scholarly journals Second-Order Integrals for Systems inE2Involving Spin

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
İsmet Yurduşen
Keyword(s):  
Hamiltonian Systems ◽  
Euclidean Plane ◽  
Second Order ◽  
Two Dimensional ◽  
Integrals Of Motion ◽  
Integrable Hamiltonian Systems

In two-dimensional Euclidean plane, existence of second-order integrals of motion is investigated for integrable Hamiltonian systems involving spin (e.g., those systems describing interaction between two particles with spin 0 and spin 1/2) and it has been shown that no nontrivial second-order integrals of motion exist for such systems.

Stochastic Averaging for Quasi-Integrable Hamiltonian Systems With Variable Mass

2013 ◽  
Vol 81 (5) ◽  
Author(s):  
Yong Wang ◽  
Xiaoling Jin ◽  
Zhilong Huang
Keyword(s):  
Hamiltonian Systems ◽  
Variable Mass ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
Integrals Of Motion ◽  
Integrable Hamiltonian Systems ◽  
Hamiltonian Equations ◽  
Averaging Technique ◽  
Variable Mass Systems ◽  
Random Responses

Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

scholarly journals Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

2019 ◽  
Vol 874 ◽  
pp. 891-925 ◽  
Author(s):  
A. I. Dyachenko ◽  
S. A. Dyachenko ◽  
P. M. Lushnikov ◽  
V. E. Zakharov
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Half Plane ◽  
Second Order ◽  
Zero Gravity ◽  
Two Dimensional ◽  
Branch Points ◽  
Integrals Of Motion ◽  
Complex Half Plane ◽  
Constants Of Motion

We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$. We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.

scholarly journals A finite-dimensional integrable system associated with a polynomial eigenvalue problem

2006 ◽  
Vol 2006 ◽  
pp. 1-9
Author(s):  
Taixi Xu ◽  
Weihua Mu ◽  
Zhijun Qiao
Keyword(s):  
Eigenvalue Problem ◽  
Integrable System ◽  
Hamiltonian Systems ◽  
Second Order ◽  
Integrable Hamiltonian Systems ◽  
Hamiltonian Structures ◽  
Polynomial Eigenvalue Problem ◽  
Constrained Hamiltonian Systems ◽  
Finite Dimensional ◽  
Order Polynomial

M. Antonowicz and A. P. Fordy (1988) introduced the second-order polynomial eigenvalue problemLφ=(∂2+∑i=1nviλi)φ=αφ(∂=∂/∂x,α=constant)and discussed its multi-Hamiltonian structures. Forn=1andn=2, the associated finite-dimensional integrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is,nis arbitrary, provide the constrained Hamiltonian systems associated with the above-mentioned second-order polynomial ergenvalue problem, and prove them to be completely integrable.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

scholarly journals Cutoff AdS3 versus $$ T\overline{T} $$ CFT2 in the large central charge sector: correlators of energy-momentum tensor

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Yi Li ◽  
Yang Zhou
Keyword(s):  
Field Theory ◽  
Euclidean Plane ◽  
Energy Momentum Tensor ◽  
Central Charge ◽  
Momentum Tensor ◽  
Two Dimensional ◽  
Conformal Field ◽  
Operator Identity ◽  
Pure Gravity ◽  
Charge Sector

Abstract In this article we probe the proposed holographic duality between $$ T\overline{T} $$ T T ¯ deformed two dimensional conformal field theory and the gravity theory of AdS3 with a Dirichlet cutoff by computing correlators of energy-momentum tensor. We focus on the large central charge sector of the $$ T\overline{T} $$ T T ¯ CFT in a Euclidean plane and a sphere, and compute the correlators of energy-momentum tensor using an operator identity promoted from the classical trace relation. The result agrees with a computation of classical pure gravity in Euclidean AdS3 with the corresponding cutoff surface, given a holographic dictionary which identifies gravity parameters with $$ T\overline{T} $$ T T ¯ CFT parameters.

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