Canonical Systems Of Differential Equations Research Articles

EQUATIONS OF PLANETARY SYSTEMS MOTION

The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.

On the deformation of coated functionally graded structures with delamination

The paper considers problems for extensional inhomogeneous layered structures with functionally graded coatings in the presence of delamination. An approach that allows, from a single point of view, to study the problem of equilibrium of a strip non-uniform along the vertical coordinate with a delaminated coating and the problem of steady-state oscillations of a cylindrical waveguide with a ring crack is developed. Using the Fourier transform, the problems for the strip and waveguide are reduced to canonical systems of differential equations with variable coefficients with respect to transformants of the stress tensor components and displacement vector. Based on the formulation of several auxiliary Cauchy problems, a scheme for constructing a system of integral equations with hypersingular kernels with respect to jumps of displacements on the banks of delamination is presented. The asymptotic behavior of the Fourier symbols of kernels of integral operators for large values of the transformation parameter is investigated. A computational scheme for solving the system of integral equations based on the boundary element method is developed. The results of computational experiments on the determination of the disclosure functions of the delamination banks for a strip and a waveguide in the presence of a discontinuity in the elastic moduli at the boundary of the main body and the coating are presented.

Effect of material inhomogeneity on characteristics of a functionally graded hollow cylinder

In this paper, we present a statement of the axisymmetric problem on steady-state oscillations of an inhomogeneous elastic cylinder on the basis of the general relations of the linear theory of elasticity. The material parameters and density are considered to be variable along the radial coordinate. The oscillations are caused by a distributed load applied to the outer part of the cylinder boundary. We reduce the direct problem on determination of the radial and longitudinal displacement components to the numerical solution of a number of canonical systems of differential equations with variable coefficients. On the basis of the solution obtained, we carry out the computational experiments to analyze the influence of the variable Lamé parameters on the displacement field components, the stress tensor components, the amplitude-frequency characteristics and the values of resonant frequencies.

Analytical Solution for Optimal Low-Thrust Limited-Power Transfers Between Non-Coplanar Coaxial Orbits

In this paper, an analytical solution for time-fixed optimal low-thrust limited-power transfers (no rendezvous) between elliptic coaxial non-coplanar orbits in an inverse-square force field is presented. Two particular classes of maneuvers are related to such transfers: maneuvers with change in the inclination of the orbital plane and maneuvers with change in the longitude of the ascending node. The optimization problem is formulated as a Mayer problem of optimal control with the state defined by semi-major axis, eccentricity, inclination or longitude of the ascending node, according to the class of maneuver considered, and a variable measuring the fuel consumption. After applying Pontryagin’s maximum principle and determining the maximum Hamiltonian, short periodic terms are eliminated through an infinitesimal canonical transformation. The new maximum Hamiltonian resulting from this canonical transformation describes the extremal trajectories for long duration transfers. Closed-form analytical solution is then obtained through Hamilton-Jacobi theory. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition. Simplified solution is determined for transfers between close orbits. The analytical solution is compared to the numerical solution obtained by integration of the canonical system of differential equations describing the extremal trajectories for some sets of initial conditions. Results show a great agreement between these solutions for the class of maneuvers considered in the analysis. The solution of the two-point boundary value problem of going from an initial orbit to a final orbit, based on the analytical solution, is also discussed.

Two-loop master integrals for the leading QCD corrections to the Higgs coupling to a W pair and to the triple gauge couplings ZW W and \u03b3\u2217W W

We compute the two-loop master integrals required for the leading QCD corrections to the interaction vertex of a massive neutral boson X0, e.g. H, Z or γ∗, with a pair of W bosons, mediated by a SU(2)L quark doublet composed of one massive and one massless flavor. All the external legs are allowed to have arbitrary invariant masses. The Magnus exponential is employed to identify a set of master integrals that, around d = 4 space-time dimensions, obey a canonical system of differential equations. The canonical master integrals are given as a Taylor series in ϵ = (4 − d)/2, up to order four, with coefficients written as combination of Goncharov polylogarithms, respectively up to weight four. In the context of the Standard Model, our results are relevant for the mixed EW-QCD corrections to the Higgs decay to a W pair, as well as to the production channels obtained by crossing, and to the triple gauge boson vertices ZWW and γ∗WW.

Two-loop master integrals for the mixed EW-QCD virtual corrections to Drell-Yan scattering

We present the calculation of the master integrals needed for the two-loop QCDxEW corrections to $ q + \bar{q} \to l^- + l^+$ and $ q + \bar{q}' \to l^- + \overline{\nu} \, , $ for massless external particles. We treat W and Z bosons as degenerate in mass. We identify three types of diagrams, according to the presence of massive internal lines: the no-mass type, the one-mass type, and the two-mass type, where all massive propagators, when occurring, contain the same mass value. We find a basis of 49 master integrals and evaluate them with the method of the differential equations. The Magnus exponential is employed to choose a set of master integrals that obeys a canonical system of differential equations. Boundary conditions are found either by matching the solutions onto simpler integrals in special kinematic configurations, or by requiring the regularity of the solution at pseudo-thresholds. The canonical master integrals are finally given as Taylor series around d=4 space-time dimensions, up to order four, with coefficients given in terms of iterated integrals, respectively up to weight four.

Localization of Zeros for Cauchy Transforms

We study the localization of zeros of Cauchy transforms of discrete measures on the real line. This question is motivated by the theory of canonical systems of differential equations. In particular, we prove that the spaces of Cauchy transforms having the localization property are in one-to-one correspondence with the canonical systems of special type, namely, those whose Hamiltonians consist only of indivisible intervals accumulating on the left. Various aspects of the localization phenomena are studied in details. Connections with the density of polynomials and other topics in analysis are discussed.

Limit properties of monotone matrix functions

The basic objects in this paper are monotonically nondecreasing n×n matrix functions D(·) defined on some open interval ı=(a,b) of R and their limit values D(a) and D(b) at the endpoints a and b which are, in general, selfadjoint relations in Cn. Certain space decompositions induced by the matrix function D(·) are made explicit by means of the limit values D(a) and D(b). They are a consequence of operator inequalities involving these limit values and the notion of strictness (or definiteness) of monotonically nondecreasing matrix functions. This treatment provides a geometric approach to the square-integrability of solutions of definite canonical systems of differential equations.

Square‐integrable solutions and Weyl functions for singular canonical systems

AbstractBoundary value problems for singular canonical systems of differential equations of the form are studied in the associated Hilbert space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2_\Delta (\imath )$\end{document}. With the help of a monotonicity principle for matrix functions their square‐integrable solutions are specified. This yields a direct treatment of defect numbers of the minimal relation and simultaneously makes it possible to assign certain boundary values to the elements of the maximal relation induced by the system of differential equations in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2_\Delta (\imath )$\end{document}. The investigation of boundary value problems for these systems and their spectral theory can be carried out by means of abstract boundary triplet techniques. This paper makes explicit the construction and the properties of boundary triplets and Weyl functions for singular canonical systems. Furthermore, the Weyl functions are shown to have a property similar to that of the classical Titchmarsh‐Weyl coefficients for singular Sturm‐Liouville operators: they single out the square‐integrable solutions of the homogeneous systems of canonical differential equations. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Comparison theorems for symplectic systems of difference equations

In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of conjoined bases of two symplectic systems with matrices W i and \( \hat W_i \) as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number of eigenvalues λ of a symplectic boundary value problem on the interval (λ 1, λ 2]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes the well-known reciprocity principle for discrete Hamiltonian systems.

Dynamics aspects in a galactic type Hamiltonian

We are interested in the investigation of the dynamics of the canonical system of differential equations deriving from the galactic type Hamiltonian given by the quartic function H = 1 2 ( y 1 2 + ω 1 2 x 1 2 ) + δ 2 ( y 2 2 + ω 2 2 x 2 2 ) − ϵ x 1 2 x 2 2 from an analytical point of view according to the real parameters δ and ϵ . Similar Hamiltonians have been considered by several investigators in order to study the motion in the central parts of elliptical galaxies.

Comparative index for solutions of symplectic difference systems

We introduce the comparative index of two conjoined bases of a symplectic difference system, which generalizes difference analogs of canonical systems of differential equations. We consider the main properties of the comparative index and its relation to the number of focal points of a conjoined basis of the symplectic system. We prove a formula relating the number of focal points (including multiplicities) of two bases in the interval (i, i + 1] and corollaries of this formula such as an estimate for the difference of the numbers of focal points of two conjoined bases in the interval (0, N + 1], the equality of the numbers of focal points of principal solutions for the primal and reciprocal systems, sufficient conditions for the solvability of the Riccati equation for a disconjugate symplectic system, etc.

Optimization of Low-Thrust Limited-Power Trajectories in a Noncentral Gravity Field—Transfers between Orbits with Small Eccentricities

Numerical and first-order analytical results are presented for optimal low-thrust limited-power trajectories in a gravity field that includes the second zonal harmonicJ2in the gravitational potential. Only transfers between orbits with small eccentricities are considered. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin Maximum Principle, successive canonical transformations are performed and a suitable set of orbital elements is introduced. Hori method—a perturbation technique based on Lie series—is applied in solving the canonical system of differential equations that governs the optimal trajectories. First-order analytical solutions are presented for transfers between close orbits, and a numerical solution is obtained for transfers between arbitrary orbits by solving the two-point boundary value problem described by averaged maximum Hamiltonian, expressed in nonsingular elements, through a shooting method. A comparison between analytical and numerical results is presented for some maneuvers.

A note on the theorems of M.G. Krein and L.A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge in the upper half-plane in the case of L 2 coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L p for any p > 2 , the spectral measure is absolutely continuous and the Szegö–Kolmogorov–Krein condition is satisfied. Thus, we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.

Canonical systems of differential equations with self-adjoint interface conditions on graphs

For n canonical systems of differential equations, the corresponding n copies of their domain (0, ∞) are thought of as a graph with vertex 0. An interface condition at 0 is given by a so-called Nevanlinna pair. Explicit formulae are deduced for the spectral representation of the corresponding underlying self-adjoint relation and the generalized Fourier transformation. Furthermore, results on compressions of the Fourier transformation to closed linear subspaces and the multiplicity of the eigenvalues if the spectrum is discrete are presented

Two-Dimensional Trace-Normed Canonical Systems of Differential Equations and Selfadjoint Interface Conditions

The class of two-dimensional trace-normed canonical systems of differential equations on $$\mathbb{R}$$ is considered with selfadjoint interface conditions at 0. If one or both of the intervals around 0 are H-indivisible the interface conditions which give rise to selfadjoint relations (multi-valued operators) are determined. It is shown that the corresponding generalized Fourier transforms are partially isometric. Compression to the halfline (0, ∞) results in boundary conditions which depend on the eigenvalue parameter involving a fractional linear transform of the Titchmarsh-Weyl coefficient of the halfline (−∞, 0). The corresponding generalized Fourier transforms are isometric except possibly on a one-dimensional subspace where they are contractive.

A new approach to the theory of the stability of linear canonical systems of differential equations with periodic coefficients

The main element of the proposed approach to constructing a theory of the stability of canonical systems is an index function (defined below), which contains all the necessary information on the system. The fundamental results of existing theory, in particular, the necessary and sufficient condition for strong stability, are expressed in new terms. The corresponding proofs only use simple mathematical means; moreover, they are much shorter than existing proofs. A number of new assertions are established, in particular, a simple sufficient condition for strong stability is obtained, which essentially generalizes the well-known Yakubovich theorem [1] of the directed width of the stability regions, and the necessary and sufficient condition for their directed convexity is obtained. Using them, certain non-local qualitative results on the regions of stability of parametric oscillations of canonical systems are established (which enable, in particular, the existing practice of constructing stability regions in accordance with their boundaries to be justified), and the conditions for high-frequency parametric stabilization of unstable systems are obtained.

A Characterization of Intermediate Weyl Coefficients

In connection with an indefinite analogue of canonical systems of differential equations some subclasses N0/Δ (Δ ∈ N0) of the Nevanlinna class N0 come up. We give a criterion for a function to belong to N0/Δ in terms of its poles and residues.

Subordinate solutions and spectral measures of canonical systems

The theory of 2×2 trace-normed canonical systems of differential equations on ℝ+ can be put in the framework of abstract extension theory, cf. [9]. This includes the theory of strings as developed by I.S. Kac and M.G. Kreįn. In the present paper the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrodinger operators on the halfline ℝ+, was originally developed D.J. Gilbert and D.B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions of the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.

J-Inner matrix functions, interpolation and inverse problems for canonical systems, II: The inverse monodromy problem

This is the second of a planned sequence of papers on inverse problems for canonical systems of differential equations. It is devoted to the inverse monodromy problem for canonical integral and differential systems. In this part, which focuses on the case of a diagonal signature matrixJ, a parametrization is obtained of the set of all solutionsM (t) for the inverse problem for integral systems in terms of two chains of entire matrix valued inner functions. Special classes of solutions correspond to special choices of these chains. This theme will be elaborated upon further in a third part of this paper which will be published in a subsequent issue of this journal. There the emphasis will be on symmetries and growth conditions all of which serve to specify or restrict the chains alluded to above, from the outside, so to speak.