y' is the velocity vector for the curve y and the “dot” indicates the dot (or scalar) product of two vectors. Differentiating the dot product equation, we have y" ¦ Y + Y ¦ Y' = 0, so Y' is perpendicular to Y (or else is the zero vector).
The statement that Y' is perpendicular to the surface says that y is going as “straight” as it can in the surface, and that the surface is exerting no force which would make the curve bend away from its path. Such a curve is a geodesic. See the book by Do Carmo for a full explanation of why geodesics defined as above minimize the distance between nearby points.
Example 13: On a sphere, the parallels of latitude y(s) yield vectors y" (s) in the plane of the parallel and perpendicular to the parallel (but not in general perpendicular to the surface), whereas any great circle yields vectors y"(s) pointing toward the center of the sphere and hence perpendicular to both the great circle and to the surface. The great circles are geodesics, but the parallels (except for the equator) are not.
¦ Geodesics on a Surface of Revolution
X = f (u), z = g(u)
is a parametrization by arc length u of a curve in the xz-plane. One consequence of this parametrization is that ( f'(u))2 + (g(u))2 = 1. Let’s rotate the curve by an angle â around the z-axis, to find the surface parametrized by
P(u, â) =
f(u)cos â f (u) sin â
Let’s suppose that curves y on the surface are parametrized by arc length s and, hence, these curves have the parametric equation
f (u(s)) cos â(s) f(u(s)) sin â(s) g(u(s))
and we need to differentiate this twice to find
f(u(s))u'(s) cos â(s) — f(u(s)) sin â^)â' (s) f (u(s))u'(s) sin â(s) + f(u(s)) cos â^)â' (s) g(u(s))u'(s)
Geodesics on a Surface of Revolution
/ (s) = d'
f'(u) cos â frr(u) cos â --- f (u) sin â
f ( u) sin â + (u )2 f '( u) sin â + 2d â' f( u) cos â
g (u) g' (u) 0
f ( u) cos â --- f (u) sin â
(â' )2 f (u) sin â + â" f (u) cos â
This array is pretty terrifying, but the two equations
^ = o.
y" ¦ — = u, and y" ¦ — = 0 (21)
which express the fact that ó" is perpendicular to the surface, give
d' — (â')2 f(u) f (u) = 0 and 2d f(u) f (u)B + â"( f(u))2 = 0 (22)
That ó" is perpendicular to the surface if formulas (21) hold follows because the vectors äP/du and äP/äâ span the tangent plane to the surface at the point (â, u). Formulas (22) follow from formulas (21), from the fact that ( f(u))2 + (g(u))2 = 1, and from the formulas
f' ( u) cos â f( u) sin â
— f (u) sin â f ( u) cos â 0
M = (f (u))2B is conserved along a trajectory of system (22) since
— Uf(u))2e' 1 = (f(u))2e" + 2d6' ¥ f=0 ds
The integral M behaves like angular momentum (see Exploration 10.5 for the central force context that first gave rise to this notion).
Substituting â' = M/( f (u))2 into the first ODE of equations (22) gives
Using equations (23) and (24), we obtain a system of first-order ODEs for the geodesics on a surface ofrevolution:
(f(u))2 d = w , M2 f (u)
W ~ (f(u))3
We recognize that ODE (24) is of the form of ODE (4), so