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== Converses and generalizations ==
It is natural to ask whether the [[Theorem#Converse|converse]] of the shell theorem is true, namely whether the result of the theorem implies the law of universal gravitation, or if there is some more general force law for which the theorem holds. MoreIf specificallywe require only that the force outside of a spherical shell is the same as for an equal point mass at its center, then there is one mayadditional askdegree of freedom for force laws.<ref name=Gurzadyan>{{cite journal| last=Gurzadyan |first=Vahe |authorlink=vahe Gurzadyan|title=The cosmological constant in McCrea-Milne cosmological scheme|journal=The Observatory|date= 1985|volume=105|pages=42–43|bibcode=1985Obs...105...42G}} https://adsabs.harvard.edu/full/1985Obs...105...42G&lang=en</ref><ref name=Arens>{{cite journal| last=Arens| first=Richard| authorlink=Richard Friederich Arens|title=Newton's observations about the questionfield of a uniform thin spherical shell|journal=Note di Matematica|date=January 1, 1990|volume=X|issue=Suppl. n. 1|pages=39–45}}</ref> The most general force, as given by the [[Gurzadyan theorem]], is:<ref name="Gurzadyan"/>
:<math> F (r) = -\frac{G M m}{r^2} + \frac{\Lambda m c^2 r}{3} </math> ▼{{block indent|Suppose there is a force <math>F</math> between masses ''M'' and ''m'', separated by a distance ''r'' of the form <math>F = M m f(r)</math> such that any spherically symmetric body affects external bodies as if its mass were concentrated at its center. Then what form can the function <math>f</math> take?}}
where <math>G</math> and <math>\Lambda</math> can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the [[cosmological constant]] term in [[general relativity]]. However, the inverse-square potential is the only potential such that the net force inside the shell is also zero.<ref name=Gurzadyan/>▼In fact, this allows exactly one more class of force than the (Newtonian) inverse square.<ref name=Gurzadyan>{{cite journal| last=Gurzadyan |first=Vahe |authorlink=vahe Gurzadyan|title=The cosmological constant in McCrea-Milne cosmological scheme|journal=The Observatory|date= 1985|volume=105|pages=42–43|bibcode=1985Obs...105...42G}} http://adsabs.harvard.edu/full/1985Obs...105...42G</ref><ref name=Arens>{{cite journal| last=Arens| first=Richard| authorlink=Richard Friederich Arens|title=Newton's observations about the field of a uniform thin spherical shell|journal=Note di Matematica|date=January 1, 1990|volume=X|issue=Suppl. n. 1|pages=39–45}}</ref> The most general force as derived by [[Vahe Gurzadyan]] in <ref name="Gurzadyan"/> [[Gurzadyan theorem]] is:
The force described by the [[Yukawa potential]]
▲:<math> F = -\frac{G M m}{r^2} + \frac{\Lambda m c^2 r}{3} </math>
:<math> U(r) = -\frac{G M m}{r} e^{-\lambda r}</math>
has the property that the force outside of a spherical shell is also a Yukawa potential with the same range <math>1/\lambda</math> and centered at the shell's center, but for <math>\lambda > 0</math> the equivalent point mass is not the same as the mass of the shell.<ref>{{cite web |url=https://wp.ufpel.edu.br/pskuhn/files/2024/09/debye-huckel-interaction-or-yukawa-in-different-geometries.pdf |access-date=14 February 202}}</ref><ref>{{cite web |url=http://kirkmcd.princeton.edu/examples/yukawa.pdf |access-date=14 February 2025}}</ref><ref>{{cite web |title=Shell theorem for a general potential |url=https://math.stackexchange.com/questions/296180/shell-theorem-for-a-general-potential |website=Mathematics Stack Exchange |access-date=14 February 2025 |language=en}}</ref> For a shell of radius <math>R</math> and mass <math>M</math>, the equivalent point mass is:
▲where <math>G</math> and <math>\Lambda</math> can be constants taking any value. The first term is the familiar law of universal gravitation; the second is an additional force, analogous to the [[cosmological constant]] term in [[general relativity]].
:<math>M_\text{eff} = M \frac{\sinh \lambda r}{\lambda r}</math>.
If we further constrain the force by requiring that the second part of the theorem also holds, namely that there is no force inside a hollow ball, we exclude the possibility of the additional term, and the inverse square law is indeed the unique force law satisfying the theorem.
On the other hand, if we relax the conditions, and require only that the field everywhere outside a spherically symmetric body is the same as the field from some point mass at the center (of any mass), we allow a new class of solutions given by the [[Yukawa potential]], of which the inverse square law is a special case.
Another generalization can be made for a disc by observing that
:<math>dM=\frac{R^2}{2} \frac{d\theta \, \sin^2(\theta)}{\pi R^2}M=\frac{ \sin^2(\theta)}{2 \pi}M \, d\theta </math>
so:
:<math>F_r = \frac{GMm}{2 \pi} \int \frac{ \sin^2 (\theta) \cos(\varphi)} {s^2} \, d\theta,</math>
where {{nowrap|<math>M=\pi R^2 \rho</math>,}} and <math>\rho</math> is the density of the body.
Doing all the intermediate calculations we get:
:<math>F(r) = \frac{G m \rho}{8r^3} \int_{R-r}^{R+r} { \frac{\left(r^2 + s^2 - R^2\right)\sqrt{2\left(r^2 R^2 + r^2 s^2 + R^2 s^2\right) - s^4 - r^4 - R^4} }{s^2} } \, ds</math>
== Newton's proofs ==
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