Re: Can you solve my gravity-maths problem in Mathematica?

*To*: mathgroup at smc.vnet.net*Subject*: [mg62856] Re: Can you solve my gravity-maths problem in Mathematica?*From*: dh <dh at metrohm.ch>*Date*: Tue, 6 Dec 2005 23:10:34 -0500 (EST)*References*: <dn373m$32r$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Hi Paul, your function does not have a minimal value for r>0. To show this, we first simplify your function by defining a new variable x= m r and we use a definite integral from zero: f[r] == F[x] == m Exp[x] Integrate[Exp[-2x1]/x1^2,{0,x}] We may drop m as it does not change the position of a minimum. If this function would have a minimum, a x0 would exists so that F'[x0] = Exp[x0] Integrate[Exp[-2x1]/x1^2,{0,x0}] +Exp[x0] Exp[-2x0]/x0^2== 0 from this we get: Integrate[Exp[-2x1]/x1^2,{0,x0}] = - Exp[-2x0]/x0^2 For positive x0 the left side is positive (positive integrand), but the right side is negative. Therefore, no x0 exists. Daniel Paul wrote: > Hello, I have been working on a simple model of quantum gravity. Like > atoms have a minimum electron shell distance caused by the quantisation > of the electromagnetic force my approximation for the average distance > between particles held together by a graviational force, using my > model, becomes : > > "Find r at the maximum value of: > > f(r) = exp( m*r ) * integrate( exp(-2*m*r)/r^2 , r ) > > (r>0) " > > Where we use constants such that c=h=G=1. In this system for example > electron mass is something like 10^(-23) I believe. (The graviational > bond enegy in this model becomes m^5 in my units.) > > So I need to find f'(r)=0. > > Can anyone find a good way to solve this for r, maybe as a series in > terms of m? Also, I would be interested in seeing a graph of f(r). >