Motion of a particle under central force (Planetary motion)
We consider the central force, f k / r with k = 1 . Now, x and ycomponents of the force can be written as f x = xf / r = − x / r and f y yf / r y / r , where r x y . Next we consider consider Newton’s law: 2
= −
3
3
=
2
= −
f = m
dv dt
with
m
Thus we have
=1
f x
=
2
+
2
.
=
dv x dt
⇒
dv x
=
f x dt
x / r 3 dt
= −
and
dv y
=
3
f y dt = − y / r dt .
We
implement the last two steps in the following program and evaluate velocity components in successive times and wherefrom we obtain obtain the position x, y. y. [Source: Book - “From Newton to Mandelbrot” Mandelbrot” by D. Stauffer and H.E. H.E. Stanley (Pub. Spinger) ] C
Motion of a particle in the central force field open(1,file='planet.dat') write(*,*)'Give vx,vy,dt' read(*,*)vx,vy,dt x=0.0 y=1.0 ncount=0 r2=x*x+y*y r3=dt/(r2*sqrt(r2)) vx=vx-x*r3 vy=vy-y*r3 x=x+dt*vx y=y+dt*vy write(1,*)x,y ncount=ncount+1 if(ncount.lt.1000)go to 10 stop end
10
•
The data stored in the file file ‘planet.dat’ can be plotted (through (through Origin or any such software) to see the closed elliptical planetary oribit.
Abhijit Kar Gupta,
[email protected]
