Chapter 9: Fundamental Physics

Section 15: The Phenomenon of Gravity


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.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]