Liouville Transformation Research Articles

The solar proton event of April 16, 1970, 2. Transformation of proton pitch angle distributions from the solar wind into the magnetosheath

During the event of April 16, 1970, energetic (∼0.7 MeV) protons were observed by Vela 6B in the solar wind and by Vela 5B in the magnetosheath. During one 4-hour period the omnidirectional flux measured by 5B was a factor of 3 larger than that measured by 6B. The pitch angle distribution observed at this time by 6B was characterized by a strong peak at 0° pitch angle. Transformation of this distribution according to the Liouville equation accounts for the enhanced omnidirectional flux and the shape of the pitch angle distributions observed at 5B. At a later time in the event, distorted pitch angle distributions, which were observed in the magnetosheath, are accounted for by a Liouville transformation of an assumed unidirectional distribution in interplanetary space. We conclude that particle entry into the magnetosheath was approximately adiabatic.

Second-order linear differential equations with two turning points

Differential equations of the form d 2 w / d x 2 = { u 2 f ( u , a , x ) + g ( u , a , x ) } w are considered for large values of the real parameter u . Here x is a real variable ranging over an open, possibly infinite, interval ( x 1 , x 2 ), and a is a bounded real parameter. It is assumed that f {u, a, x) and g{u,a, x) are free from singularity within ( x 1 , x 2 ), and f (u, a, x) has exactly two zeros, which depend continuously on a and coincide for a certain value of a . Except in the neighbourhoods of the zeros, g(u,a,x) is small in absolute value compared with u 2 f(u, a, x ). By application of the Liouville transformation, the differential equation is converted into one of four standard forms, with continuous coefficients. Asymptotic approximations for the solutions are then constructed in terms of parabolic cylinder functions. These approximations are valid for large u , uniformly with respect to x ε ( x 1 , x 2 ) and also uniformly with respect to a . Each approximation is accompanied by a strict and realistic error bound. The paper also includes some new properties of parabolic cylinder functions.

Asymptotic Structure in Torsional Free Oscillations of the Earth-I. Overtone Structure

Summary By means of a Liouville transformation, the torsional eigenfrequencies nσl of a spherically symmetric, non-rotating, elastic and isotropic (SNREI) Earth can be defined as the eigenvalues of a second order differential equation with no first derivative term. From this equation, an asymptotic formula for nσl for fixed l can be derived in terms of the radial travel time for shear waves as well as other structural parameters for the mantle. This formula has a number of geophysical applications which yield new constraints on the rigidity structure of the mantle. In particular, the asymptotic structure of torsional free oscillation data contains information about the size, sharpness and position of discontinuities in the upper mantle.

Asymptotic behavior of solutions of differential-difference equations.

Abstract : In this paper, the problem was considered of determining the asymptotic behavior of solutions of linear differentialdifference equations whose coefficients possess asymptotic series. Although the problem is considerably more complicated than the corresponding problem for ordinary differential equations, by means of a sequence of transformations the problem was reduced to a form where the standard techniques of ordinary differential equation theory could be employed. The differential-difference equation was transformed into an integral equation which was trans formed into an integro- differential equation. At this point the Liouville transformation plays a vital role. Although the guiding ideas were simple, the analysis became formidable. For this reason, only some of the more immediate aspects of the problem were considered.