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With the accuracy of the available P₁₁ amplitudes of πΔ scattering, we show that two resonance poles near the pi Delta threshold, obtained in several analyses, are stable against large variations of parameters within a dynamical coupled-channels analysis. The number of poles in the 1.5 GeV < W < 2 GeV region could be more than one, depending on how the structure of the single-energy solution of SAID is fitted. Lastly, our results indicate the need of more accurate πN scattering data in the W > 1.6 GeV region for high precision resonance extractions.

Here a new method is proposed for constructing examples in relativistic classical particle mechanics, starting with the Poincaré group and space reflections as transformations of position and velocity variables for two free particles and deriving equations for Lorentz covariant position variables for two interacting particles. The advantages are that the equations do not couple the position variables of the two particles, and solutions yield the particle world lines directly. Solutions are obtained for a special case.

For a classical-mechanical system of any fixed number of particles it is observed that space-translation invariance and conservation of angular momentum imply conservation of momentum. For three particles it is shown, as previously for two, that Poincaré invariance implies that the total kinematic momentum cannot be a constant of motion unless the accelerations are zero. Finally, the equations involved make it appear most likely that this is true for any number of particles.

The purpose of the present paper is to elucidate the relationship between the time dependence of quantum operators in the Heisenberg picture and the time dependence of the corresponding dynamical variables in the underlying classical theory. This problem is studied in the nonrelativistic particle mechanics and in the field theory. Further, it is shown how the operator solutions of the quantum equations of motion are related to the corresponding solutions of the classical equations of motion. An explicit formula is given, which expresses quantum operators in the Heisenberg picture in terms of their classical counterparts. This formula is particularly usefulmore » in the study of the classical limit of the quantum theory. The dependence on $$\hbar$$ of the matrix elements of the coordinate operators and the field operators is explicitly given, which enables one to study the quantum corrections to the classical theory in all orders. Coherent states of the quantum system play an essential role in the formalism.« less

Here, for a classical-mechanical system of two particles, the conditions for Lorentz-invariant equations of motion are expressed in terms of relativistic momentum variables, and are shown to imply that neither the conventional total kinematic particle momentum nor the conventional total kinematic particle angular momentum is a constant of the motion unless the accelerations are zero. This is compared with a theorem of Van Dam and Wigner.

A cylindrically symmetric parallel bundle of magnetic lines of force, in equilibrium under their mutual gravitational attraction ("magnetic universe"), has recently received attention. While a Newtonian analysis suggests that the equilibrium is unstable, the complete general relativity analysis shows that the equilibrium is stable. This discrepancy may have to do with the unusually slow falloff of the gravitational field at large distances in this geometry. In order to understand the gravitational field of the static magnetic universe somewhat better, we have studied its timelike and lightlike geodesics (i.e., the orbits in it of electromagnetically neutral test particles with unit ormore » zero rest mass). Since the density of magnetic flux-and energy and stress and, therefore, "gravitating mass"-is approximately uniform in the vicinity of the axis, the motion of test particles there is like that in a Newtonian simple harmonic oscillator field. "Vicinity" here means within a small fraction ρ of the range radius $$\bar{a}$$ = (6.96/B₀) × 10²⁴ cm (B₀ is the magnetic field on the axis measured in gauss). As is to be expected from the universality of the angular frequency ω0 in the harmonic oscillator field and the relation: orbital velocity ≅ ω₀ρ, no motion can get too far from the axis. Otherwise the physical orbital velocity would exceed the speed of light. It is in this way that the strength of the attractive field, though it does not remain strictly of the harmonic oscillator type as one proceeds outward, implies that there is a critical straddling radius ρ = 1/√3. Circular or circular helical light tracks occur only at the critical radius, and with B₀ = 10⁵ G, the time required for light to circumnavigate the critical circle is about 200 years. The cylinder marked out by this radius plays a unique limiting role: All particles, whether of zero or nonzero mass, and no matter what their initial positions and velocities (except in the one singular subcase of light tracks parallel to the cylindrical axis), must have their orbits lying wholly or partially within the cylindrical region ρ < 1/√3; hence the use of the adjective "straddling." Constants of motion which correspond closely to ζ-component linear momentum, angular momentum, and energy in Newtonian mechanics are defined. Bounds are placed on these dynamical constants and on the apsidal radii by the requirement that the range of motion be real. Finally, the magnetic universe is complete in the sense that "no news can enter or leave''-all orbits are of infinite duration.« less

The motion of frontal disturbances in the atmosphere is studied by the numerical solution of differential equations based upon a two-layer model of an incompressible fluid on a rotating earth. The density of each layer is assumed to be constant. The upper and lower fluids correspond respectively to warm and cold air. In this first attempt, only the motion of the lower cold air layer is studied by assuming, in effect, that the dynamics of the perturbations in the upper warm air layer can be neglected. The numerical study of this simple mechanical model shows that even though thermodynamic processesmore » have been ignored, the occlusion process, characteristic for warm and cold fronts, develops from an initially sinusoidal frontal pattern. Two cases of different initial conditions are examined. Case A: Only the east-west component of wind velocity is initially geostrophic. Case B: Both east-west and north-south components are initially geostrophic. In both cases, computations indicate that the cold front propagates faster than the warm front and that a relatively strong mass convergence zone appears behind the cold front only. This fact suggests the occurrence of severe storms associated with cold fronts, but not with warm fronts in the atmosphere. The numerical method developed here to calculate the movement of the front is based on following the motion of the material “particles” at the front. This method has applications to the numerical solution of a certain class of hydrodynamic flow problems in which the entire boundary of the domain of integration is not given a priori, but must be determined (so-called free-boundary problems).« less