Liouville Systems Research Articles

New coupled Liouville system: Prolongation structure, soliton solution, and complete integrability

We suggest a new coupled Liouville equation which is exactly solvable. We obtain the Lax pair through a prolongation analysis and also obtain the exact one-soliton-like solution by a direct procedure. We confirm our result through a Painleve analysis of the similarity reduced systems.

On the motion of chaplygin's sledge

The plane motion of Chaplygin's sledge is studied. In his original studies of this non-holonomic system, Chaplygin /1/ assume that the support plane is horizontal, and used a reduction factor to reduce the problem to the study of a Hamiltonian system with two degrees of freedom and one cyclical coordinate (i.e., a completely Liouville integrable system). A smooth reversible replacement of the phase variables is used below for the reduction. The motion is studied in detail by the methods of Hamiltonian mechanics, and the motion on an inclined plane is studied by the averaging method. The problem was earlier studied in /2–4/ for certain constraints on the position of the sledge centre of gravity. Chaplygin's equations of motion on an inclined plane were integrated in /3/ on the assumption that the centre of gravity lies on a line through the blade and perpendicular to the blade.

Solution of generalized non-singular Sturm–Liouville systems via an approximation method in a Hilbert space

The approximation method in a Hilbert space is applied to the analysis of a generalized non-singular Sturm-Liouville eigenvalue problem. The variable coefficients in the system equation and the solution are approximated by a suitable linear independent basis, the expansion coefficients are explicitly expressed as a backward recursive formula and the eigenvalues are obtained by solving an algebraical characteristic equation. The computation is simple and straightforward.

Symmetries of scaler fields. III. Two-dimensional integrable models

The symmetry of the relativistically invariant two-dimensional system /phi//sub 12/ = f(/phi/) is investigated. Necessary and sufficient conditions for the system to belong to the Liouville type are obtained. Simple examples of Liouville systems are given. It is shown that to establish all integrable systems it is sufficient to assume that the elements of the Lie-Backlund algebra are polynomials in /phi/ /sub i/. Determining equations for the recursion operator of the system are obtained, and two examples of recursion operators are given. Some general arguments about methods of calculating conserved densities are presented.

On the product solutions of Storm—Liouville equations with many parameters in a space of the type L2

In this paper the Storm—Liouville system of differential equations of many parameters is considered. The existence of the solutions which belong to a space of the type L 2 under some conditions is proved. Moreover, an estimation is given from below and from above for the number of linearly independent solutions of the system in some singular end.

The Stieltjes Summability Method and Summing Sturm–Liouville Expansions

The Stieltjes summability method for divergent integrals is defined. The name is derived from its close relationship with the Stieltjes transform. After proving some basic properties, the Stieltjes summability method is compared with Cesàro and Abel summability methods. The main result proves the Stieltjes summability of eigenfunction expansions associated with a certain class of singular Sturm–Liouville systems, and develops, using the Stieltjes means, a stable method of summing such expansions under perturbations of the coefficients.

Branching from Large Eigenvalues of Nonlinear Sturm–Liouville Systems

The behavior of solutions branching from large eigenvalues of certain nonlinear Sturm–Liouville systems is studied. A perturbation method is presented which permits the solution branches to be extended beyond the region of validity of standard perturbation and iteration techniques. Numerical calculations support the asymptotic results.

Domain of feasible motions in certain mechanical systems

Domains of feasible motions — projections of integral manifolds onto a configuration space — are studied. It is shown that in systems with symmetry and in Liouville systems the reorganization of such domains is possible only simultaneously with the reorganization of the integral manifolds. An example is presented of a system with gyroscopic forces, in which the domains of feasible motions change topological type outside a bifurcation set.

Basic Fourier series

The concept of basic number is applied to the development of a simple analogue of the Sturm–Liouville system of the second order. This is then employed to deduce a family ofq-orthogonal functions, which leads to a generalization of the Fourier and Fourier–Bessel expansions. The numerical approximation of basic integrals is discussed and some aspects of the evaluation ofCa(q; x)are mentioned. A few of the zeros of this function are listed, and, in conclusion, an indication is given of the possibility of applying the analysis presented in this paper to thé study of stochastic processes and time-series.

On the Distribution of the Eigenvalues of a Two-Parameter System of Ordinary Differential Equations of the Second Order

In this paper two simultaneous Sturm–Liouville systems are considered, the first defined for the interval $0 \leqq x_1 \leqq 1$, the second for the interval $0 \leqq x_2 \leqq 1$, and each containing the parameters $\lambda $ and $\mu $. Denoting the eigenvalues and eigenfunctions of the simultaneous systems by $(\lambda _{j,k} ,\mu _{j,k} )$ and $\psi _{j,k} (x_1 ,x_2 )$, respectively, $j,k = 0,1, \cdots $, the principle of the argument and asymptotic methods are employed to derive asymptotic formulas for these expressions, as $j^2 + k^2 \to \infty $, when $(j,k)$ is restricted to lie in each of several sectors of the $(x,y)$-plane. These results partially resolve a problem posed by Atkinson concerning the behavior of the eigenvalues and eigenfunctions of multiparameter Sturm–Liouville systems and constitute a further stage in the development of the theory related to these questions.

Estimating the Eigenvalues of Sturm–Liouville Problems by Approximating the Differential Equation

This paper is concerned with computing accurate approximations to the eigenvalues and eigenfunctions of regular Sturm–Liouville differential equations. The method consists of replacing the coefficient functions of the given problem by piecewise polynomial functions and then solving the resulting simplified problem. Error estimates in terms of the approximate solutions are established and numerical results are displayed. Since the asymptotic properties for Sturm–Liouville systems are preserved by the approximation, the relative error in the higher eigenvalues is much more uniform than is the case for finite difference or Rayleigh–Ritz methods.

On the problem op n-stationary centers

Among the various problems of celestial mechanics related to the n-body problem, a certain amount of interest attaches to the specific situation wherein a passive gravitational point mass M moves under the assumption that the relative disposition of the other active gravitational masses experiences no large changes. If the attracting masses are regarded as points and if changes in the relative disposition of the attracting bodies are neglected, one arrives at the problem of the motion of the point M in a field produced by n-stationary attracting centers (the point M here represents the ( n+ l)-th body). In addition to the problem of central motion ( n = 1), soluble dynamics problems of this category include Euler's case [1] of two ( n= 2) stationary Newtonian attracting centers. This problem, which for a long time was of solely theoretical Interest as an example of an integrable Liouville system [2], has recently been attracting attention in connection with the mechanics of artificial satellites, particularly after it was shown that the potential of an oblate spheroid can be approximated by the potential of two specifically chosen stationary Newtonian attracting centers [3 and 4]. The solution of the problem for n-attracting centers for n ≥ 3 is unknown, except for a single special case of three centers pointed out by Lagrange and considered In greater detail by J.A. Serre [5]. We shall concern ourselves here with problems on the existence of periodic trajectories in the case of n-attracting centers. An arbitrary and not necessarily Newtonian gravitational law will be assumed. Our analysis is based on the theory of quasiindices of singular force field points as set forth in [60].

Ordinary Differential Equations.

First--Order of Differential Equations. Second--Order Linear Equations. Linear Equations with Constant Coefficients. Power Series Solutions. Plane Autonomous Systems. Existence and Uniqueness Theorems. Approximate Solutions. Efficient Numerical Integration. Regular Singular Points. Sturm--Liouville Systems. Expansions in Eigenfunctions. Appendices. Bibliography. Index.