Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type

Symmetry constraints for (2+1)-dimensional dispersionless integrable equations are considered. It is demonstrated that they naturally lead to certain systems of hydrodynamic type which arise within the reduction method. One also easily obtains an associated complex curve (Sato function) and corresponding generating equations. Dispersionless KP and 2DTL hierarchy are considered as illustrative examples.     

Second order Hamiltonian formalism and constraints algebra for non-polynomial supergravities

The second order Hamiltonian formalism for a non-polynomial N = 1D = 10 supergravity coupled to super Yang-Mills theory is developed. This is done by starting from the first order canoncial covariant formalism on group manifold. The Hamiltonian, generator of time evolution, is found as a functional of the first class constraints of this coupled system. These contraints close the constraint algebra and they are the generators of all the Hamiltonian gauge symmetries.     

Lie algebra structures for four-component Hamiltonian hydrodynamic type systems

Four-component Hamiltonian systems of hydrodynamic type induce, through the Haantjes tensor, a Lie algebra structure on tangent planes in the space of dependent variables. We show that this Lie algebra is either reductive or solvable with a nilpotent three-dimensional subalgebra. We demonstrate how the precise Lie algebraic structure is determined by the Hamiltonian structure of the system. An application to perturbations of the Benney system is presented.     

Complexifications and real forms of Hamiltonian structures

We consider generalizations of the standard Hamiltonian dynamics to complex dynamical variables and introduce the notions of real Hamiltonian form in analogy with the notion of real forms for a simple Lie algebra. Thus to each real Hamiltonian system we are able to relate several nonequivalent ones. On the example of the complex Toda chain we demonstrate how starting from a known integrable Hamiltonian system (e.g. the Toda chain) one can complexify it and then project onto different real forms. Received 18 October 2001 / Received in final form 24 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: gerjikov@inrne.bas.bg     

On the vierbein formalism of general relativity

Both the Einstein–Hilbert action and the Einstein equations are discussed under the absolute vierbein formalism. Taking advantage of this form, we prove that the “kinetic energy” term, i.e., the quadratic term of time derivative term, in the Lagrangian of the Einstein–Hilbert action is non-positive definitive. And then, we present two groups of coordinate conditions that lead to positive definitive kinetic energy term in the Lagrangian, as well as the corresponding actions with positive definitive kinetic energy term, respectively. Based on the ADM decomposition, the Hamiltonian representation and canonical quantization of general relativity taking advantage of the actions with positive definitive kinetic energy term are discussed; especially, the Hamiltonian constraints with positive definitive kinetic energy term are given, respectively. Finally, we present a group of gauge conditions such that there is not any second time derivative term in the ten Einstein equations.     

Hamiltonian structure and gauge symmetries of Poincaré gauge theory

This is a review of the constrained dynamical structure of Poincaré gauge theory which concentrates on the basic canonical and gauge properties of the theory, including the identification of constraints, gauge symmetries and conservation laws. As an interesting example of the general approach, we discuss the teleparallel formulation of general relativity.     

Monodromy in non-integrable systems on certain compact classical phase spaces

On the example of bending vibrational polyads of the acetylene molecule (C₂H₂) in the approximation of the $1:1:1:1$ resonant oscillator with axial symmetry, whose geometry is similar to the n-shell approximation of the perturbed hydrogen atom, we show how remaining invariant tori of the underlying classical non-integrable system form a nontrivial continuous family with monodromy. We read this monodromy off the quantum energy spectrum which was observed experimentally by spectroscopists, and we uncover its origins through the particular topology, geometry, and symmetry. We explain how monodromy characterizes the chaotic region surrounded by the tori. We detail the explicit correspondence between the bending polyads of C₂H₂ and the n-shells of the hydrogen atom, and uncover the dynamical SO(3) symmetry of the bending polyads and the corresponding spherically localized vibrational states.     

On the spectra of periodic waves for infinite-dimensional Hamiltonian systems

We consider the problem of determining the spectrum for the linearization of an infinite-dimensional Hamiltonian system about a spatially periodic traveling wave. By using a Bloch-wave decomposition, we recast the problem as determining the point spectra for a family of operators J_γL_γ, where J_γ is skew-symmetric with bounded inverse and L_γ is symmetric with compact inverse. Our main result relates the number of unstable eigenvalues of the operator J_γL_γ to the number of negative eigenvalues of the symmetric operator L_γ. The compactness of the resolvent operators allows us to greatly simplify the proofs, as compared to those where similar results are obtained for linearizations about localized waves. The theoretical results are general, and apply to a larger class of problems than those considered herein. The theory is applied to a study of the spectra associated with periodic and quasi-periodic solutions to the nonlinear Schrödinger equation, as well as periodic solutions to the generalized Korteweg-de Vries equation with power nonlinearity.     

Necessary conditions for super-integrability of Hamiltonian systems

We formulate a general theorem which gives a necessary condition for the maximal super-integrability of a Hamiltonian system. This condition is expressed in terms of properties of the differential Galois group of the variational equations along a particular solution of the considered system. An application of this general theorem to natural Hamiltonian systems of n degrees of freedom with a homogeneous potential gives easily computable and effective necessary conditions for the super-integrability. To illustrate an application of the formulated theorems, we investigate: three known families of integrable potentials, and the three body problem on a line.     

Matrix factorization method for the Hamiltonian structure of integrable systems

We demonstrate that the process of matrix factorization provides a systematic mathematical method to investigate the Hamiltonian structure of non-linear evolution equations characterized by hereditary operators with Nijenhuis property.     

Hamilton系统Noether理论的新型逆问题

研究Hamilton系统Noether理论新型的逆问题,得到利用Noether理论从已知的第一积分构建Hamilton函数和对称性的一般解法和若干特殊解法,提出由Hamilton函数直接导出守恒量的两条推论.举例说明所得结果的应用.     

A set of Lie symmetrical non－Noether conserved quantity for the relativistic Hamiltonian systems

For the relativistic Hamiltonian system, a new type of Lie symmetrical non-Noether conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing special infinitesimal transformations for q_s and p_s, we construct the determining equations of Lie symmetrical transformations of the system, which only depend on the canonical variables. A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.     

Deformations of Dispersionless KdV Hierarchies

The obstructions to the existence of a hierarchy of hydrodynamic conservation laws are studied for a multicomponent dispersionless KdV system. It is proved that if the lowest order obstruction vanishes then all higher obstructions automatically vanish, if and only the underlying algebra is a Jordan algebra. Deformations of these multicomponent dispersionless KdV-type equations are also studied. It is shown that no new obstructions appear and, hence, that the existence of a fully deformed hierarchy depends only on the existence of a single purely hydrodynamic conservation law.     

Integrable Hamiltonian N-body problems of goldfish type featuring N arbitrary functions

A simple application of a neat formula relating the time evolution of the N zeros of a (monic) time-dependent polynomial of degree N in the complex variable w to the time evolution of its N coefficients allows to identify integrable Hamiltonian N-body problems in the plane featuring N arbitrary functions, the equations of motions of which are of Newtonian type: accelerations equal forces nonlinearly dependent on the coordinates of the N particle. The motions generally take place in the complex z-plane, or, equivalently, in the Cartesian xy-plane with z = x + iy. It is also easy to identify qualitative features of special subclasses of these models, for instance cases in which all the motions starting from an arbitrary real set of initial data are confined and multiply periodic. It is also indicated how to generate from these models hierarchies of analogous models with analogous properties.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Some applications of stochastic averaging method for quasi Hamiltonian systems in physics

Many physical systems can be modeled as quasi-Hamiltonian systems and the stochastic averaging method for quasi-Hamiltonian systems can be applied to yield reasonable approximate response statistics. In the present paper, the basic idea and procedure of the stochastic averaging method for quasi Hamiltonian systems are briefly introduced. The applications of the stochastic averaging method in studying the dynamics of active Brownian particles, the reaction rate theory, the dynamics of breathing and denaturation of DNA, and the Fermi resonance and its effect on the mean transition time are reviewed. Supported by the National Natural Science Foundation of China (Grant Nos. 10772159 and 10802074), the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20060335125), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y7080070)     

Symmetry of Hamiltonian and conserved quantity for a system of generalized classical mechanics

This paper focuses on a new symmetry of Hamiltonian and its conserved quantity for a system of generalized classical mechanics.The differential equations of motion of the system are established.The definition and the criterion of the symmetry of Hamiltonian of the system are given.A conserved quantity directly derived from the symmetry of Hamiltonian of the generalized classical mechanical system is given.Since a Hamilton system is a special case of the generalized classical mechanics,the results above are equally applicable to the Hamilton system.The results of the paper are the generalization of a theorem known for the existing nonsingular equivalent Lagrangian.Finally,two examples are given to illustrate the application of the results.