• 대한수학회지
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• 제53권4호
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• pp.795-834
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• 2016

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^{\Omega}$ ($D^2$, ${\partial}D^2$) of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism Cal : $Diff^{\Omega}$ ($D^1$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to a homomorphism ${\bar{Cal}}$ : Hameo($D^2$, ${\partial}D^2$)${\rightarrow}{\mathbb{R}}$ to that of the vanishing of the basic phase function $f_{\underline{F}}$, a Floer theoretic graph selector constructed in [9], that is associated to the graph of the topological Hamiltonian loop and its normalized Hamiltonian ${\underline{F}}$ on $S^2$ that is obtained via the natural embedding $D^2{\hookrightarrow}S^2$. Here Hameo($D^2$, ${\partial}D^2$) is the group of Hamiltonian homeomorphisms introduced by $M{\ddot{u}}ller$ and the author [18]. We then provide an evidence of this vanishing conjecture by proving the conjecture for the special class of weakly graphical topological Hamiltonian loops on $D^2$ via a study of the associated Hamiton-Jacobi equation.

• Korean Journal of Mathematics
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• 제17권4호
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• pp.507-519
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• 2009

We give a theorem of the existence of the multiple solutions of the Hamiltonian system with the square growth nonlinearity. We show the existence of m solutions of the Hamiltonian system when the square growth nonlinearity satisfies some given conditions. We use critical point theory induced from the invariant function and invariant linear subspace.

• 대한수학회보
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• 제44권3호
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• pp.415-418
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• 2007

In this note we show the relationship between the period of a isochronous center of a planar Hamiltonian system and the Gauss curvature of the surface S = (x, y, H(x, y)) where H is the energy function of the system.

• 대한수학회지
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• 제46권2호
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• pp.363-447
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• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

• 대한수학회보
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• 제53권3호
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• pp.667-680
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• 2016

We get a theorem which shows the multiple weak solutions for the bifurcation problem of the superquadratic nonlinear Hamiltonian system. We obtain this result by using the variational method, the critical point theory in terms of the $S^1$-invariant functions and the $S^1$-invariant linear subspaces.

• 대한수학회보
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• 제54권6호
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• pp.2043-2051
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• 2017

Consider a sequence of compactly supported Hamiltonian diffeomorphisms ${\phi}_k$ of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the ${\phi}_k$ $C^0$-converge to the identity, then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the ${\phi}_k$. We discuss connections to the open problem of whether the Calabi homomorphism extends to the Hamiltonian homeomorphism group. The proof is based on a relationship between the Calabi invariant of a $C^0$-small Hamiltonian diffeomorphism and the generalized phase function of its graph.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2004년도 ICCAS
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• pp.1623-1628
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• 2004

In the 3-dimensional cyclic Lotka-Volterra equations, we show the solution on the invariant hyperplane. In addition, we show the existence of the invariant hyperplane by the center manifold theorem under the some conditions. With this result, we can lead the hyperplane of the n-dimensional cyclic Lotka-Volterra equaions. In other section, we study the 3- or 4-dimensional Hamiltonian Lotka-Volterra equations which satisfy the Jacobi identity. We analyze the solution of the Hamiltonian Lotka- Volterra equations with the functions called the split Liapunov functions by [4], [5] since they provide the Liapunov functions for each region separated by the invariant hyperplane. In the cyclic Lotka-Volterra equations, the role of the Liapunov functions is the same in the odd and even dimension. However, in the Hamiltonian Lotka-Volterra equations, we can show the difference of the role of the Liapunov function between the odd and the even dimension by the numerical calculation. In this paper, we regard the invariant hyperplane as the important item to analyze the motion of Lotka-Volterra equations and occur the chaotic orbit. Furtheremore, an example of the asymptoticaly stable and stable solution of the 3-dimensional cyclic Lotka-Volterra equations, 3- and 4-dimensional Hamiltonian equations are shown.

• 대한화학회지
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• 제41권3호
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• pp.123-129
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• 1997

1,3-CHD의 비평면 고리반전 진동운동에 미치는 환산질량 효과를 고찰하기 위하여 원자간의 결합 벡터 모델을 이용한 컴퓨터 프로그램을 작성하였고, 이로부터 큰 진폭으로 운동하는 고리반전 진동운동에 대한 운동에너지 팽창함수를 계산하였다. 1,3-CHD의 운동에너지 팽창함수의 계산에 필요한 구조 파라미터들은 순이론적 HF/6-31G** 방법에 의하여 결정하였으며, 이 화합물의 비평면 고리반전 진동운동에 대한 위치에너지 함수는 운동에너지 팽창함수와 이전에 보고된 저주파 라만 데이터를 이용하여 결정하였다. 그 결과 1,3-CHD의 비평면 진동좌표에 대한 운동에너지 팽창함수를 고려하여 진동 Hamiltonian을 계산하였을 때 1,3-CHD의 고리반전 진동운동에 대한 보다 신빙성 있는 비평면 위치에너지 함수를 결정할 수 있었다.

• 한국산학기술학회논문지
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• 제19권2호
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• pp.608-616
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• 2018

일반적으로 비선형 시스템은 1차와 2차 시스템의 곱의 형태로 선형화되며, 시스템의 근은 1차 시스템의 근과 2차 시스템의 중근, 서로 다른 두 실근, 복소근으로 구성된다. 그리고 LQ(Linear Quadratic) 제어는 성능지수함수를 최소화하는 제어법칙을 설계하는 방법으로 시스템의 안정성을 보장하는 장점과 가중행렬 조정으로 시스템의 근의 위치를 조정하는 극배치 기능이 있다. 가중행렬에 의해 LQ 제어는 시스템의 근의 위치를 임의로 이동시킬 수 있지만 시행착오 방법으로 가중행렬을 설정하는 어려움이 있다. 이것은 해밀토니안(Hamiltonian) 시스템의 특성방정식을 이용하여 해결 할 수 있다. 또한 제어가중행렬이 상수의 대칭행렬이면 제어법칙을 반복적으로 적용하여 시스템의 여러 근을 원하는 폐루프 근으로 이동시킬 수 있다. 이 논문은 해밀토니안 시스템의 특성방정식을 이용하여 조단 블록을 갖는 시스템의 중근을 두 실근으로 이동시키는 상태가중행렬과 제어법칙을 계산하는 방법을 제시한다. 삼각함수로 표현된 상태가중행렬로 해밀토니안 시스템의 특성방정식을 구한다. 그리고 이동된 두 실근이 특성방정식의 근이라는 조건에서 중근과 상태가중행렬의 관계식(${\rho},\;{\theta}$)을 유도한다. 상태가중행렬이 양의 반한정행렬이 될 조건에서 중근의 이동범위를 구한다. 그리하여 이동범위에서 선택한 두 실근을 관계식에 대입하여 상태가중행렬과 제어법칙을 계산한다. 제안한 방법을 간단한 3차 시스템의 예제에 적용해본다.

• Journal of Magnetics
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• 제16권2호
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• pp.108-113
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• 2011

The electron spin resonance lineshape (ESRLS) function for the electron spin resonance linewidth (ESRLW) of $Cr^{3+}$ (S = 3/2) in ferroelectric lithium niobate single crystals doped with 0.05 wt% of Cr, is obtained by using the projection operator technique (POT), developed by Argyres and Sigel. The ESRLS function is calculated to be axially symmetric about the c - axis and analyzed by using the spin Hamiltonian $H_{SP}={\mu}_B(B{\cdot}{^\leftrightarrow_{g}}{\cdot}S)+S{\cdot}{^\leftrightarrow_{D}}{\cdot}S$ with the parameters g = 1.972 and D = $0.395\;cm^{-1}$. In the ca plane, the linewidths show a strong angular dependence, whereas in the ab plane, they are independent of the angle. This result implies that the resonance center has an axial symmetry along the c - axis. Further, from the temperature dependence of the linewidths that is shown, it can be seen that the linewidths increase as the temperature increases, at a frequency of v = 9.27GHz. This result implies that the scattering effect increases with increasing temperature. Thus, the POT is considered to be more convenient to explain the scattering mechanism as in the case of other optical resonant systems.