Parabolic co-ordinates

It is easy to see that the Hamilton-Jacobi differential equation is separable neither in rectangular nor in polar co-ordinates. It may, however, be made separable by introducing parabolic co-ordinates. We put... 

The parabolic co-ordinates f and tj execute librations between the zero points of /x(f) and /2(ij) in (6). We will consider the character of the motion first for the case in which J, and consequently C, does not vanish. Here the region in which/ f) and/a( j) are positive does not extend to the positions f=0 and 0 the zero points miu an(i Vmia are different from 0. The third co-ordinate in this case performs a rotation. The path is confined to the interior of a ring having the direction of the field as axis of symmetry and the... 

The calculation of the Stark effect by parabolic co-ordinates allows us to illustrate by an example some previous considerations regarding the restriction of the quantum conditions to nondegenerate action variables. 

For E =0 the motion of the Stark effect passes over into the simple Kepler motion. This is separable in polar co-ordinates as well as in parabolic co-ordinates. From the separation in polar co-ordinates ( 22) we obtain the action variables Jr, Je, J0, and the quantum condition... 

If now we calculate the Kepler motion in parabolic co-ordinates, we have only to put E=0 in the above calculations. We obtain the action variables J(, J, and (the last has the same significance as in polar co-ordinates) and the quantum condition... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

The parabolic co-ordinates used in the separation method to determine the motion of an electron in the hydrogen atom under the influence of an electric field are a special case of elliptic co-ordinates. The latter are the appropriate separation variables for the more general problem of the motion of a particle attracted to two fixed centres of force by forces obeying Coulomb s law. If one centre of force be displaced to an infinite distance, with an appropriate simultaneous increase in the intensity of its field, we get the case of the Stark effect at the same time the elliptic eo-ordinates become parabolic. 

The electrons within the atom are actually not quantised in parabolic coordinates, but instead, on account of the central field of the atom core, in polar co-ordinates. It would, then, not be logical to attempt to select favoured values of m and n3. Instead, we shall calculate the quantity... 

Added February 10, 1927.—J. H. Van Vleck in Proc. Nat. Acad. America, vol. 12, p. 662 (December, 1926), has discussed the mole refraction and the diamagnetic susceptibility of hydrogen-like atoms with the use of the wave mechanics, obtaining results identical with our equations (24) and (34). He also considered the effect of the relativity corrections (which is equivalent to the effect of a central field) and concluded that equation (24), derived by the use of parabolic instead of spherical co-ordinates, is not invalidated.]... 

The Kf matrix is related to the conventional K matrix of MQDT by a frame transformation from parabolic to spherical co-ordinates the K matrix is then related by a further frame transformation to the quantum defects [43, 45], The first term in Eq. (4) gives a contribution to the photoionization intensity borrowed from the bound-state spectrum. The dlyom term represents direct photoionization, and the overall expression allows Fano-type interference between these terms. In Eq. (5) A is a phase shift in the parabolic rep-... 

In all isotherms plotted within the region of low surfactant concentrations (2.5 10 6 to 3 I O 6 mol dm 3) and at A a - 0.5 mN m l there is a linear part corresponding to r = kC dependence (r is the adsorption). This part of the isotherm for curve 1 is presented in linear co-ordinates on the top left side of Fig. 3.77. A short plateau follows where dAo/d gC = 0. The further increase in surfactant concentration leads to a parabolic increase in Act until the next flexion of the curve is reached at Act - 8-10 mN m 1. Similar change in the course of Ao(0 isotherms has been found also for potassium and sodium oleates solutions [369,370], decane and undecane acid solutions [366] and aqueous solutions of saturated fatty alcohols [367]. It is worth to note that the measurements were carried out with purified substances so that any inoculations by a second surfactant are excluded. With the increase in surfactant concentration the parabolic part gradually transforms into a second linear part of the isotherm. 

The dependence of the radius of border curvature on the co-ordinate of the direction of flow involved in the calculation of flow rate through a foam is determined from Leonard-Lemlich s equation (5.2) as well from that of Laplace while for the drainage process an independent equation is proposed (for example, parabolic equation). 

The first model of a parabolic profile (with constant co-ordinate at the parabola top) gives the following expressions about the pressure gradient at border mouth and the t(r) dependence for an arbitrary border cross-section... 

To compute unsteady flows, the time derivative terms in the governing equations need to be discretized. The major difference in the space and time co-ordinates lies in the direction of influence. In unsteady flows, there is no backward influence. The governing equations for unsteady flows are, therefore, parabolic in time. Therefore, essentially all the numerical methods advance in time, in a step-by-step or marching approach. These methods are very similar to those applied for initial value problems (IVPs) of ordinary differential equations. In this section, some of the methods widely used in the context of the finite volume method are discussed. 

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