Separation of the variables

Unfortunately the actual Hamiltonian (Eq. II.3) also contains a two-electron part, which prevents the separation of the variables mentioned above. Since the two-electron operator may be written in the form... 

Clerk Maxwell (South Kensington Conferences, 1876), in discussing the work of Willard Gibbs, remarked that the existence of a system depends on the magnitudes of the system, which are the quantities of the components, the volumes, the entropies, as well as on the intensities of the system, viz., the temperature and the potentials of the components (cf. 143). In his Theory of Heat he also refers to a separation of the variables in terms of which the state can be defined into two classes, one of which includes what are called intensities (pressure, temperature), and the other magnitudes (volume, entropy). 

This equation may be solved by separation of the variables, which results in formula (A4.12) of Appendix 4 at t = 0. By Fourier transformation of the latter we find the spectrum... 

When the reaction temperature does not change with time, the rate coefficients remain constant, so that separation of the variables and integration gives... 

Appropriate substitutions, separation of the variables and integration gives... 

This separation of the variables allows a vast reduction in the number of calculations. This savings becomes more significant as the number of data points increases. Further, note that the final summation in this equation is in the form of a one-dimensional Fourier transform. This implies that the considerable calculational advantage of the fast Fourier transform (FFT) algorithm may be used here. The entire summation may be performed by repeated application of the one-dimensional FFT. This implies that for any data set that it is practical to apply the FFT, it would be also practical to apply the nonlinear constraints for improvement. 

This assumption can be tested using the integral method. Integrating Eq. 13.12 from the starting concentration [A]o at time 0 to the concentration [A] at time t, we get by separation of the variables... 

The V may be interpreted as a gravitational potential, velocity potential or, in the present context, as a circulation potential. The differential equation may be solved by separation of the variables under the assumption that the potential may be defined as the product of three one-dimensional potentials, or cartesian components of V = X (x) -Y(y) Z(z). This solution is substituted into the equation and after differentiation, divided by V, to give... 

To apply Eq. (2.8) to the hydrogen atom it is first transformed into polar coordinates (r,9,4>) and then solved by the method of separation of the variables. This involves writing the solution in the form... 

The first equation is much easier to manipulate than the second one. A simple check for separability of the variables is that curves of log strain vs. log time with stress as... 

Although no attempt will be made to solve this very complicated equation, it should be pointed out that in this form the separation of the variables is possible, and equations that are functions of r, 9, and 4> result. Each of the simpler equations that are obtained can be solved to give solutions that are functions of only one variable. These partial solutions are described by the functions R(r), 0(9), and ), respectively, and the overall solution is the product of these partial solutions. 

Separation of the variables to enable solution of the electronic wave equation of hJ requires clamping of the nuclei and hence imposing cylindrical symmetry on die system. The calculated angular momentum eigenfunctions are artefacts of diis approximation and do not reflect die full symmetry of die quantum-mechanical molecule. 

Suppose that K x > yt over the interval of integration from Z = 0 (the surface of the liquid on plate j) to Z = ZT (the floor of the plate on plate j). Separation of the variables in Eq. (13-44) followed by integration yields... 

We consider two equations for the three functions Pci(x) C, (x) and can express the ion concentration distribution through the potential distribution. By separation of the variables one can carry out the integration which yields... 

The differential equation (5) is said in this case to be soluble by separation of the variables, or, for short, to be separable. 

The Hamilton-Jacobi equation is solved by separation of the variables on putting... 

It may happen that the Hamiltonian function does not consist o a sum of terms depending on only one pair of variables qkpk, but that the Hamilton-Jacobi equation may be solved by separation of the variables, i.e. on the assumption that... 

Hence in the absence of identical commensurabilities all systems of co-ordinates, in which separation of the variables is possible, are connected by a transformation of the form... 

Hitherto wc have applied the quantum theory only to mechanical systems whose motion may be calculated by separation of the variables. We proceed now to deal in a general manner with the question of when it is possible to introduce the angle and action variables wk and Jfc so admirably suited to the application of the quantum theory. For this purpose it is necessary, in the first place, to fix the J s by suitable postulates so that only integral linear transformations with the determinant 1 are possible for it is only in such cases that the quantum conditions (1) Jk=nkh... 

Even the three-body problem, to say nothing of those involving more bodies, belongs to that class of mechanical problems which have not been solved by the method of separation of the variables, and, indeed, are hardly likely to be. In all such cases one is compelled to fall back on methods which give the motion to successive degrees of approximation. These methods are applicable if a parameter A can be introduced into the Hamiltonian function in such a way that for A=0 it degenerates into the Hamiltonian function H0 of a problem soluble by the method of separation, provided also that it may be expanded in a series... 

We have considered this problem in detail in 18. If the equation (1) is soluble by separation of the variables, we obtain new angle and action variables wk°, J/c°. If the generator of the transformation is... 

Our considerations show that in the case of one degree of freedom the motions for which phase relations hold are the only ones possible according to the quantum theory. The same is true if the equation (5) is soluble by separation of the variables or can be made so by a transformation of the wp° s. Equations of the form (6) are then obtained for the individual terms of So and all conclusions which follow from this equation can be arrived at in the same manner. 

We shall consider a system whose motion may be found by the method of separation of the variables when unperturbed. As we have seen ( 14), in separable systems the trajectory in the g-space is bounded by a series of surfaces, each of the separation co-ordinates oscillating backwards and forwards between two surfaces of such a series. In certain cases these surfaces may coincide. The number of dimensions of the region filled by the path is then decreased by 1. 

We have invariably collected all the x s on one side, the t s, on the other, before proceeding to the integration. This separation of the variables is nearly always attempted before resorting to other artifices for the solution of the differential equation, because the integration is then comparatively simple. 

E. Clapeyron s equation, previously discussed on page 453, may be solved by the method of the separation of the variables. 

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