Related papers: Closed, spirograph-like orbits in power law centra…
The Bertrand theorem concluded that; the Kepler potential, and the isotropic harmonic oscillator potential are the only systems under which all the orbits are closed. It was never stressed enough in the physical or mathematical literature…
We present new families of bound, closed, nonelliptical orbits that are supported by various spherical potentials in clear contradiction to Newton's and Bertrand's theorems. We calculate analytically some typical closed orbits of…
Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In the present…
The nature of boundedness of orbits of a particle moving in a central force field is investigated. General conditions for circular orbits and their stability are discussed. In a bounded central field orbit, a particle moves clockwise or…
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 $V(r)=W(r)+b/r^2$…
The closedness of orbits of central forces is addressed in a three dimensional space in which the Poisson bracket among the coordinates is that of the SU(2) Lie algebra. In particular it is shown that among problems with spherically…
A celebrated result of Bertrand states that the only central force potentials on the plane with the property that all bounded orbits are periodic are the Kepler potential and the potential of the harmonic oscillator. In this paper, we…
In the present work, metrics which lead to projected closed orbits are found by comparing the relativistic differential equation of orbits with the corresponding classical differential equation. Physical and geometrical properties of these…
Newton famously showed that a gravitational force inversely proportional to the square of the distance, $F \sim 1/r^2$, formally explains Kepler's three laws of planetary motion. But what happens to the familiar elliptical orbits if the…
A useful crude approximation for Abelian functions is developed and applied to orbits. The bound orbits in the power-law potentials A*r^{-alpha} take the simple form (l/r)^k = 1 + e cos(m*phi), where k = 2 - alpha > 0 and 'l' and 'e' are…
We consider the geometric Titius-Bode rule for the semimajor axes of planetary orbits. We derive an equivalent rule for the midpoints of the segments between consecutive orbits along the radial direction and we interpret it physically in…
For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits…
It can be easily shown that bound orbits around a static source can exist only in 4 dimension and in none else for any long range force. This is so not only for Maxwell's electromagnetic and Newton's gravity but also for Einstein's…
We discuss the existence and stability of circular orbits of a relativistic point particle moving in a central force field. The stability condition is somewhat more restrictive in Special Relativity. In the particular case of attractive…
Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently extended Bertrand theorem. He designed new static…
The classical orbits of a test string in the transverse space of a singular heterotic fivebrane source are classified. The orbits are found to be either circular or open, but not conic because the inverse square law is not satisfied at long…
We prove a generalization of the Poincar\'e-Birkhoff theorem for the open annulus showing that if a homeomorphism satisfies a certain twist condition and the nonwandering set is connected, then there is a fixed point. Our main focus is the…
In this paper, we consider the motion of a particle on a surface of revolution under the influence of a central force field. We prove that there are at most two analytic central potentials for which all the bounded, nonsingular orbits are…
We study closed universes in holographic theories of quantum gravity. We argue that within any fixed theory, factorization implies there is one unique closed universe state. The wave function of any state that can be prepared by the path…
The generalization of Bertrand's theorem to the case of the motion of point particle on the surface of a cone is presented. The superintegrability of such models is discussed. The additional integrals of motion are analyzed for the case of…