Plane polar coordinates

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

By using the plane polar coordinates defined in Eq. (D.l), one obtains... 

Consider a particle of mass m confined to move on a circle of radius a. Express the Flamiltonian operator in plane polar coordinates and then determine the energy levels and wave functions. 

Here A is the distance of the foci, which are found on the. s 12-axis. For = 0we have plane polar coordinates. Varying v e [0,2n at constant u describes an elliptical orbit with a = yjA2 + u2 and b = u its semimajor and semiminor axis, respectively. 

It has two components, (x, y) in cartesian coordinates, and these can be transformed into plane polar coordinates to give... 

We can evaluate the integral in (4.73) by transforming to plane polar coordinates and using the integral representation of the Bessel function J0,... 

Solve the Schrodinger equation for the two-dimensional isotropic harmonic oscillator using plane polar coordinates. First show that... 

We have considered only one particular degenerate vibrational level. The general case is dealt with by solving the isotropic two-dimensional harmonic oscillator in plane polar coordinates (Problem 6.19). The result is... 

A simple example is provided by the coordinates that may be used to describe the motion of a particle in a plane. As generalized coordinates one can use either Cartesian coordinates qi = x, q2 = y, or plane polar coordinates qi = r, q2 = central force the coordinate p becomes cyclic, while both x and y are non-cyclic. The number of cyclic coordinates thus depends on the choice of generalized coordinates, and for each problem there may be one particular choice for which all coordinates are cyclic. 

The differential dxdy represents an element of area in cartesian coordinates, with the domain of integration extending over the entire xy-plane. An alternative representation of the last integral can be expressed in plane polar coordinates r, 6. The two coordinate systems are related by... 

Recognize the real and imaginary parts of a complex number expressed in either cartesian or plane polar coordinates... 

Very recently, SC-SCF in hyperspherical coordinates was tested for the coupled stretching vibrations of H20 and C02.25 The hyperspherical coordinates are highly collective and are thus expected to be very suitable for SCF. Also, for the states considered here the hyperspherical system provides a good frequency separation between the coordinates involved. For a linear, nonbending molecule A-B-C, the Hamiltonian of the stretching vibrations can be written as follows in hyperspherical coordinates (which are just plane polar coordinates in this case) ... 

Figure 2.6 A position vector, p, in a plane, with plane polar coordinates and Cartesian coordinates.

Changing from plane polar coordinates to Cartesian coordinates is an example of transformation of coordinates, and can be done by using the equations... 

Express the equation y = b, where bis a constant, in plane polar coordinates. 

Find the values of the plane polar coordinates that correspond to x = 2, y = 4. 

Transform this to plane polar coordinates and find the value of B such that the integral of over the entire x-y plane is equal to unity. 

In transforming from Cartesian to plane polar coordinates, the factor p, which is used with the product of the differentials d[Pg.216]

Here, r and 6 are plane polar coordinates of the projectile referred to the target at the origin, and p is the reduced mass. L is the angular momentum about the centre of mass and the second term in eqn. (50) is, of course, the centrifugal potential. 

For many potential functions it is easier to work in plane polar coordinates than in rectangular coordinates. In polar coordinates Eq. 10.7 has the form... 

Cartesian coordinates locate a point (x, y) in a plane by specifying how far east (x coordinate) and how far north (y coordinate) it lies from the origin (0, 0). A second popular way to locate a point in two dimensions makes use of plane polar coordinates, (r, 6), which specifies distance and direction from the origin. As shown in Fig. 10.4, the direction is defined by an angle 0, obtained by counterclockwise rotation from an eastward heading. Expressed in terms of Cartesian variables x and y, the polar coordinates are given by... 

It is more economical to use hyperspherical coordinate systems " for HLH systems. For collinear configurations, these coordinates are also plane polar coordinates, but the turning center is located at the origin. These coordinates have had a wide application to collinear re-actions,especially those of the HLH variety. The hyperspherical radius p is independent of the arrangement channel index... 

In our earlier theory of a planar interface we took to be independent of the position in any plane parallel to the interface, and to depend only on the distance z in the direction perpendicular to such planes. Now, instead, 4 depends in an essential way on two coordinates, viz. those in any plane perpendicular to the three-phase line but all sudi planes are equivalent, as is independent of position along the three-phase line. For example, let r, 6, x be cylindrical coordinates in the cylinder of Fig. 8.13, with X the axial coordinate, so that r, 0 are plane polar coordinates in any plane perpendicular to the three-phase line. Since the axis of the cylinder is parallel to the three-phase line. 

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