Geodesics

On a sphere all geodesics are arcs of great circles. On a surface of constant negative curvature (like (c)) geodesics diverge exponentially, as noted in early work on chaos theory (see page 971). The path of a geodesic can in general be found by requiring that the analog of acceleration vanishes for it. In the case of a surface defined by z f[x, y] this is equivalent to solving

x''[t] -(f^{ (1, 0)}[x[t], y[t]](y'[t]^{2} f^{ (0, 2)}[x[t],y[t]] + 2 x'[t]y' [t]f^{ (1, 1)}[x[t], y[t]] + x'[t]^{2} f^{ (2, 0)}[x[t], y[t]]))/(1 + f^{ (0, 1)}[x[t], y[t]]^{2} + f^{ (1, 0)}[x[t], y[t]]^{2})

together with the corresponding equation for y'', as already noted by Leonhard Euler in 1728 in connection with his development of the calculus of variations.