相关论文: A proof of Bertrand's theorem using the theory of …

We discuss an alternative non-perturbative proof of Bertrand's theorem that leads in a concise way directly to the two allowed fields: the newtonian and the isotropic harmonic oscillator central fields.

We study a specific family of uniformly isochronous polynomial systems. Our results permit to solve a problem about centers of such systems.

We present an alternative proof of Perron's theorem, which is probabilistic in nature. It rests on the representation of the Perron eigenvector as a functional of the trajectory of an auxiliary Markov chain.

In this paper we give an elementary proof for Bertrand's postulate also known as Bertrand-Chebyshev theorem.

We derive an expression for the determination of the apsidal angles that holds good for arbitrary central potentials. Then we discuss under what conditions the apsidal angles remain independent of the mechanical energy and angular momentum…

We show that Isserlis' theorem follows as a corollary to the invariant tensor theorem for isotropic tensors.

We give a counting based proof of the Graham Pollak Theorem

In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $\ddot q(t)=-\nabla U(q(t))$ in the presence of symmetries of a compact Lie group $\Gamma$ acting…

The Bertrand's theorem is extended, i.e. closed orbits still may exist for other central potentials than the power law Coulomb potential and isotropic harmonic oscillator. It is shown that for the combined potential $V(r)=W(r)+b/r^2$…

In this note, we give an alternate proof of the multinomial theorem using a probabilistic approach. Although the multinomial theorem is basically a combinatorial result, our proof may be simpler for a student familiar with only basic…

We give a short proof of Urabe's criteria for the isochronicity of periodical solutions of the equation $\ddot{x}+g(x)=0$. We show that apart from the harmonic oscillator there exists a large family of isochronous potentials which must all…

Revisiting and extending an old idea of Michel H\'enon, we geometrically and algebraically characterize the whole set of isochrone potentials. Such potentials are fundamental in potential theory. They appear in spherically symmetrical…

An technically interesting proof of a known theorem.

We revisit in this note the H\'enon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the H\'enon's parabolae and get all isochrone central potentials under mild…

An elementary proof of Bertrand's theorem is given by examining the radial orbit equation, without needing to solve complicated equations or integrals.

This is an expository note discussing how the Erdos--Ramanujan proof of Bertrand's postulate may be adapted to show the existence of finite fields.

Isochrone potentials, as defined by Michel H\'enon in the fifties, are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Isochrone potentials encompass the…

We present in this work a new and simple proof of the false centre theorem.

We prove the theorems which are equivalent to the Roland's results such that a new form of them allows to consider some generalizations. In particular, we give generators of primes more than a fixed prime.

We give a new proof of Brooks' theorem that immediately implies a strengthening of Brooks' theorem, known as Catlin's theorem.