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A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances $a,b$ $(a>b)$, then find the equation of the path.

I came across this question on MSE. While solving I could only proceed to the step
$$ h^2(du/dθ^2+u^2)=(k/u^2)+C $$
where $C$ is constant of integration and the force is considered as $$\mathbf F=-k\mathbf r.$$ I also know that at an apse $du/dθ=0$. How do I proceed further?

EDIT: I have found out the pedal equation of the orbit and it seems to be an ellipse. How do I find out in terms of polar coordinates $u(=1/r),θ$? It is required to prove that $$ u^2=(sin^2θ)/b^2+(cos^2θ)/a^2 $$ Here's a link to the original question Orbits under central forces

Sharmi C
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    You should link to the original question so people can see the solution you are talking about. – Ross Millikan Jun 28 '18 at 05:04
  • Can you elaborate a bit more. What is the $h^2$ step and can you formulate your question using the appropriate MathJax. – Rumplestillskin Jun 28 '18 at 05:54
  • The motion under the gravity potential $GMm/r$ has the equation $\ddot{\mathrm r}=-\frac{GM}{r^3}\mathbf r$. Is that also "a force, towards a fixed centre, proportional to the distance"? – Lutz Lehmann Jun 28 '18 at 09:36

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The system $$ m\ddot{\mathbf r} = -k\mathbf r $$ that you seem to describe consists of two decoupled harmonic oscillators. It has solutions $x(t)=A\sin(\omega t+\phi_1)$, $y(t)=B\sin(ωt+ϕ_2)$, $ω^2=\frac{k}m$.

If you set $A=a$, $B=b$, $ϕ_1=0$ and $ϕ_2=\frac\pi2$, then you get a horizontal ellipse with main axes with half-lengths $a$ and $b$.

Lutz Lehmann
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  • I do Not know about Decoupled harmonic oscillators. Could you provide me a link where I could study about it? – Sharmi C Jun 28 '18 at 09:43
  • You get $m\ddot x+kx=0$ where you have no $y$ involved and $m\ddot y+ky=0$ where you have no $x$ involved, the two differential equations, even though they belong to a system, can be separated as scalar ODE, the system is decoupled, it decomposes into smaller isolated sub-systems, here the smallest, scalar ones (scalar = 1D). – Lutz Lehmann Jun 28 '18 at 09:50