Totally Geodesic Invariant Subm anifolds of Contact Metric Manifold

The purpose of this paper is to give a sufficient and necessary condition of totally geodesic on invariant submanifold of contact metric manifold and is to generalize the results in [3] and [4].

INVARIANT FORM AND INTEGRAL INVARIANTS ON KAHLER MANIFOLD

The important notions and results of the integral invariants of Poincard and Cartan-Poincard and the relationship between integral invariant and invariant form established first by E. Cartan in the classical mechanics are generalized to Hamilton mechanics on Kiihler manifold, by the theory of modern geometry and advanced calculus, to get the corresponding wider arid deeper results. differential

THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRDINGER EQUATION

In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.

Lunar landing trajectory design based on invariant manifold

The low-energy lunar landing trajectory design using the invariant manifolds of restricted three-body problem is studied. Considering angle between the ecliptic plane and lunar orbit plane, the four-body problem of sun-earth-moon-spacecraft is divided into two three-body problems, the sun-earth-spacecraft in the ecliptic plane and the earth-moon-spacecraft in the lunar orbit plane. Using the orbit maneuver at the place where the two planes and the invariant manifolds intersect, a general method to design low energy lunar landing trajectory is given. It is found that this method can save the energy about 20% compared to the traditional Hohmann transfer trajectory. The mechanism that the method can save energy is investigated in the point of view of energy and the expression of the amount of energy saved is given. In addition, some rules of selecting parameters with respect to orbit design are provided. The method of energy analysis in the paper can be extended to energy analysis in deep space orbit design.

INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.

INVARIANT MANIFOLDS AND THEIR STABILITY IN A THREE－DIMENSIONAL MEASURE－PRESERVING MAPPING SYSTEMS

In the paper researches on a three-dimensional measure-preserving mappingsystem are made,which is the three-dimensional extension of the Keplerian mapping.With formal series method the expressions of the invariant curves and invarianttori are obtained,Finally the stability of these in variant manifolds is also discussed.

INVARIANT SUB-MANIFOLDS AND MODES OF NONLINEAR AUTONOMOUS SYSTEMS

A definition of the modes of a nonlinear autonomous system was developed. The existence conditions and orbits' nature of modes are given by using the geometry theory of invariant manifolds that include stable manifold theorem, center maifold theorm and sub-center manifold theorem. The Taylor series expansion was used in order to approach the sub-manifolds of the modes and obtain the motions of the mods on the manifolds. Two examples were given to demonstrate the applications.

THE INVARIANT MANIFOLD METHOD AND THE CONTROLLABILITY OF NONLINEAR CONTROL SYSTEM

The problem of controllability of nonlinear control system is a significant field which has an extensive prospect of application. A. M. Kovalev of Ukraine Academy of Science applied the oriented manifold method developed in dynamics:of rigid body to nonlinear control system for the first time,and obtained a series of efficient results. Based on Kovalev's oriented manifold method, firstly, by invariant manifold method the problem of controllability of nonlinear control system was studied and the necessary condition of the controllability of a kind of affine nonlinear system was given out. Then the realization of the necessary condition wars discussed. At last, the motion of a rigid body with two rotors,vas, investigated and the necessary condition which is satisfied by this system,vas proved.

Invariant manifold growth formula in cylindrical coordinates and its application for magnetically confined fusion

For three-dimensional vector fields,the governing formula of invariant manifolds grown from a hyperbolic cycle is given in cylindrical coordinates.The initial growth directions depend on the Jacobians of Poincarémap on that cycle,for which an evolution formula is deduced to reveal the relationship among Jacobians of different Poincarésections.The evolution formula also applies to cycles in arbitrary finite n-dimensional autonomous continuous-time dynamical systems.Non-Möbiusian/Möbiusian saddle cycles and a dummy X-cycle are constructed analytically as demonstration.A real-world numeric example of analyzing a magnetic field timeslice on EAST is presented.

Even and Odd Coherent States for Time-Dependent Harmonic Oscillator

The dynamical invariant for a general time-dependent harmonic oscillator is constructed by making use of two linearly independent solutions to the classical equation of motion. In terms of this dynamical invariant we define the time-dependent creation and annihilation operators and relevantly introduce even and odd coherent states for time dependent harmonic oscillator. The mathematical and quantum statistical properties of these states are discussed in detail. The harmonic oscillator with periodically varying frequency is treated as a demonstration of our general approach.

Influence of Auxiliary Equation on Wave Functions for Time-Dependent Pauli Equation in Presence of Aharonov-Bohm Effect

Invariant operator method for discrete or continuous spectrum eigenvalue and unitary transformation approach are employed to study the two-dimensional time-dependent Pauli equation in presence of the Aharonov-Bohm effect （AB） and external scalar potential. For the spin particles the problem with the magnetic field is that it introduces a singularity into wave equation at the origin. A physical motivation is to replace the zero radius flux tube by one of radius R, with the additional condition that the magnetic field be confined to the surface of the tube, and then taking the limit R → 0 at the end of the computations. We point that the invariant operator must contain the step function θ（r - R）. Consequently, the problem becomes more complicated. In order to avoid this dimculty, we replace the radius R by ρ（t）R, where ρ（t） is a positive time-dependent function. Then at the end of calculations we take the limit R →0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation satisfied by p（t）, i.e., C 〉 0, C = 0, and C 〈0. Following the C＇s values the spectrum of quantum states is discrete （C 〉 0） or continuous （C ≤ 0）.

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.

PERSISTENCE OF RESONANT INVARIANT TORI WITH NON-HAMILTONIAN PERTURBATION

A persistence theorem for resonant invariant tori with non-Hamiltonian perturbation is proved. The method is a combination of the theory of normally hyperbolic invariant manifolds and an appropriate continuation method. The results obtained are extensions of Chicone’s for the three dimensional non-Hamiltonian systems.

Propagator for a Time-Dependent Damped Harmonic Oscillator with a Force Quadratic in Velocity

The propagator for a time-dependent damped harmonic oscillator with a force quadratic in velocity is obtained by making a specific coordinate transformation and by using the method of time-dependent invariant.

Testing the validity of the Ehrenfest theorem beyond simple static systems: Caldirola–Kanai oscillator driven by a time-dependent force

The relationship between quantum mechanics and classical mechanics is investigated by taking a Gaussian-type wave packet as a solution of the Schr o¨dinger equation for the Caldirola–Kanai oscillator driven by a sinusoidal force. For this time-dependent system, quantum properties are studied by using the invariant theory of Lewis and Riesenfeld. In particular,we analyze time behaviors of quantum expectation values of position and momentum variables and compare them to those of the counterpart classical ones. Based on this, we check whether the Ehrenfest theorem which was originally developed in static quantum systems can be extended to such time-varying systems without problems.

A road map to higher genus Gromov-Witten invariants of Calabi-Yau quintics

This is a survey of using NMSP method to study higher genus Gromov-Witten invariants of Calabi-Yau quintics.It emphasizes on how and why the various methods are introduced to solve several important conjectures for higher genus Gromov-Witten invariants of Calabi-Yau quintics.

On Some Recent Progress in Complex Geometry-the Area Related to Homogeneous Manifolds

In this article,we give a survey of some progress of the complex geometry,mostly related to the Lie group actions on compact complex manifolds and complex homogeneous spaces in the last thirty years.In particular,we explore some works in the special area in Di erential Geometry,Lie Group and Complex Homogeneous Space.Together with the special area in nonlinear analysis on complex manifolds,they are the two major aspects of my research interests.

A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD

In this paper,for any local area-minimizing closed hypersurface∑with RcΣ=RΣ/ngΣ,immersed in a(n+1)-dimension Riemannian manifold M which has positive scalar curvature and nonnegative Ricci curvature,we obtain an upper bound for the area of∑.In particular,when∑saturates the corresponding upper bound,∑is isometric to S^(n)and M splits in a neighborhood of∑.At the end of the paper,we also give the global version of this result.

Including Space-Time in the Extended Group Cl3* of Relativistic Form-Invariance

The inclusion of space-time in the extended group of relativistic form-invariance, Cl3*, is specified as the inclusion of the whole space-time manifold in this multiplicative Lie group. First physical results presented here are: the geometric origin of the time arrow, a better understanding of the non-simultaneity in optics and a mainly geometric origin for the universe expansion, and its recent acceleration.

Totally Umbilical Screen Transversal Lightlike Submanifolds of Semi-Riemannian Product Manifolds

We study totally umbilical screen transversal lightlike submanifolds immersed in a semi-Riemannian product manifold and obtain necessary and sufficient conditions for induced connection ▽ on a totally umbilical radical screen transversal lightlike submanifold to be metric connection. We prove a theorem which classifies totally umbilical ST-anti-invariant lightlike submanifold immersed in a semi-Riemannian product manifold.