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Joseph Bertrand. In classical mechanics, Bertrand's theorem states that among central-force potentials with bound orbits, there are only two types of central-force (radial) scalar potentials with the property that all bound orbits are also closed orbits.
In mathematics, Bertrand's postulate (now a theorem) states that, for each , there is a prime such that < <.First conjectured in 1845 by Joseph Bertrand, [1] it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.
In number theory, Bertrand's postulate is the theorem that for any integer >, there exists at least one prime number with n < p < 2 n − 2. {\displaystyle n<p<2n-2.} A less restrictive formulation is: for every n > 1 {\displaystyle n>1} , there is always at least one prime p {\displaystyle p} such that
Bertini's theorem (algebraic geometry) Bertrand–Diquet–Puiseux theorem (differential geometry) Bertrand's ballot theorem (probability theory, combinatorics) Bertrand's postulate (number theory) Besicovitch covering theorem (mathematical analysis) Betti's theorem ; Beurling–Lax theorem (Hardy spaces) Bézout's theorem (algebraic geometry)
In the second edition, Goldstein corrected all the errors that had been pointed out, added a new chapter on perturbation theory, a new section on Bertrand's theorem, and another on Noether's theorem. Other arguments and proofs were simplified and supplemented. [3]
They are the only two problems that have closed orbits for every possible set of initial conditions, i.e., return to their starting point with the same velocity (Bertrand's theorem). [1]: 92 The Kepler problem also conserves the Laplace–Runge–Lenz vector, which has since been generalized to include other interactions.
Bertrand's box paradox – Mathematical paradox; Bertrand's postulate – Existence of a prime number between any number and its double; Bertrand's theorem – Physics theorem; Bertrand's ballot theorem – Theorem that gives the probability that an election winner will lead the loser throughout the count; Bertrand–Edgeworth model ...
In Bertrand's original paper, he sketches a proof based on a general formula for the number of favourable sequences using a recursion relation. He remarks that it seems probable that such a simple result could be proved by a more direct method.