Volume 16, Issue 7, July 1975
Index of content:

The triad‐interaction representation of homogeneous turbulence
View Description Hide DescriptionThe triad‐interaction representation has been presented for the 2D and 3D homogeneous flows. This has several advantages over the usual Fourier‐amplitude representation: (i) The incompressibility is built into the equation as in the vorticity equation. (ii) For a given wave vector, the number of dynamic equations is one less than that of the Fourier‐amplitude equations. (iii) In the inviscid limit, energy and enstrophy are conserved in 2D, whereas the 3D flow conserves energy and helicity. (iv) The entire family of triad interactions is categorized into two classes in 2D and four classes in 3D, according to the geometry of triad wave vectors. Lastly, (v) the necessary conditions for isotropy in 3D emerge as the reflexional, rotational, and spherical symmetries in the wave vector space, whereas polar symmetry is only the requirement in 2D. The triad‐interaction representation has proved very useful in the investigation of isolating constants of motion and the statistical theory of nonisotropic turbulence.

Isolating constants of motion for the homogeneous turbulence of two and three dimensions
View Description Hide DescriptionFor the inviscid eddy motion in a finite‐dimensioned Fourier space, it is stated that energy and enstrophy are the isolating constants of motion for the 2D homogeneous turbulence. In contrast, the 3D isotropic turbulence has energy as the only constant of motion. If we relax the reflexional invariance, however, helicity emerges as another invariant; hence energy and helicity are said to be the isolating constants of motion for the helical turbulence. Although these are the key assumptions in the construction of equilibrium distributions, they have heretofore been accepted, without proof, as a natural property of the Navier–Stokes dynamics. This paper provides the proof. We have shown here that quadratic constants of motion for the individual triad‐interactions collapse to energy–enstrophy in 2D, but to energy and helicity in 3D.

An invariance theory for second‐order variational problems
View Description Hide DescriptionThis paper investigates the invariance properties of second‐order variational problems when the configuration space is subjected to an r‐parameter local Lie group of transformations. In particular, the recent results of Hanno Rund on first‐order problems are extended to the higher order case: A new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of the Zermelo conditions and Noether’s theorem are presented. The results are applied to a variational problem whose second‐order Lagrangian depends upon a scalar field in Minkowski space, and some conformal identities are obtained.

Simplified calculations for radiation reaction forces
View Description Hide DescriptionThe Lorentz–Dirac equation of motion for the electron is derived by a new method which makes tedious power series expansions unnecessary.

Analysis of the dispersion of low frequency uniaxial waves in heterogeneous periodic elastic media
View Description Hide DescriptionThe dispersion of harmonic uniaxial waves in heterogeneous periodic elastic media is investigated. The frequency dependence of the wave phase velocity is obtained in the form of a power series valid for small frequencies and arbitrary spatial variations of the heterogeneities. The dominant dispersion term is always negative and proportional to the square of the frequency. Near the static limit of zero frequency the dispersion is of the normal type—the group velocity, which is also a quadratic decreasing function of the frequency, decreases faster than the phase velocity.

The influence of linear damping on nonlinearly coupled positive and negative energy waves
View Description Hide DescriptionThe linearly damped response to the nonlinear resonant mixing of two monochromatic coherent waves, involving modes of different energy sign, is shown to be always explosively unstable. Degeneration theory, modified to encompass explosively unstable solutions, is then applied to distinguish regions of negligible and strong damping, where the equations can be solved analytically. Effective damping, characterized by a damping rate ν, much higher than the (normalized) initial excitation U _{0}, of the source waves, increases the explosion time by a factor of ν/U _{0}.

The Hamiltonian H = (−1/2) d ^{2}/d x ^{2} + x ^{2}/2 + λ/x ^{2} reobserved
View Description Hide DescriptionThe Schrödinger problem for the title Hamiltonian is considered as a perturbed one‐dimensional harmonic oscillator. Exact bound statesolutions can be derived from a classical differential equation in the theory of Laguerre polynomials. These solutions are valid and analytically dependent on λ only in a limited range of the perturbation strength. Within this region the oscillator Hamiltonian restricted to odd and even paritysubspaces is unitary equivalent to H restricted over the respective perturbed subspaces. It is shown that due to the singular nature of the perturbation the allowed λ range is narrowed if side conditions are imposed to make the wavefunctions ’’physically interpretable.’’

On convergent iterations and their applications to Hartreee–Fock equations
View Description Hide DescriptionA method for solving a restricted class of nonlinear equations is presented and applied in detail to solution of the Hartree–Fock (HF) equations.

Construction of a meromorphic many‐channel p‐wave S matrix
View Description Hide DescriptionThe simplest possible nonrelativistic many‐channel p‐wave S matrix which is meromorphic on its energy Riemann surface, as well as the underlying potential matrix, is explicitly constructed by means of the many‐channel Marchenko equations. The results suggest that, contrary to the case in which no coupling between channels is present, such an S matrix necessarily increases in complexity with increasing angular momentum.

On angular momentum and channel coupling for a meromorphic many‐channel S matrix
View Description Hide DescriptionWe consider a class of possible (i.e., not known a p r i o r i to be unitary) nonrelativistic many‐channel S matrices meromorphic on their energy Riemann surfaces whose general form is suggested by the inverse scattering problem. These possible S matrices are associated with n (spinless) coupled channels of the same angular momentem l, and they reduce, in the one‐channel limit, to the quotient of two polynomials of degree 2m in the wavenumber k. It is shown that the assumptions which imply unitarity for real energies of the open‐channel submetrix also imply that no channels can be coupled when m?l.

On the projective unitary representations of connected Lie groups
View Description Hide DescriptionWe show here the possibility of finding a unique Lie group associated with each connected Lie groupG such that every projective unitary representation of can be lifted to a unitary representation of that is to say, all PUR of G can be found from the UR of only one group This method is applied to the research of PUR of the Galilei group and compared with the preceding ones.

Neutron transport and diffusion in inhomogeneous media. I
View Description Hide DescriptionThe asymptotic solution of the neutron transport equation is obtained for large near‐critical domains D which possess a cellular, nearly periodic structure. A typical mean free path in D is taken to be of the same order of magnitude as a cell diameter, and these are taken to be small (of order ε) compared to a typical diameter of D. The solution is asymptotic with respect to the small parameter ε. It is a product of two functions, one determined by a detailed cell calculation and the other obtained as the solution of a time dependent diffusion equation. The diffusion equation contains precursor (delayed neutron) densities, equations for which are derived. The coefficients in the diffusion equation, which are determined using the results of the cell calculation, differ from those now used in engineering applications. The initial condition for the diffusion equation is derived, and the problem of determining the boundary condition is discussed.

Bäcklund transformation solutions of the Toda lattice equation
View Description Hide DescriptionBäcklund transformation, superposition formula, and multisoliton solutions are constructed for the Toda lattice difference–differential equation with some discussion of its generalizations.

Harmonic mappings of Riemannian manifolds and stationary vacuum space–times with whole cylinder symmetry
View Description Hide DescriptionWe consider stationary, cylindrically‐symmetric gravitational fields in the framework of harmonic mappings of Riemannian manifolds. In this approach the emphasis is on a correspondence between the solution of the Einstein field equations and the geodesics in an appropriate Riemannian configuration space. Using Hamilton–Jacobi techniques, we obtain the geodesics and construct the resulting space–time geometries. We find that the light cone structure of the configuration space delineates the distinct exterior fields of Lewis and van Stockum which together form the most general solution with whole cylinder symmetry.

Orthogonal polynomials. II
View Description Hide DescriptionA class of orthogonal polynomials defined by a weight function of compact support is considered. These are known to satisfy three‐term recursion relations. It is shown, under rather weak restrictions, that the traces of powers of the Jacobi matrices formed from the coefficients in the recursion relations are simply related to Fourier coefficients of the logarithm of the weight function.

Analytic T matrices for Coulomb plus rational separable potentials
View Description Hide DescriptionThe l=0 partial wave projected Coulomb off‐shell T matrix T _{ c,l=0} in momentum representation is obtained in closed form. Problems existing in the literature concerning the half‐ and on‐shell behavior of T _{ c } and T _{ c,l } are discussed and clarified by means of explicit formulas. The remaining derivations in this paper are based on T _{ c,l=0}. We consider the class of N‐term separable potentials where the form factors are rational functions of p ^{2} (in momentum representation). We prove that the l=0 T matrix corresponding to the Coulomb potential plus any such so‐called rational separable potential has a very simple form, namely, it can be written in terms of rational functions and the (simple) hypergeometric function with parameters (1, iγ; 1+iγ), where γ is the well‐known Coulomb parameter. Explicit analytic formulas are derived for a number of simple members of the class, the Yamaguchi potential being one of them. In this particular case the expressions of Zachary and of Bajzer are reproduced who used a method based on the O _{4} symmetry.

Is something missing in the Boltzmann entropy?
View Description Hide DescriptionA representation theorem for entropyfunctionals on the set of probability densities on the space R _{ n } is proved. The important feature of the theorem is that the representation contains, in addition to the Boltzmann term, the continuous analog of the Hartley entropy as well as a term that, in statistical mechanics, corresponds to the chemical potential and is usually introduced ad hoc into the expression for entropy.

Evaluation of a class of lattice sums in arbitrary dimensions
View Description Hide DescriptionStarting from the Poisson summation formula in m dimensions, a class of lattice sums is evaluated analytically. The resulting formulas are applicable to the electronic‐structure studies of crystalline solids, the analysis of stability of quantized vortex arrays in extreme type‐II superconductors and in rotating superfluidhelium, and the investigation of Bose–Einstein condensation in finite systems.

Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation
View Description Hide DescriptionIt was established by Levi‐Civita that in n dimensions there exist n+1 types of coordinate systems in which the Hamilton–Jacobi equation is separable, n of which are in general nonorthogonal; the form of the separated equations was given by Burgatti and Dall’Acqua. In this paper first the general forms of the n+1 types of metric tensors of the corresponding corresponding Riemannian spaces V _{ n } are determined. Then, sufficient conditions are given for coordinate systems in which the Schrödinger, Helmholtz, and Laplace equation are separable. It is shown that there again exist n+1 types of such systems, whose metric tensors are of the same form as those of the Hamilton–Jacobi equation. However, except for the ’’essentially geodesic case’’ of Levi‐Civita they are further restricted by a condition on the determinant of the metric; this condition is a generalization of that found by Robertson for orthogonal systems in the case of the Schrödinger equation.

Integrals of motion and resonances in a dipole magnetic field
View Description Hide DescriptionA method is developed for deriving a third integral of motion, besides the Hamiltonian and the angular momentum, of a charged particle, in a dipole magnetic field. This method is particularly useful in resonance cases, where the usual adiabatic invariants are not applicable. First the Hamiltonian is reduced to a ’’regular’’ form, i.e., its lowest order terms are written as H _{2}=Φ_{12} + Φ_{22}, where Φ_{12}= (1/2)(a ^{2}+ p ^{2} _{ a }), Φ_{22}= (1/2)(ω^{2} b ^{2}+ p ^{2} _{ b }). Then the third integral can be constructed step by step as a series; in every resonance case a different form of the integral is derived. In the nonresonant cases the Hamiltonian is written in a normal form H=H* (Φ_{1}, Φ_{2}), where Φ_{1}, Φ_{2} are canonical variables introduced by using von Zeipel’s method. The nonresonant orbits are quasiperiodic with frequencies ω_{1}=∂H*/∂Φ_{1}, ω_{2}=ω∂H*/∂Φ_{2} and rotation number Rot=ω_{1}/ω_{2}. As an example the location of a particular resonance is found. The comparison with the numerical integrations is satisfactory.