We applied our variational methods (including Jacobi integrals) to the following example problem: on a sphere, find the path between any two points a and b which minimizes the distance traveled between a and b.
We wrote this as a variational problem, by calculating the
infinitesimal distance ds along the curve as 
goes to
and 
goes to 
. We integrated up
this infinitesimal distance along the curve to get the total distance
s traveled along the curve, which was the functional 
, where 
. We chose as our integration variable so that
we would get an integrand with no direct dependence on its integration
variable, which would give us a conserved Jacobi integral J. This
is indeed what we got: a functional of the path 
, whose
integrand depended only on and 
. Thus extremizing this
functional leads to an Euler-Lagrange equation, which can be
integrated to get a constant Jacobi integral J (since the integrand
has no explicit -dependence).
We then calculated the Jacobi integral, which determined an equation
for 
. This equation determined an integral equation for
, just as we obtained integral equations for the duration
of a motion when Newton II told us that 
. We
evaluated the integral to obtain .
We then massaged the result a bit until its geometric meaning became
clear. Our equation for implied that, along the path
, the dot product of 
-- the
instantaneous position on the path -- with some constant vector
must vanish. The set of all points orthogonal to is
a plane through the origin. Thus our path must always lie in the
intersection of the sphere with some plane through the origin -- that
is, it must lie along a great circle of the sphere.
-- KB