// falling in from infinity; small impact parameters.
// gcc p.c fall.c -fnested-functions -O3
#include "h.h"
int main(){assert(N==2);
  int j = 11; while(--j){
  pt z[] = {{{{1, -sqrtl(2)},{0,(5.5-j)*2.e-6}},1e-20},{{{0,0},{0,0}},1}};
  spin(z, 2.*0.4714045207910317, 1, 1000);
  prx("fall", z); printf("j=%d\n", j); getchar();}
  return 0;}
/* 
mv2/2 = ke = - pe = 1/r ; -1/r is the potential energy
v = dr/dt; m=1
(dr/dt)^2 = 2/r 
(dt/dr)^2 = r/2  ; reciprocal
dt/dr = √(r/2) = 2^(-½)r^(½) ; square root
t(r) = (2/3)r^(3/2)2^(-½) ; integrate by r
t = 2^(½)3^(-1)r^(3/2) ; simplify constants
t^(⅔) = 2^(⅓)3(-⅔)r ; take ⅔ power of both sides
r = 2^(-⅓)3^(⅔)t^(⅔) ; solve for r
(define (r t) (* (expt 2 -1/3) (expt 3 2/3) (expt t 2/3)))
(define (dr/dt t) (* (expt 2 2/3) (expt 3 -1/3) (expt t -1/3)))
(define (t r) (* (expt 2 1/2) (expt 3 -1) (expt r 3/2)))
*/
/*
 for dt = 0.00005*ds*df (in p.c)
 1.000000032992724  1.414213656711674  0.000026870059478  0.000028500004129 
 for dt = 0.0001*ds*df 
 0.999999985306545  1.414213521730140  0.000025455843391  0.000026999998328 
 0.999999962196816  1.414213456003915  0.000019798988384  0.000020999996582 
 0.999999951591204  1.414213425735185  0.000014142134257  0.000014999996861 
 1.000000482640594  1.414214927588523  0.000008485289566  0.000009000018825 
 1.000002847578684  1.414221616548282  0.000002828443233  0.000003000037019 

 for dt = 0.0002*ds*df
 1.000000017877259  1.414213613854030  0.000025455845050  0.000027000002139 
 1.000000094350792  1.414213829791800  0.000019798993618  0.000021000008609 
 1.000000048089514  1.414213698673623  0.000014142136987  0.000015000003134 
 0.999999821835685  1.414213058550118  0.000008485278352  0.000008999993054 
 0.999999974968357  1.414213491584231  0.000002828426984  0.000002999999675 

 for dt = 0.0004*ds*df
 1.000000050191025  1.414213705251159  0.000025455846702  0.000027000005930 
 1.000000070826186  1.414213763254170  0.000019798992693  0.000021000006478 
 1.000000202827711  1.414214136339311  0.000014142141371  0.000015000013202 
 1.000000310193797  1.414214439835409  0.000008485286647  0.000009000012110 
 1.000003959087410  1.414224760365129  0.000002828449528  0.000003000051479 
*/