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Bertrand
Pythagoreana
Sarkovskii 





BERTRAND'S
THEOREM 


Page under
construction. 





This astonishing theorem about central force motion
asserts that the observed periodicity of all bounded
celestial motions implies that the law of
gravitation governing the motions must be Newton's
"Inversesquare" law. The original
argument was brilliant but, as with most physics
proofs at the time, lacked rigor at several stages.
The casual assumptions were that anything that
looked continuous was continuous, even improper
integrals depending on parameters. Further,
the integrals and their asymptotic limits were
interchanged with abandon. 


Since the original proposal, proofs have been
constructed using HamiltonJacobi theory and
analytic functions, but these tend to rely on more
smoothness conditions on the force law than
necessary. Here, we give a
modification of the original proof, historically
interesting because wherever possible we use
mathematical methods known at the time the theorem
was first proposed. 


We will begin with a precise statement of the
theorem. A second section will give a quick
overview of the basics of central force motion.
Remaining sections will complete the proof  one I
consider a real gem of mathematics! 





Mnauscript
in preparation. 
