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

1. Homework Statement
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. Homework 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.

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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...

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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.

siddharth
Homework Helper
Gold Member
What tim_lou is pointing out is that, going from

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