The least-curvature principle of Gauss and Hertz and geometric dynamics.

Section 1 shows that the torse forming vector ¯eld re°ects eco- nomical phenomena via the S-shaped ¯eld lines. Section 2 analyze the least-curvature principle of Gauss and Hertz with application to geometric dynamics and particularly to economic dynamical systems. Section 3 explores the controllability of the neoclassical growth geo- metric dynamics. Mathematics Subject Classi¯cation: JEL CODE: C61, E37, O30, O39. Key words: Gauss-Hertz least curvature, S-shaped curves, torse forming vector ¯eld, geometric dynamics, economic growth. 1 S-shaped time evolutions S-shaped (or sigmoid) time evolutions [1] are frequently observed in some dynamic economic phenomena (product life cycles, the gradual di®usion of technological innovations or long-term °uctuations in income, productivity growth etc). It is also usefull in biology and demogrphics dynamics. These S-shaped evolutions are usually incorporated into formal models by ODE system such as x_ (t) = X(x(t)); (1) (°ow) where a solution x(t) has an in°ection point x0 = x(t0), i.e., xÄ(t0) = ¹x_ (t0) 1