Noncommutative Geometry and Classical Orbits of Particles in a Central Force Potential

Abstract

We investigate the effect of the noncommutative geometry on the classical orbits of particles in a central force potential. The relation is implemented through the modified commutation relations [x i , x j ] = iθ ij . Comparison with observation places severe constraints on the value of the noncommutativity parameter.

Orbits of particles in noncommutative Schwarzschild spacetime

“…The last two terms in (20) show the noncommutative correction and we defined eccentricity e as in commutative case. The general form of this result is in agreement with the one obtained by a different method in reference [9]. Since noncommutativity parameter is extremely small, the noncommutative corrections will be very small.…”

Noncommutative geometry and the stability of circular orbits in a central force potential

“…The Poisson bracket should possess the same properties as the quantum mechanical commutator, namely, it should be bilinear , anti-symmetric and should satisfy the Leibniz rules and the Jacobi identity . The general form of the Poisson brackets for this deformed version of classical mechanics can be written as follows [8,9] :…”

“…The effect of space noncommutativity on orbital motions of particles in a central force potential has been investigated by Mirza and Dehghani [9]. For an inverse squared force(Coulomb force) as F = − k r 2 , the Hamiltonian up to the first order of α becomes…”

“…We assume that the time variation of the L is so small that in a short time interval (for example one century for Mercury) it could be taken as a constant of the motion. In this way, one can put a bound on the value of α by comparing the results of the perturbing term with the experimental value of the precession of the perihelion of Mercury [8,9]. For the Kepler problem it is well known that the bounded orbits are closed, which is a result of the following integral,…”

“…The modifications induced by the generalized uncertainty principle on the classical orbits of particles in a central force potential firstly has been considered by Benczik et al [8]. The same problem has been considered within noncommutative geometry by Mirza and Dehghani [9]. The main consequence of these two investigation is the constraint imposed on the minimal observable length and noncommutativity parameter in comparison with observational data of Mercury.…”