Cartesian Frame Research Articles

Observer frame rotation rates and magnetic fields in spatially homogeneous universes

We discuss how the rotation of an observer’s Cartesian reference frame is related to the precession of the shear tensor’s principal axes and to the rotation vector of the fluid. An approximate solution of Einstein’s equations illuminates this relationship further. In the case with a magnetic field and fluid flow a vorticity is allowed in Bianchi I cosmologies.

The tidal physics of headland flows and offshore tidal bank formation

The general tidal characteristics in the neighbourhood of a promotory are derived using a numerical model developed in plane polar coordinates. These same tidal features are found for a model of the tides off the Start Point promontory using a Cartesian reference frame. It is shown that promontories are characterised by strong tidal flows and it is suggested that vorticity generated near the headland plays a key role in the dynamics of offshore tidal bank formation.

Time dependent compressible flow about a spherically symmetric polymer in solution

We obtain the general solution of the linear Navier-Stokes equation for time dependent compressible viscous flow about a spherically symmetric polymer molecule. The solution is presented in a covariant form valid in a general cartesian coordinate frame. In the course of deriving the solution we obtain a general decomposition of the unperturbed flow in the absence of the polymer. Our solution generalizes the earlier solution derived by Schmitz and Felderhof for the case of creeping flow.

Absolute cross sections for collision-induced depolarized scattering of light in krypton and xenon

Depolarized Raman spectra of binary collisional pairs of atoms in krypton and xenon are obtained at gas densities of 1–10 amagat. Absolute intensities relative to a known rotational transition of nitrogen are determined. For light of 4880 A wavelength incident in the x-direction, polarized in the z-direction and scattered in the y-direction of a cartesian frame x, y, z, at a frequency shift of -12 cm-1, the differential scattering cross section per unit wavenumber band times volume, is found to be 1·10 × 10-52 cm6 ± 10 per cent for krypton, and 4·76 × 10-52 cm6 ± 10 per cent for xenon, if the sum of both polarizations is considered. Wave-mechanical and classical computations reproduce both the shape and the intensity of the experimental spectra if the so-called point-dipole model of the anisotropy of the polarizability of collisional pairs of atoms is used. Other models of the anisotropy are seen to be overcorrected by these criteria.

On Asymmetric Periodic Solutions of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

Previous work on the plane circular restricted problem of three bodies (Message 1953, 1959, 1970, and Fragakis 1973) has shown the existence, in association with each of the commensurabilities 2:1 and 3:1 of the orbital periods, of a pair of families of asymmetric periodic solutions, branching from the stable series of symmetric periodic solutions of Poincaré’s second sort associated with that commensurability. (Each solution of either family is the mirror image, in the line of the two finite bodies, of a member of the other family of solutions associated with the commensurability.) The stability is transferred at the bifurcation to the two series of asymmetric orbits, each of which is therefore stable. Recent numerical integrations carried out by one of us (P.J.M.) have found such asymmetric periodic orbits associated also with the 4:1 commensurability, and quantities describing orbits of one of the two series are given in Table 1, showing the run of such orbits up to a second bifurcation with the same series of symmetric periodic orbits from which it sprang. Quantities describing some members of this series of symmetric orbits are given in Table 2. It is seen that stability is transferred back to the symmetric series at the second bifurcation. (The unit of distance is the distance between the two finite bodies, the unit of speed is the speed, of their relative motion, and the initial conditions given (x°, ẋ°, ẏ°) are for a crossing of the line of the two finite bodies, this line being taken as axis of “x” in a rotating Cartesian frame in the usual way. The mean values of the major semi-axis and eccentricity are denoted by ā and ē, respectively, C is Jacobi’s constant, and ȳ2 is the mean value of the critical argument ȳ2 = 4λ – λ′ – 3ω. The mass ratio used is 0.000954927, T is the period of the solution in units of the period of the motion of the two finite bodies, and 2π c/T is the non-zero characteristic exponent.)

Lorentz transformations as space-time reflections

A rank-two tensor is built out of the 4-velocities of two inertial observers, which corresponds precisely to the most general Lorentz matrix connecting the two Cartesian frames of the observers. The Lorentz tensor is then factorized as the product of two ’’complementary’’ space-time reflections. It is shown that the first tensorial factor performs the very essential tasks (i.e., FitzGerald contraction and time dilation) of the corresponding Lorentz transformation, while the second factor is just an internal reflection performed in one and the same intertial frame. Thus, in its essential features, a Lorentz transformation between two different inertial frames obtains upon performing just one space-time reflection. It is also shown that the (same) Lorentz tensor of the two inertial observers can be factorized into ’’complementary’’ reflections either by two hyperplanes with spacelike normals, or else by tow hyperplanes with timelike normals, which geometric meaning is rather simple. An application of the presented formalism to Dirac’s 4-spinor transformation law is also briefly discussed.

The Cartesian frame of reference: a structure unifying the description of dyslexia.

It is shown that six phenomena often associated with dyslexia may be attributed to the lack of a visual, Cartesian frame of reference. These phenomena are (1) reading errors due to letter reversal, inversion, or rotation, (2) form recognition of all classes of mono-oriented objects which is independent of the figure's egocentric orientation, (3) defective visual sequential scanning, resulting in confused letter and word order (4) poor visual balance performance, (5) the failure to acquire nonspatial ordering relations, including temporal relations and tenses, and (6) the characteristics intellectual profile of severely retarded readers, namely, PIQ/VIQ greater than 1, where PIQ is the performance IQ score and VIQ the verbal IQ score on the Wechsler Intelligence Scale for Children (WISC). The latter phenomenon is dependent on (2) and (5) above. A strategy for further research is outlined.

Free Molecular Flow over a Rotating Sphere

6= freestream temperature =|5| = the x, y, and z components of the mass velocity of the freestream molecules with respect to the local coordinates, respectively = Cartesian stationary frame of reference = local Cartesian coordinates, fixed on the moving differential area = unit vectors in the x, y, and z directions, respectively = azimuth, measured from negative X axis = mn^, freestream density = momentum accommodation coefficients, Eq. (12) and Eq. (13) — tangential component of momentum flux, Eqs. (6b) and (6c), or shearing stress acting on the differential area, Eqs. (14b) and (14c) = latitude, measured from positive Y axis SubscriptsK = components in the X, Y, and Z directions, respectively i = quantity impinging to the sphere r = quantity reflected from the sphere w = value at sphere surface x, y, z = components in the x, y, and z directions, respectively.

Parametric Representation of Probability in Two DimensionsߞA New Approach in System Reliability Evaluation

The evaluation of system reliability becomes very diffilcult when the number of components in a system is large or the system configuration is complex. This note presents a straightforward method for system reliability evaluation. Probability is treated as a point in a Cartesian frame and formulas are derived to evaluate the reliability of series, parallel, series-parallel, and bridge networks. The values obtained by the classical method agree closely with those obtained by this parametric method.

Drawing Vibrational Modes by Computer

Spectroscopists by necessity utilize group theory and normal coordinate treatment (NCT) to describe the normal modes of vibration of molecules. As discussed previously, the normal coordinates can be transformed to a Cartesian reference frame yielding a table of Cartesian displacement coordinates. Using known bond distances and bond angles, the equilibrium Cartesian coordinates can be calculated. However, equilibrium and displacement coordinates are very difficult to comprehend unless the graphical equivalent (or plot) is drawn.

Invariant Piezoelectric Coefficients for Crystals

AbstractIn writing out tensors which represent physical properties of crystals, it is customary to choose a Cartesian frame of reference which has a specific orientation with respect to the crystallographic axes. A new formulation is proposed in which the physical properties (in particular the piezoelectric property) of crystals can be specified without reference to any coordinate system.

The incorporation of spirit-levelling into a three-dimensional cartesian reference frame

Abstract In modern first-order geodetic work it is customary to measure horizontal and vertical angles, spatial distances and astronomic azimuths, latitudes and longitudes. These are projected onto an ellipsoid of reference and are then adjusted to yield positional co-ordinates on that ellipsoid. The results of spirit-levelling, which yield heights above the geoid, are commonly adjusted as a separate entity. Thus the classical method produces two distinct sets of unrelated quantities. A method for rigorously incorporating the results of spirit-levelling into a three-dimensional Cartesian frame of reference is presented. The use of the geoid and of an ellipsoid is avoided.

Angular Dependence of Zeeman Spectra ofGd3+in CaF2at Cubic Sites

The optical Zeeman spectra of $^{6}P_{\frac{5}{2}}$ and $^{6}P_{\frac{7}{2}}$ multiplets of ${\mathrm{Gd}}^{3+}$ in Ca${\mathrm{F}}_{2}$ at a site of cubic symmetry show a pronounced angular variation with rotation of the sample in the magnetic field. This occurs even when the initial and final magnetic sublevels are rotationally invariant. This phenomenon may be explained by considering that the $^{8}S_{\frac{7}{2}}$ ground state, split only 0.149 ${\mathrm{cm}}^{\ensuremath{-}1}$ by the crystal field, is described at magnetic fields greater than about 4 kG by the projection quantum number ${J}_{z}$ referred to the magnetic field axis; and that the excited states are most conveniently described with the basis along the fourfold crystal ax is. The calculation of the dependence of the Zeeman patterns on crystal orientation, therefore, ${\mathrm{require}}^{\mathrm{S}}$ a transformation of the basis to a common Cartesian reference frame. This may be accomplished by the rotation matrix ${{R}_{{{J}_{z}}^{\ensuremath{'}}{J}_{z}}}^{J}$ acting on the ground-state basis, or by transformation of the crystal-field Hamiltonian for the excited states from the fourfold basis to the magnetic field. The crystal-field quantum numbers (irreducible representations) assigned to the $^{6}P_{\frac{7}{2}}$ and $^{6}P_{\frac{5}{2}}$ multiplets are in order of increasing energy: $^{2}\ensuremath{\Gamma}_{6}$, $^{4}\ensuremath{\Gamma}_{8}$, $^{2}\ensuremath{\Gamma}_{7}$ and $^{4}\ensuremath{\Gamma}_{8}$, $^{2}\ensuremath{\Gamma}_{7}$. The measured $g$ factors are 1.61\ifmmode\pm\else\textpm\fi{}0.06 and 1.76\ifmmode\pm\else\textpm\fi{}0.1 for the $^{6}P_{\frac{7}{2}}$ and $^{6}P_{\frac{5}{2}}$ levels, respectively.

Dynamical Content of Differential Geometry

A formulation of the notion of a Cartesian basis for a vector space suitably adapted to the local differential vector space at a space-time point leads to this description of and derivation for the law of motion of a mass point: There exists a class of equivalent 4-dimensional Cartesian frames in which the trajectory is a straight line, the velocity being constant along the trajectory. The lived-in coordinate system, in which the trajectory is in general not a straight line, is in general not a Cartesian coordinate system. The equation for the motion of the point is purely a statement of differential geometry—an integrability condition—and the forces which are responsible for the deviation from constant velocity are immediately derivable from the metric tensor of the non-Cartesian system. The equation of motion and the relations among dynamical variables are more relativistic than Newtonian, but the metric tensor is positive definite; no indefinite metric is needed. A Newtonian equation of motion is an approximation based on small velocity and it covers the general case of motion in a noninertial system in an external field of force. Intrinsic mass is identified as a constant of the motion and, although the relevancy of mass is somewhat limited in this description of the motion of a single point in an immutable force field, it is seen that the effective mass of the point contains a contribution by the potential energy.

Thermodynamic potentias of elastic dielectrics at finite fields and deformations

The paper is a continuation of Hajicek, Czech. J. Phys.B 17 (1967), 613. The so-called reduced tensors are defined, a special case of which, the scalar densities referred to volume unit of an undeformed body, has often been used in the elasticity theory. Relations written by means of the former retain the simplicity they possessed in the convected system, and in addition they are invariant under any transformation of coordinates, thus allowing a record also in the Cartesian frame. The symmetry of the relations obtained enables us to define all eight reduced densities of the thermodynamic potentials without any approximation, i.e. for finite fields and deformations, and to write the respective Maxwell's relations. If the latter are expressed through the unreduced tensors, relations differing from those commonly used are obtained, if the quadratic and higher order phenomena are not neglected. Already in the theory of quadratic phenomena, therefore, it is necessary either to make use of the reduced tensors, or to consider the modified Maxwell's relations.

An algebra for the point groups

A concept of composite numbers to denote coordinates of points in a cartesian frame of reference is developed in order to describe certain symmetry relationships between points in a crystal lattice. The laws of an algebra for use in the study of the crystallographic point groups are stated and are established on the basis of conventional matrix algebra.

General Theory for Thick Shell Analysis

Applications of shell structures to many engineering fields have introduced the problem of analyzing thick elastic shells. A useful tool for describing shells is tensor analysis. By using such a tool, a formulation based on power series expansions for displacements is developed. The resulting equations may be considered as a recurrence group, permitting the obtaining of the series coefficients in terms of six principal variables. These principal variables are the three middle surface displacements, and the three rotations about the base vectors. The equations for the thick shell theory are developed for the geometrical nonlinear case. The engineering solution of such equations can be obtained by using a cartesian reference frame.

THE CORRECTED ORTHOGONAL ELECTROCARDIOGRAM AND VECTORCARDIOGRAM IN 510 NORMAL MEN (FRANK LEAD SYSTEM).

Corrected orthogonal electrocardiographic records were taken from 510 normal male subjects. Their age ranged from 19 to 84 years. Data processing and analysis were performed by means of a digital computer. Measurements were taken from the three scalar orthogonal leads and from a variety of spatial vectors in a Cartesian reference frame. Main emphasis was put on series of instantaneous vectors of the QRS complex, the ST segment, and the T wave. In order to eliminate inter-individual variability in electrocardiographic wave durations, time normalization was applied by dividing QRS, ST, and T into equal parts in time. Results for instantaneous vectors were also obtained in absolute time at intervals of 0.01 second. In addition, time integrals, polar vectors, and Eigenvectors were determined for P, QRS, and T. Normal ranges were computed for both normally and abnormally distributed results.

Matter tensors in the crystallographic groups of cartesian symmetry

I. Many recent publications in crystals physics give the components of matter tensors in the various crystallographic groups. We think it is useful to complete the known results with a table of the components of general matter tensors of polar and axial nature, of second, third and fourth order, for the 20 Cartesian groups: triclinic, monocl~ic, orthorhombic, tetragonal and cubic. Our results are derived by means of the method of direct inspection, which has been explained elsewhere 1), i). We shall give here a very simple example. In Cartesian frames, under coordinate transformations of the first kind, tensor components transform as (non-commutative) products of coordinates: under coordinate transformations of the second kind, the components of polar tensors transform also as coordinate products, however, the transformation equations for components of axial tensors differ from those for products of coordinates by a factor -1. Let us consider two second-order tensors, respectively of polar and axial nature, in the point group m. We choose a Cartesian reference frame with the y axes parallel to the inversion diad: the symmetry transformation which corresponds to the inversion diad is then

Rectilinear flow of non-Bingham plastic solids and non-Newtonian viscous liquids. I

In a previous paper on steady rectilinear plastic flow of a Bingham solid (1), the point of view was adopted that the study of all possible velocity distributions was equivalent to a study of the geometry of velocity contours in a section normal to the flow. This led to an analysis of the differential geometry of velocity contours in rectilinear plastic flow under zero pressure gradient, and the problem of finding all possible velocity distributions was eventually reduced to the problem of finding the general integral of a single linear second-order partial differential equation with two independent variables. But the form of this differential equation (equation (25) of (1)) was such that only a very limited number of solutions could be found by standard methods. The particular solutions which were obtained corresponded to those flow patterns in which the velocity contours could be identified with a family of parametric curves in a curvilinear coordinate system obtainable by conformal transformation of a cartesian frame of reference. In effect, this means that the results so far achieved by using the new approach to the problem could have been obtained by more direct methods.