Lagrange Manifold Research Articles

PARALLELISM OF DISTRIBUTIONS AND GEODESICS ON F(±a2; ±b2)-STRUCTURE LAGRANGIAN MANIFOLD

This paper deals with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying (Fv2-a2)(Fv2+a2)(Fv2 - b2)(Fv2 + b2) = 0. In this paper, the authors have proved that if an almost product structure P on the tangent space of a 2n-dimensional Lagrange manifold E is defined and the F(±a2; ±b2)-structure on the vertical tangent space TV (E) is given, then it is possible to define the similar structure on the horizontal subspace TH(E) and also on T(E). In the next section, we have proved some theorems and have obtained conditions under which the distribution L and M are r-parallel, r¯ anti half parallel when r = r¯ . The last section is devoted to proving theorems on geodesics on the Lagrange manifold

Universal lowest energy configurations in a classical Heisenberg model describing frustrated systems with wheel geometry

We minimize the energy function of the classical Heisenberg model describing the frustrated wheel shape systems consisting of an odd number of spin vectors, where a single vector is located at the center and the remaining vectors occupy $n$ sites on a ring. Using the Lagrange manifold method developed recently, we find the exact geometrical configurations of the spin vectors corresponding to the global energy minima. We reveal two subsets of collinear $n/2$ spin vectors which are tilted by an angle $\ensuremath{\psi}$ in the opposite direction with respect to that fixed by the central vector. We prove that $\ensuremath{\psi}$ does not depend on $n$ and all the spin vectors are collinear, if the system is nonfrustrated or it is subject to weak frustration, otherwise the sublattices start to rotate. In this frustration region, the configurations are double degenerate and differ by chirality. Our findings confirm that the classification of spin frustration holds in the classical limit and allows us to discriminate different regions by the proper configurations. We demonstrate the correspondence between the total spin in the ground state of the quantum model and a component of the net total spin vector which can be exploited in analysis of the quantum models and physical complexes.

Ergodic Deformations of Nonlinear Hamilton Systems and Local Homeomorphism of Metric Spaces

The orbits of slowly perturbed Hamilton systems and the associated ergodic deformations of Lagrange manifolds are studied. The main results are based on the Mather approach [18, 19] to the construction of the homologies of invariant probabilistic measures, which minimize some Lagrange functionals, and on the elliptic Gromov–Salamon–Zehnder–Floer theory [7, 9, 12, 20, 26] of the construction of invariant manifolds. We have constructed the invariant submanifolds, which are the supports of invariant ergodic measures and have a structure of locally homeomorphic metric spaces. We analyze the problem of construction of efficient criteria of their global homeomorphism, which was posed by Professor A. M. Samoilenko during the study of ergodic deformations of nonlinear Hamilton systems and their adiabatic invariants. It is established that the mapping f : X → Y from a linearly connected Hausdorff space X onto a simply connected (in particular, contractible) space Y is a homeomorphism iff f is local and homeomorphic, and the preimage f−1(y) of every point y ∈ Y is a nonempty compact subset in X.

Prediction and characterization of multiple extremal paths in continuously monitored qubits

We examine most-likely paths between initial and final states for diffusive quantum trajectories in continuously monitored pure-state qubits, obtained as extrema of a stochastic path integral. We demonstrate the possibility of "multipaths" in the dynamics of continuously-monitored qubit systems, wherein multiple most-likely paths travel between the same pre- and post-selected states over the same time interval. Most-likely paths are expressed as solutions to a Hamiltonian dynamical system. The onset of multipaths may be determined by analyzing the evolution of a Lagrange manifold in this phase space, and is mathematically analogous to the formation of caustics in ray optics or semiclassical physics. Additionally, we develop methods for finding optimal traversal times between states, or optimal final states given an initial state and evolution time; both give insight into the measurement dynamics of continuously-monitored quantum states. We apply our methods in two systems: a qubit with two non-commuting observables measured simultaneously, and a qubit measured in one observable while subject to Rabi drive. In the two-observable case we find multipaths due to caustics, bounded by a diverging Van-Vleck determinant, and their onset time. We also find multipaths generated by paths with different "winding numbers" around the Bloch sphere in both systems.

Kelley condition and structure of Lagrange manifold in a neighborhood of a first-order singular extremal

The paper considers optimal control problems linearly depending on the scalar control parameter in which there exist first-order singular extremals. The author proves a theorem on the structure of a generic Lagrange manifold (field of extremals) in a neighborhood of first-order singular extremals. As a consequence of this theorem, the author proves the optimality of singular extremals and nonsingular extremals in problems with fixed endpoints on small intervals of time. As an illustration, the paper presents constructions of Lagrange manifolds for the general linear-quadratic control problem with completely integrable linear system of differential constraints and for a certain problem of mathematical economics, a two-factor economic growth model with production function of the Cobb-Douglas type.

Metallic structure on Lagrangian manifold

In this paper, the author convention with the Lagrange vertical structure on the vertical space TV (E) endowed with a non null (1,1) tensor field FV satisfying metallic structure F2 − αF − βI = 0. The horizontal subspace TH(E) is applied on the same structure. Next, some theorems are proved and obtained conditions under which the distribution L and M are ∇-parallel, ¯∇ anti half parallel when ∇ = ¯∇. Lastly, certain theorems on geodesics on the Lagrange manifold are deduced.

A Phase-Space Beam Summation Formulation for Ultrawide-Band Radiation

A new discrete phase space Gaussian beam (GB) summation representation for ultrawide-band (UWB) radiation from an aperture source distribution is presented. The formulation is based on the theory of the windowed Fourier transform (WFT) frames, wherein we introduce a novel relation between the frequency and the frame overcompleteness. With this procedure, the discrete lattice of beams that are emitted by the aperture satisfies the main requirement of being frequency independent, so that only a single set of beams needs to be traced through the medium for all the frequencies in the band. It is also shown that a properly tuned class of iso-diffracting (ID) Gaussian-windows provides the "snuggest" frame representation for all frequencies, thus generating stable and localized expansion coefficients. Furthermore, due to the ID property, the resulting GBs propagators are fully described by frequency independent matrices whose calculation in the ambient environment need to be done only once for all frequencies. Consequently, the theory may also be expressed directly in the time-domain as will be presented elsewhere. The localization implied by the new formulation is demonstrated numerically for an UWB focused aperture. It is shown that the algorithm extracts the local radiation properties of the aperture source and enhances only those beams that conform with these properties, i.e., those residing near the phase space Lagrange manifold. Further localization is due to the fact the algorithm accounts only for beams that pass within a few beamwidths vicinity of the observation point. It is thus shown that the total number of beams is much smaller than the Landau Pollak bound on the aperture's degrees of freedom.

On the time‐dependent parabolic wave equation

One approach to the study of wave propagation in a restricted domain is to approximate the reduced Helmholtz equation by a parabolic wave equation. Here we consider wave propagation in a restricted domain modelled by a parabolic wave equation whose properties vary both in space and in time. We develop a Wentzel‐Kramers‐Brillouin (WKB) formalism to obtain the asymptotic solution in noncaustic regions and modify the Lagrange manifold formalism to obtain the asymptotic solution near caustics. Associated wave phenomena are also considered.

Caustic consideration of long planetary wave packet analysis in the continuously stratified ocean

The wave packet method, one form of the WKB technique, recently has been employed to investigate the evolution of long planetary wave packets in relation to the complex climate variability in the world oceans. However, such a method becomes invalid near the caustics. Here, the Lagrange manifold formalism is used to extend this analysis to include the caustic regions. We conclude that even though the wave packet method fails near the caustics, the equations derived from this method away from caustics are identical to ones from the Lagrange manifold formalism near caustics

On caustics associated with the linearized vorticity equation

The linearized vorticity equation serves to model a number of wave phenomena in geophysical fluid dynamics. One technique that has been applied to this equation is the geometrical optics, or multi-dimensional WKB technique. Near caustics, this technique does not apply. A related technique that does apply near caustics is the Lagrange Manifold Formalism. Here we apply the Lagrange Manifold Formalism to determine an asymptotic solution of the linearized vorticity equation and to study associated wave phenomena on the caustic curve.

The time-dependent Schrödinger equation at turning points

The Lagrange Manifold (WKB) formalism enables the determination of the asymptotic series solution of linear hyperbolic and parabolic differential equations at turning points. Here this formalism is applied to determine the asymptotic solution of the time-dependent Schrödinger equation at turning points. We also use this approach to study some physical phenomenology associated with the time-dependent Schrödinger equation at caustics, the locus of turning points.

Phase-space localization and discrete representations of wave fields

The high-frequency behavior of wave fields in free space is characterized by localization in phase space, exemplified by the description of geometrical optics in which distinct wave vectors are associated with the rays passing through each spatial point. I explore the manifestation of phase-space localization in discrete representations of the wave field, in particular the discrete Fourier transform (DFT) and the Gabor representation. A number of auxiliary concepts, such as spectral truncation, the Lagrange manifold, and the Landau–Pollak (LP) theorem, are described and exploited in the process of understanding the behavior of high-frequency fields, and it is shown that the Lagrange manifold is the source in phase space of the dominant contributions to these discrete representations in a number of specially selected examples. The LP theorem specifies the number of discrete degrees of freedom required for a given field to be approximated to a prescribed accuracy, and the theorem controls both the number of samples for DFT implementation and the number of Gabor coefficients required. The behavior of the Gabor coefficients away from the Lagrange manifold is studied for a Fresnel wave (quadratic phase variation on a line), and it is shown that these coefficients decay exponentially, signifying the localization of the Gabor representation. The number of operations required for computing the Gabor representation is compared with the fast-Fourier-transform (FFT) implementation of the DFT, with the LP dimension used as the common cardinality of the two representations, the result being that the FFT is asymptotically more efficient than the Gabor representation even after one allows for the localization of the latter. However, the Gabor representation can be used in circumstances in which the FFT is inapplicable, when the explicit localization is a significant advantage.

On caustics associated with horizontal rays and vertical modes

Wave propagation in a medium with slowly varying inhomogeneities can sometimes be modelled by a combination of normal mode theory in one direction and ray theoretical methods in the other directions. Caustics can occur in the ray theory ansatz. Here we consider those caustics using a Lagrange manifold approach.

On caustics associated with hyperbolic systems

The Lagrange manifold formalism is adapted to find asymptotic solutions for a class of hyperbolic systems near caustics.

Caustics and the parabolic wave equation

Caustics associated with the parabolic approximation to the reduced Helmholtz equation are considered using the modified Lagrange manifold formalism.

Transport Equations for Some Interesting Media

High frequency wave propagation in inhomogeneous media is often treated using the geometrical optics technique. Near caustics, this approach requires modification. One such modification is the Lagrange Manifold formalism. Transport equations associated with the Lagrange Manifold formalism valid near caustics in several media of geophysical interest are developed. Some energy and polarization considerations at caustics are also discussed.

Time‐evolution of a caustic

The Lagrange manifold formalism is adapted to study the time‐evolution of caustics associated with high frequency wave propagation in media with both spatial and temporal inhomogenities.

Caustics associated with the asymptotic solution of the diffusion equation

The Lagrange manifold technique for finding asymptotic solutions of wave equations at turning points is adapted to diffusion-type equations.

On caustics related to several common indices of refraction

In the Lagrange manifold approach to the asymptotic solution of a differential equation, the caustic curve is determined from the phase of an oscillatory integral. Here we present the phase functions of such integrals for several types of stratified media of geophysical interest.

Space‐time caustics

The Lagrange manifold (WKB) formalism enables the determination of the asymptotic series solution of linear differential equations modelling wave propagation in spatially inhomogeneous media at caustic (turning) points. Here the formalism is adapted to determine a class of asymptotic solutions at caustic points for those equations modelling wave propagation in media with both spatial and temporal inhomogeneities. The analogous Schrodinger equation is also considered.