共查询到20条相似文献,搜索用时 46 毫秒
1.
A. V. Kurov 《Moscow University Physics Bulletin》2016,71(4):375-380
We show that, as distinct from completely integrable Hamiltonian systems, a commutative partially integrable system admits different compatible Poisson structures on a phase manifold that are related by a recursion operator. The existence of action–angle coordinates around an invariant submanifold of such a partially integrable system is proved. 相似文献
2.
We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra \bar{E} of the Lie algebra E and the reduced self-dual Yang-Mills equations, we obtain an expanding integrable model of the Giachetti-Johnson (GJ) hierarchy whose Hamiltonian structure can
also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra gN. As an application, we apply the loop algebra \tilde{E} of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup-Newell (KN) hierarchy which, consisting of two arbitrary parametersα andβ, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra \tilde{F} of the Lie algebra F to obtain an
expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R3 based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations. 相似文献
3.
Oleg I. Bogoyavlenskij 《Communications in Mathematical Physics》1996,180(3):529-586
This paper develops a new theory of tensor invariants of a completely integrable non-degenerate Hamiltonian system on a smooth manifoldM
n. The central objects in this theory are supplementary invariant Poisson structuresP
c which are incompatable with the original Poisson structureP
1 for this Hamiltonian system. A complete classification of invariant Poisson structures is derived in a neighbourhood of an invariant toroidal domain. This classification resolves the well-known Inverse Problem that was brought into prominence by Magri's 1978 paper deveoted to the theory of compatible Poisson structures. Applications connected with the KAM theory, with the Kepler problem, with the basic integrable problem of celestial mechanics, and with the harmonic oscillator are pointed out. A cohomology is defined for dynamical systems on smooth manifolds. The physically motivated concepts of dynamical compatibility and strong dynamical compatibility of pairs of Poisson structures are introduced to study the diversity of pairs of Poisson structures incompatible in Magri's sense. It is proved that if a dynamical systemV preserves two strongly dynamically compatible Poisson structuresP
1 andP
2 in a general position then this system is completely integrable. Such a systemV generates a hierarchy of integrable dynamical systems which in general are not Hamiltonian neither with respect toP
1 nor with respect toP
2. Necessary conditions for dynamical compatibility and for strong dynamical compatibility are derived which connect these global properties with new local invariants of an arbitrary pair of incompatible Poisson structures.Supported by NSERC grant OGPIN 337. 相似文献
4.
T. V. Skrypnyk 《Physics of Atomic Nuclei》2002,65(6):1108-1112
We construct a new family of infinite-dimensional Lie algebras on hyperelliptic curves. Using them, we find new integrable Hamiltonian systems, which are direct higher rank generalizations of the Steklov-Liapunov integrable systems associated with the e(3) algebra and the Steklov-Veselov integrable systems associated with the so(4) algebra. 相似文献
5.
6.
We construct a class of integrable generalization of Toda
mechanics with long-range interactions. These systems are
associated with the loop algebras L(Cr) and L(Dr) in the sense that their Lax matrices can be realized in
terms of the c=0 representations of the affine Lie algebras
C(1)r and
D(1)r and the interactions pattern involved
bears the typical characters of the corresponding root systems. We
present the equations of motion and the Hamiltonian structure.
These generalized systems can be identified unambiguously by
specifying the underlying loop algebra together with an ordered
pair of integers (n,m). It turns out that different systems
associated with the same underlying loop algebra but with
different pairs of integers
(n1,m1) and (n2,m2) with
n2<n1 and
m2<m1 can be related by a nested Hamiltonian
reduction procedure. For all nontrivial generalizations, the extra
coordinates besides the standard Toda variables are Poisson
non-commute, and when either $n$ or
m≥3, the Poisson
structure for the extra coordinate variables becomes some Lie
algebra (i.e. the extra variables appear linearly on the
right-hand side of the Poisson brackets). In the quantum case, such
generalizations will become systems with noncommutative variables
without spoiling the integrability. 相似文献
7.
A. Lesfari 《Journal of Geometry and Physics》1999,31(4):165-286
During the last few decades, algebraic geometry has become a tool for solving differential equations and spectral questions of mechanics and mathematical physics. This paper deals with the study of the integrable systems from the point of view of algebraic geometry, inverse spectral problems and mechanics from the point of view of Lie groups. Section 1 is preliminary giving a little background. In Section 2, we study a Lie algebra theoretical method leading to completely integrable systems, based on the Kostant-Kirillov coadjoint action. Section 3 is devoted to illustrate how to decide about the algebraic complete integrability (a.c.i.) of Hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sense of the phase space being foliated by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). Adler-van Moerbeke's method is a very useful tool not only to discover among families of Hamiltonian systems those which are a.c.i., but also to characterize and describe the algebraic nature of the invariant tori (periods, etc.) for the a.c.i. systems. Some integrable systems, such as Kortewege—de Vries equation, Toda lattice, Euler rigid body motion, Kowalewski's top, Manakov's geodesic flow on S O (4), etc. are treated. 相似文献
8.
The trace identity is extended to the general loop algebra. The
Hamiltonian structures of the integrable systems concerning vector
spectral problems and the multi-component integrable hierarchy can be
worked out by using the extended trace identity. As its
application, we have obtained the Hamiltonian structures of the Yang
hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the
multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the
multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell
(M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy
separately. 相似文献
9.
10.
XU Xi-Xiang 《理论物理通讯》2012,57(6):953-960
A family of integrable differential-difference equations is derived from a new matrix spectral problem. The Hamiltonian forms of obtained differential-difference equations are constructed. The Liouville integrability for the obtained integrable family is proved. Then, Bargmann symmetry constraint of the obtained integrable family is presented by binary nonliearization method of Lax pairs and adjoint Lax pairs. Under this Bargmann symmetry constraints, an integrable symplectic map and a sequences of completely integrable finite-dimensional Hamiltonian systems in Liouville sense are worked out, and every integrable differential-difference equations in the obtained family is factored by the integrable symplectic map and a completely integrable finite-dimensional Hamiltonian system. 相似文献
11.
This paper surveys the classification of integrable evolution equations whose field variables take values in an associative
algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems
possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the
Painlevé transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed
and the biHamiltonian structures for several examples are found.
Received: 12 March 1997 / Accepted: 27 August 1997 相似文献
12.
V. L. Golo 《Letters in Mathematical Physics》1981,5(2):155-159
We study an algebra of Poisson brackets of the Hamiltonian system defined by the nonlinear Leggett equations of spin dynamics in the A- and the B-phases of superfluid 3He. For the A-phase the Poisson algebra results in a special case of the equations of motion of a rigid body in ideal fluid; for the B-phase, in the absence of magnetic field, it allows for a reduction to a smaller Poisson algebra that provides exact solutions for the Leggett equations. 相似文献
13.
We construct the Poisson algebra associated to a singular mapping into symplectic space and show that this is an algebra of smooth functions generating solvable implicit Hamiltonian systems. 相似文献
14.
Based on semi-direct sums of Lie subalgebra \tilde{G}, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derivedfrom a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity --- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we provethat the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems. 相似文献
15.
A one-dimensional discrete conservative Hamiltonian with a generalized form of the Schmidt potential, is constructed with the help of a non-integrable discrete Hamiltonian whose parametrized double-well potential can be reduced to the ?4 potential. The new conservative Hamiltonian is completely integrable in the discrete static regime, and the associate exact nonlinear solution is shown to coincide with the continuum nonlinear periodic solution of the non-integrable Hamiltonian. Numerical simulations and nonlinear stability analysis suggest that the discrete mapping derived from the completely integrable Hamiltonian undergoes a bifurcation which does not leads to the chaotic phase with randomly pinned states, but instead to a phase where real solutions become rare forming a cluster of periodic points around an elliptic fixed point. 相似文献
16.
A new simple loop algebra is constructed, which is devote to establishing an isospectral problem. By making use of Tu scheme,
NLS-MKdV hierarchy is obtained. Again via expanding the loop algebra above, another higher-dimensional loop algebra is presented.
It follows that an integrable coupling of NLS-MkdV hierarchy is given. Also, the trace identity is extended to the quadratic-form
identity and the Hamiltonian structures of the NLS-MKdV hierarchy and integrable coupling of NLS-MkdV hierarchy are obtained
by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings
or multi-component hierarchies. 相似文献
17.
In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity. 相似文献
18.
Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra
are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e. by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order
in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. for each case — in the ambient space
n
, the sphere and the ellipsoid — the Schrödinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lamé type.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and the Fonds FCAR du Québec. 相似文献
19.
A 2+1-dimensional discrete is presented, which is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems, with aid of the nonlineaxization of Lax pairs. The system is completely integrable in the Liouville sense. 相似文献
20.
An algebraic analysis of the Hamiltonian formulation of the model two-dimensional gravity is performed. The crucial fact is
an exact coincidence of the Poisson brackets algebra of the secondary constraints of this Hamiltonian formulation with the
SO(2,1)-algebra. The eigenvectors of the canonical Hamiltonian H
c
are obtained and explicitly written in closed form. 相似文献