# Determing a polar orbit for a non-inverse square central force

1. Jan 20, 2007

### ^_^physicist

1. The problem statement, all variables and given/known data
A particle of unit mass is projected with a velocity v-(sub 0), at right angle to the radius vector at a distance 'a' from the origin of a center of attractive force given by:

f(r)= -k*(4/(r^3)+ (a^2)/(r^5)).

If (v-(sub 0))^2 = (9*k)/(2*(a^2)) find the polar equation of the resulting orbit.

2. Relevant equations
F=ma

Treat 1/r = u

acceleration in polar coordinates= (r*{double dot} - r*((theta) {dot})^2 * e-sub r) + (r*(theta) {double dotted} + 2*r {dotted} + (theta) {dot})e-sub theta. Where e-sub are units vectors.

theta{dot} = l*u^2, where l is angular momentum per mass.
3. The attempt at a solution

Ok this is kind of long so I may skip few steps (sorry):

m*((r{double dot}) - r*((theta){dot})^2 = f(r), since the angular componet of acceleration is zero for this situtation.

m*(r{double dot} - (theta{dot})^2) = f(u^-1)

m*(r{double dot} - (theta{dot})^2) = -k*(4/(u^3)+ (a^2)/(u^5)).

m* [ -l^2*u^2*d^2*u/d(theta)^2-1/u*(l^2*u^3)]=k*(4/u^3 +a^2/u^5)

d^2*u/d(theta)^2 + u= (k*(4/u^3 + a^2/u^5))/(m*l^2*u^2)

and then I get stuck. I have tried multiple avenues for trying to solve this diff. eq, but none of them seem to cut it.

Anyone have any ideas? If the suggestion for problem goes to using an energy relation, I have tried that too and I get stuck in a similar problem.

Last edited: Jan 21, 2007
2. Jan 21, 2007

### tim_lou

wot... gives me a headache viewing non-latexed equations... anyway,

$$\frac{-4k}{r^3}\neq \frac{-4k}{u^3}$$

so basically, you have:
$$u''=-u-\frac{m}{l^2u^2}F(u)$$

you'll have an ugly non-linear term of u^3...

Last edited: Jan 21, 2007
3. Jan 21, 2007

### ^_^physicist

Sorry about the non-latex (I am working on it, I swear...but I use maple in the lab to write up equations so don't really have a need to use latex (except here).

Thanks for help, however, could you clarify with the f(u) statement: are you referring to the f(u^(-1)) or are you actually meaning f(u).

Because when I found the general form of the differential equation of an orbit in my textbook it gave it as f(u^(-1)).

But again thanks.

4. Jan 21, 2007

### siddharth

What tim_lou is pointing out is that, going from

is incorrect, because f(u^-1)= -k*(4/((u-1)^3)+ (a^2)/((u-1)^5)), and not as you have written.