This is a comment on the recent paper by G. S. Adkins and J. McDonnell "Orbital precession due to central-force perturbations" published in Phys. Rev. D75 (2007), 082001 [arXiv:gr-qc/0702015]. We show that the main result of this paper, the formula for the precession of Keplerian orbits induced by central-force perturbations, can be obtained very simply by the use of Hamilton's vector.In the recent paper G. S. Adkins and J. McDonnell reconsidered the old problem of perihelion precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Their main result is the formula for perihelion precession in the form of a one-dimensional integral convenient for numerical calculations.The reason why this well studied and essentially textbook problem [2] came into the focus of the current research is the recent usage of this classical effect to constrain hypothetical modifications of Newtonian gravity from higher dimensional models [3], as well as the density of dark matter in the solar system [4].Traditionally the simplest way to study the perihelion motion is the use of the Runge-Lenz vector [5,6]. The Runge-Lenz vectoris the extra constant of motion originated from the hidden symmetry of the Coulomb/Kepler problem [7]. Here α = GmM , L is the angular momentum vector and v is the relative velocity of a planet of mass m with respect to the Sun of mass M . Geometrically the Runge-Lenz vector points towards the perihelion. Therefore its precession rate is just the precession rate of the perihelion [8].However, we will use not the Runge-Lenz vector but its less known cousin, the Hamilton vector [9,10,11,12] where ϕ is the polar angle in the orbit plane. This very useful vector constant of motion of the Kepler problem was well known in the past, but mysteriously disappeared from textbooks after the first decade of the twentieth century [9,11,12,13]. Of course, A and u are not independent constants of motion. The relation between them isRemembering that the magnitude of the Runge-Lenz vector is A = α e, where e is the eccentricity of the orbit [5], we get from (3) the magnitude of the Hamilton vectorIf the potential U (r) contains a small central-force perturbation V (r) besides the Coulomb binding potential,the Hamilton vector (as well as the Runge-Lenz vector) ceases to be conserved and begins to precess with the same rate as the Runge-Lenz vector, because according to (3) the two vectors are perpendicular.To calculate the precession rate of the Hamilton vector we first find its time derivativėwhere µ = mM m+M is the reduced mass. To get (5), we have used Newton's equation of motion for˙ v and the equatioṅ e ϕ = −φ e r .

Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

Observations indicate that ordinary matter, the baryons, influence the structural properties of dark matter on galactic scales. One such indication is the radial acceleration relation, which is a tight correlation between the measured gravitational acceleration and that expected from the baryons. We show here that the dark matter density profile that has been motivated by dissipative dark matter models, including mirror dark matter, can reproduce this radial acceleration relation.

The purpose of this article is to present a simple solution of the classic dog-and-rabbit chase problem which emphasizes the use of concepts of elementary kinematics and, therefore, can be used in introductory mechanics course. The article is based on the teaching experience of introductory mechanics course at Novosibirsk State University for first year physics students which are just beginning to use advanced mathematical methods in physics problems.