I am looking for the maximum of the function
$$V(r^*)=\left( 1-\dfrac{2m}{r} \right)\left( \dfrac{l(l+1)}{r^2}-\dfrac{6m}{r^3} \right)$$
in the coordinate $r^*$ for given $l$ and $m$, where $r^*=r+2m \ln(r-2m)$.
How can I do it?
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1 Answer
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The necessary condition to maximize is D[V,r]/D[r*,r] which can be solved analytically
sol=Solve[0 == D[(1 - 2 m/r) ((l (l + 1))/r^2 - (6 m)/r^3), r]/D[r + 2 m Log[r - 2 m], r], r]
(* {{r -> 2 m},
{r -> (9 m + 3 l m + 3 l^2 m -
Sqrt[-96 (l + l^2) m^2 + (-9 m - 3 l m - 3 l^2 m)^2])/(2 (l + l^2))},
{r -> (9 m + 3 l m + 3 l^2 m +
Sqrt[-96 (l + l^2) m^2 + (-9 m - 3 l m - 3 l^2 m)^2])/(2 (l + l^2))}}*)
Knowing the parameters l,m you might check the result to be a maximum!
answered Mar 1, 2020 at 16:54
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$\begingroup$ ...i'm kinda confused. The function $V(z)$ clearly diverges (to $\infty$) at $z\to0^+$, so how can it have a maximum at all? Your solutions seem to be minima (or local maxima), not a global maximum! $\endgroup$ Mar 1, 2020 at 17:38
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$\begingroup$ @AccidentalFourierTransform It's only a necessary condition for an extremum. I didn't check for maximum! The influence of the constraint shoulden'd be disregared. $\endgroup$ Mar 1, 2020 at 18:51
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1$\begingroup$ I'm not sure what you mean by "the constraint", but the function is unbounded from above for all $m,l$, so the problem has no solution... $\endgroup$ Mar 1, 2020 at 19:15
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$\begingroup$ @AccidentalFourierTransform The constraint (sorry for my wording)
r*=r+2Log[r-2m], m>0shows, that realr*only exist ifr>2m. Thus the singularityV[0+]is excluded . $\endgroup$ Mar 2, 2020 at 8:03 -
$\begingroup$ Ah, I see, that makes sense! (But do note that $r=2m$ is a boundary, so the maximum could in principle be located there. Strictly speaking, one should test that value too, together with the three you write in your answer). $\endgroup$ Mar 3, 2020 at 22:37
FindMaximumgives the maximum for the variabelrI do not know how to find the maximum inr^*$\endgroup$Vdefined in terms of the coordinater', but I need the maximum of the potential in terms of the variabler*` whch depends onr$\endgroup$