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Register a new userNewtonian approach for the Kepler-Coulomb problem from the point of view of velocity space
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The hodograph of the Kepler-Coulomb problem, that is, the path traced by its velocity vector, is shown to be a circle and then it is used to investigate other properties of the motion. We obtain the configuration space orbits of the problem starting from initial conditions given using nothing more than the methods of synthetic geometry so close to Newtons approach. The method works with elliptic, parabolic and hyperbolic orbits; it can even be used to derive Rutherfords relation from which the scattering cross section can be easily evaluated. We think our discussion is both interesting and useful inasmuch as it serves to relate the initial conditions with the corresponding trajectories in a purely geometrical way uncovering in the process some seldom discussed interesting connections.
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The magnetic moment of a particle orbiting a straight current-carrying wire may precess rapidly enough in the wires magnetic field to justify an adiabatic approximation, eliminating the rapid time dependence of the magnetic moment and leaving only the particle position as a slow degree of freedom. To zeroth order in the adiabatic expansion, the orbits of the particle in the plane perpendicular to the wire are Keplerian ellipses. Higher order post-adiabatic corrections make the orbits precess, but recent analysis of this `vector Kepler problem has shown that the effective Hamiltonian incorporating a post-adiabatic scalar potential (`geometric electromagnetism) fails to predict the precession correctly, while a heuristic alternative succeeds. In this paper we resolve the apparent failure of the post-adiabatic approximation, by pointing out that the correct second-order analysis produces a third Hamiltonian, in which geometric electromagnetism is supplemented by a tensor potential. The heuristic Hamiltonian of Schmiedmayer and Scrinzi is then shown to be a canonical transformation of the correct adiabatic Hamiltonian, to second order. The transformation has the important advantage of removing a $1/r^3$ singularity which is an artifact of the adiabatic approximation.
The inverse problem for electromagnetic field produced by arbitrary altered charge distribution in dipole approximation is solved. The charge distribution is represented by its dipole moment. It is assumed that the spectral properties of magnetic field of the dipole are known. The position of the dipole and its Fourier components are considered as the unknown quantities. It is assumed that relative increments of amplitude and phase of magnetic field in the vicinity of the observation point are known. The derived results can be used for study of phenomena concerned with occurrence and variation of localized electric charge distribution, when the position and the dynamics of a localized source of electromagnetic field are to be defined.
The fact that the capacitance coefficients for a set of conductors are geometrical factors is derived in most electricity and magnetism textbooks. We present an alternative derivation based on Laplaces equation that is accessible for an intermediate course on electricity and magnetism. The properties of Laplaces equation permits to prove many properties of the capacitance matrix. Some examples are given to illustrate the usefulness of such properties.
In this paper we study the Kepler problem in the non commutative Snyder scenario. We characterize the deformations in the Poisson bracket algebra under a mimic procedure from quantum standard formulations and taking into account a general recipe to build the noncommutative phase space coordinates (in the sense of Poisson brackets). We obtain an expression to the deformed potential, and then the consequences in the precession of the orbit of Mercury are calculated. This result allows us to find an estimated value for the non commutative deformation parameter introduced.
The paper draws the attention to the spatiotemporal symmetry of various vector-like physical quantities. The symmetry is specified by their invariance under the action of symmetry operations of the Opechowski nonrelativistic space-time rotation group O(3).{1, 1}= O(3), where 1 is time-reversal operation. It is argued that along with the canonical polar vector, there are another 7 symmetrically distinct classes of stationary physical quantities, which can be - and often are - denoted as standard three-components vectors, even though they do not transform as a static polar vector under all operations of O(3). The octet of symmetrically distinct directional quantities can be exemplified by: two kinds of polar vectors (electric dipole moment P and magnetic toroidal moment T, two kinds of axial vectors (magnetization M and electric toroidal moment G), two kinds of chiral bi-directors C and F (associated with the so-called true and false chirality, resp.) and still another two achiral bi-directors N and L, transforming as the nematic liquid crystal order parameter and as the antiferromagnetic order parameter of the hematite crystal alpha-Fe2O3, respectively.
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