Find the tunnel between two points A and B on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of radius a, and with uniform density ρ. Then the standard form Euler-Lagrange differential equation in polar coordinates is r_ϕϕ(r^3 - r a^2) + r_ϕ^2(2a^2 - r^2) + a^2 r^2 = 0, along with the boundary conditions r(ϕ = 0) = r_0, r_ϕ(ϕ = 0) = 0, r(ϕ = ϕ_A) = a, and r(ϕ = ϕ_B) = a. Integrating once gives r_ϕ^2 = (a^2 r^2)/r_0^2 (r^2 - r_0^2)/(a^2 - r^2).