d^2x/dt^2 = -1/r^3 e_r,
極座標にする
∂_t^2 r - r(∂_tφ)^2 = -1/r^3,
1/r (d/dt)(r^2 ∂_tφ) = 0.
r^2 ∂_tφ = h,
∂_tφ = h/r^2.
∂_t^2 r - h^2/r^3 = -1/r^3,
v = dr/dt,
v(∂v) = (h^2-1)(dr/dt)r^(-3),
v^2/2 = -1/2(h^2-1)r^(-2) + C,
v^2 = -(h^2-1)r^(-2) + C,
dr/dt = 1/r√(Cr^2 - (h^2-1)),
1/2(Cr^2 - (h^2-1))^(-1/2) d(r^2) = dt,
1/C√(Cr^2 - (h^2-1)) = t + D,
r^2 - (h^2-1)/C = (t + D)^2,
r = [(h^2-1)/C + (t + D)^2]^(1/2).
φ = h∫dt/r^2
= h log|√((t+D)^2 + (h^2-1)/C) + (t+D)|.
C,Dは積分定数。