By Mark Levi
Figure 1. The geodesic at \(A\) is tangent to a principal line of curvature of normal curvature \(k_1\) (not drawn).
What is a Jacobi Field?
A Jacobi vector field governs the separation of two nearby geodesics, to the leading order; the Jacobi equation is the linearization of the geodesic equation around a geodesic. In mechanical terms (and for embedded surfaces in \({\mathbb R} ^3\)), the distance \(s\) between two point masses that are sliding abreast on a surface (see Figure 1) in the absence of gravity and friction satisfies to the leading order the Jacobi equation
\[\ddot s + K ( r ) v^2 s = 0;\tag1\]
here, \(r=r(t)\) is the position of one of the masses, \(v = | \dot r |\) is the speed, and \(K\) is the Gaussian curvature. Most books on differential geometry derive \((1)\), but the derivations require some background — as well as some space and some time. Instead, I would like to give a back-of-the-envelope derivation of \((1)\) in a special case using little more than the high school formula \(F = mv ^2 /R\) for the centripetal force.
The Setup
Consider a unit point mass \(A\) with a velocity that points in the direction of a principal curvature at the instant in question; an identical particle \(B\) is near to and abreast of \(A\) (i.e., the arc \(AB\) is perpendicular to the velocity of \(A\)). Both \(A\) and \(B\) have the same constant speed \(v\), and \(B\)'s direction of motion is close to that of \(A\) by assumption (see Figure 1). As mentioned, there is no gravity or friction.
Figure 2. Restoring component of the centrifugal force. Here, \(s\) is the arc length.
A Heuristic Derivation of (1)
An observer who is sliding with the reference frame of \(A\) feels the centrifugal \(g\)-force due to the curvature of \(A\)'s path:
\[F = k_1 v ^2 .\tag2\]
This inertial force—which also acts on \(B\) from \(A\)'s point of view—has the tangential component \(F_{\rm restoring} = F \sin \theta\) that pulls \(B\) towards \(A\) (see Figure 2). But \(\theta = k_2 s +o(s)\) by the definition of curvature (see Figure 3). Therefore,
\[F_{\rm restoring} = Fk_2 s+o(s) \overset{(2)}{=} k_1k_2 v ^2 s+o(s).\]
And since \(k_1k_2 =K\) is the Gaussian curvature, this explains \((1)\) — but only for the special case when the velocity points in a principal direction of curvature.
Figure 3. Angle between the normals equals the angle between the tangents; the latter \(\approx k_2 s\).
How to Explain (1) Heuristically for an Arbitrary Direction?
I would like to leave this question as a fun problem and may address it in the next installment. It turns out that the special case that I describe here misses an interesting aspect, one that is also hidden in the formal machinery of standard derivation. For example, what if the geodesic is a straight line on a ruled surface? Considering this question yields a mechanical interpretation of the Hessian determinant that has not occurred to me before.
The figures in this article were provided by the author.
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.