From Osher Doctorow Ph.D.

I meant to put this same subject heading on my last posting a few minutes ago, but I forgot to put a subject heading on the posting. There is one typo in the posting: the last expression should read 1/[v(1 - v)] = 1/v) + 1/(1 - v) instead of reading v(1 - v) = (1/v) + 1/(1 - v). For those who deleted the previous posting because it had no subject heading, it turns out that if, instead of taking force as the time rate of change of momentum as it has been taken since Sir Isaac Newton in the later 1700s, one takes force as proportional to the time rate of change of "logistic momentum" mv (1 - v) where m is mass and v is speed or velocity, then if the proportionality is written F = kmv(1 - v), the higher or stronger the proportionality is, that is to say the larger k > 0 is, the closer v is to 1/2 for mass approximately constant. However, 1/2 is halfway between a speed of 0 ("at rest" condition) and the speed of light which is taken as 1 (ignoring superluminal speeds here). This enables us to solve the linked logistic system: 1) dv1/dt = k1v1(1 - v1) + k2v2(1 - v2) dv2/dt = k3v1(1 - v1) + k4v2(1 - v2) since with v1 approximately 1/2, we can use partial fractions to obtain an exact solution for v2 because 1/[v(1 - v)] = 1/v + 1/(1 - v) for any v. The linked logistic system (1) was first introduced by James W. Bruno and me and Chris Kappner in a 1981 publication in population growth/migration, and reflects the interrelated effects of the two populations on each other and slightly modifies the Lotka-Volterra predator-prey equations. The latter are useful when the environment contains competitors and limitations on growth (scarcity, etc.). We did not apply the equations (1) at that time to physics or analyze the case of v1 = 1/2. I introduced the equations (1) today on stardrive1@yahoogroups.com to model an Alcubierre-like warp drive in space travel with v1 the rear speed and v2 the front speed of the craft which (curiously enough) are different in the Alcubierre type scenarios. Osher Doctorow Ph.D.