Andrey V. Tsiganov
We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.
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We generalize, to some extent, the results on integrable geodesic flows on two dimensional manifolds with a quartic first integral in the framework laid down by Selivanova and Hadeler. The local structure is first determined by a direct integration of the differential system which expresses the conservation of the quartic observable and is seen to involve a finite number of parameters. The global structure is studied in some details and leads to a class of models living on the manifolds S^2, H^2 or R^2. As special cases we recover Kovalevskaya's integrable system and a generalization of it due…
Integrable two-dimensional models which possess an integral of motion cubic or quartic in velocities are governed by a single prepotential, which obeys a nonlinear partial differential equation. Taking into account the latter's invariance under continuous rescalings and a dihedral symmetry, we construct new integrable models with a cubic or quartic integral, each of which involves either one or two continuous parameters. A reducible case related to the two-dimensional wave equation is discussed as well. We conjecture a hidden D_{2n} dihedral symmetry for models with an integral of n-th order i…
In this paper, we propose integrable discretizations of a two-dimensional Hamiltonian system with quartic potentials. Using either the method of separation of variables or the method based on bilinear forms, we construct the corresponding integrable mappings for the first three among four integrable cases.
We consider the problem of motion of a rigid body about a fixed point under the action of an axisymmetric combination of potential and gyroscopic forces. We introduce a new integrable case, valid on zero level of the cyclic integral, that generalizes the known case of motion of a body in liquid due to Chaplygin and its subsequent generalization by Yehia. Apart from certain singular potential terms, the new case involves finite potential and gyroscopic forces, which admit physical interpretation as resulting from interaction of mass, magnetized parts and electric charges on the body with gravit…
We consider integrable system on the sphere $S^2$ with an additional integral of fourth order in the momenta. At the special values of parameters this system coincides with the Kowalevski-Goryachev-Chaplygin system.
In the present note we introduce a new solution of this equation, lead- ing to a new integrable system with a quartic integral, which involves 16 free parameters. A special case of the new system admits interpretation in a problem of rigid body dynamics. It gives a new integrable variation of the cases due to Kovalevskaya (1889), Chaplygin (1903) Goriatchev (1916) and Yehia (2006).
The method introduced in (Yehia H M 2006 J. Phys. A: Math. Gen. 39 5807-5824) and (Yehia H M 2012 J. Phys. A: Math. Gen. 45 395209) is extended to construct new families of several-parameter integrable systems, which admit a complementary integral quartic in the velocities. A total of 12 new systems are obtained, with a number of parameters ranging from 7 up to 16 parameters.
Several completely integrable, indeed solvable, Hamiltonian many-body problems are exhibited, characterized by Newtonian equations of motion ("acceleration equal force"), with linear and cubic forces, in N-dimensional space (N being an arbitrary positive integer, with special attention to N=2, namely motions in a plane, and N=3, namely motions in ordinary three-dimensional space). All the equations of motion are written in covariant form ("N-vector equal N-vector"), entailing their rotational invariance. The corresponding Hamiltonians are of normal type, with the kinetic energy quadratic in th…
We consider two well-known integrable systems on the plane using the concept of natural Poisson bivectors on Riemaninan manifolds. Geometric approach to construction of variables of separation and separated relations for the generalized Henon-Heiles system and the generalized system with quartic potential is discussed in detail.
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.
We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.
In this note we comment on two recently published papers by G. Valent: The 1st is the preprint "On a Class of Integrable Systems with a quartic First Integral, arXiv:1304.5859. April 22, (2013)". We show that the two integrable Hamiltonian systems introduced in this reprint as original results are not new. They are special cases of two systems introduced by the present author in 2006 in two papers [6] and [5]. The second paper is "On a Class of Integrable Systems with a Cubic First Integral, Commun. Math. Phys. 299, 631{649 (2010), In that paper two integrable Hamiltonian systems admitting a c…
We consider an integrable system in five unknowns having three quartics invariants. We show that the complex affine variety defined by putting these invariants equal to generic constants, completes into an abelian surface; the jacobian of a genus two hyperelliptic curve. This system is algebraic completely integrable and it can be integrated in genus two hyperelliptic functions.
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.
New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.
V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-Bäcklund transformations are also found for all equations.
Wolfes model aka $G_2/I_6$-rational integrable model: $g^{(2)}, g^{(3)}$ hidden algebras and quartic polynomial algebra of integrals
本文研究一维三体 Wolfes（$G_2/I_6$-有理）可积模型，借助 MAPLE-18 符号计算揭示其积分构成四次多项式代数并嵌入 $g^{(3)}$ 泛包络代数。
Taking full advantage of two independent projectively equivalent metrics on the ellipsoid leading to Liouville integrability of the geodesic flow via the well-known Jacobi-Moser system, we disclose a novel integrable system on the sphere $S^n$, namely the "dual Moser" system. The latter falls, along with the Jacobi-Moser and Neumann-Uhlenbeck systems, into the category of (locally) Stäckel systems. Moreover, it is proved that quantum integrability of both Neumann-Uhlenbeck and dual Moser systems is insured by means of the conformally equivariant quantization procedure.
The symmetry approach is used for classification of integrable isotropic vector Volterra lattices on the sphere. The list of integrable lattices consists mainly of new equations. Their symplectic structure and associated PDE of vector NLS-type are discussed.
The complete lists of vector hyperbolic equations on the sphere that have integrable third order vector isotropic and anisotropic symmetries are presented. Several new integrable hyperbolic vector models are found. By their integrability we mean the existence of vector Backlund transformations depending on a parameter. For all new equations such transformations are constructed.
We construct a constant curvature analogue on the two-dimensional sphere ${\mathbf S}^2$ and the hyperbolic space ${\mathbf H}^2$ of the integrable Hénon-Heiles Hamiltonian $\mathcal{H}$ given by $$ \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ Ω\left( q_{1}^{2}+ 4 q_{2}^{2}\right) +α\left( q_{1}^{2}q_{2}+2 q_{2}^{3}\right) , $$ where $Ω$ and $α$ are real constants. The curved integrable Hamiltonian $\mathcal{H}_κ$ so obtained depends on a parameter $κ$ which is just the curvature of the underlying space, and is such that the Euclidean Hénon-Heiles system $\mathcal{H}$ is smoothly obtained in…
We consider evolution equations for curves in the 3-dimensional sphere $S^3$ that are invariant under the group $SU(2,1)$ of pseudoconformal transformations, which preserves the standard contact structure on the sphere. In particular, we investigate how invariant evolutions of Legendrian and transverse curves induce well-known integrable systems and hierarchies at the level of their geometric invariants.
A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian $H_κ$, we will make use of a group theoretical approach in which the curvature $κ$ of the underlying space will be treated as an additional (contraction) parameter, and we will make extensive use of projective coordinates and their associated phase spaces. It turns out …
We obtain bi-Hamiltonian structure for a family of integrable systems on the sphere S with an additional integral of third order in momenta. These results are applied to the Goryachev system and Goryachev-Chaplygin top for which we give an explicit procedure to find the separated coordinates and the separated relations.
We introduce and study the Chaplygin systems with gyroscopic forces. This natural class of nonholonomic systems has not been treated before. We put a special emphasis on the important subclass of such systems with magnetic forces. The existence of an invariant measure and the problem of Hamiltonization are studied, both within the Lagrangian and the almost-Hamiltonian framework. In addition, we introduce problems of rolling of a ball with the gyroscope without slipping and twisting over a plane and over a sphere in $\mathbb R^n$ as examples of gyroscopic $SO(n)$--Chaplygin systems. We describe…
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transform…
Spherical and Planar Ball Bearings -- a Study of Integrable Cases
本文研究球面与平面上多个均质球体及一个动力学非对称球体或平面构成的无滑动滚动非完整系统，推导运动方程并证明不变测度存在，对 n=1 情形找到两类可用求积法积分的可积模型，推广了 Chaplygin 球可积问题，并显式求解了相应的平面问题。
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and co…
Two-dimensional Hamiltonian systems admitting second invariants which are quartic in the momenta are investigated using the Jacobi geometrization of the dynamics. This approach allows for a unified treatment of invariants at both arbitrary and fixed energy. In the differential geometric picture, the quartic invariant corresponds to the existence of a fourth rank Killing tensor. Expressing the Jacobi metric in terms of a Kahler potential, the integrability condition for the existence of the Killing tensor at fixed energy is a non-linear equation involving the Kahler potential. At arbitrary ener…
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa-Holm equation and the multi-component modified Camassa-Holm equation are provided. It is shown that these equations arise from non-streching invariant curve flows respectively in the three-dimensional Euclidean geometry, the two-dimensional Möbius sphere and $n$-dimensional sphere ${\mathbb S}^n(1)$. Integrability to these systems is also studied.
We discuss a family of integrable systems on the sphere $S^2$ with an additional integral of third order in momenta. This family contains the Coryachev-Chaplygin top, the Goryachev system, the system recently discovered by Dullin and Matveev and two new integrable systems. On the non-physical sphere with zero radius all these systems are isomorphic to each other.
The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the generic anisotropic oscillator Hamiltonian is presented, and its s…
From the viewpoint of integrable systems on algebraic curves, we discuss linearization of birational maps arising from the seed mutations of types $A^{(1)}_1$ and $A^{(2)}_2$, which enables us to construct the set of all cluster variables generating the corresponding cluster algebras. These birational maps respectively induce discrete integrable systems on algebraic curves referred to as the types of the seed mutations from which they are arising. The invariant curve of type $A^{(1)}_1$ is a conic, while the one of type $A^{(2)}_2$ is a singular quartic curve. By applying the blowing-up of the…
We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schroedinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework.
Integrability of the magnetic geodesic flow on the sphere with a constant 2-form
研究球面上磁测地流的可积性，证明其为Liouville可积，积分形式为二次和线性动量函数。
The Equations of motion of vortex sources (examined earlier by Fridman and Polubarinova) are studied, and the problems of their being Hamiltonian and integrable are discussed. A system of two vortex sources and three sources-sinks was examined. Their behavior was found to be regular. Qualitative analysis of this system was made, and the class of Liouville integrable systems is considered. Particular solutions analogous to the homothetic configurations in celestial mechanics are given.
We show that the following elementary geometric properties of the motion of a discrete (i.e. piecewise linear) curve select the integrable dynamics of the Ablowitz-Ladik hierarchy of evolution equations: i) the set of points describing the discrete curve lie on the sphere S^3, ii) the distance between any two subsequant points does not vary in time, iii) the dynamics does not depend explicitly on the radius of the sphere. These results generalize to a discrete context our previous work on continuous curves.
Chaos in the BMN matrix model
研究BMN矩阵模型中的经典混沌运动，通过考察三种ansatz下的两耦合非谐振子约化系统，发现前两种存在混沌而第三种可积，证明该模型在Liouville意义上不可积。
We consider a nonholonomic system describing a rolling of a dynamically non-symmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel--Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconical) coordinates on the Poisson sphere, which can be usefu…
The motion on the sphere $S^2$ with the potential $V= (x_1x_2x_3)^{-2/3}$ is considered. The Lax representation and the linearisation procedure for this two-dimensional integrable system are discussed.
We discuss the concept of natural Poisson bivectors, which allows us to consider the overwhelming majority of known integrable systems on the sphere in framework of bi-Hamiltonian geometry.
In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and a scattering of vortices and obtained the conditions of its realization. We completed the bifurcation analysis and investigated for collinear and Thomson's configurations the dependence of stability in linear approximation and frequency of rotation in relative coordinates from value of a full moment. We indicated the geometric interpretation for characteristic situations. We constructed a phase portrait and geometric projec…
We discuss two polynomial bi-Hamiltonian structures for the generalized integrable Chaplygin system on the sphere S^2 with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation, the separation relations and the transformation of the corresponding algebraic curves of genus two is considered in detail.
Integrable problem of three vorteces on a plane and sphere are considered. The classification of Poisson structures is carried out. We accomplish the bifurcational analysis using the variables introduced in previous part of the work.
Group based moving frames have a wide range of applications, from the classical equivalence problems in differential geometry to more modern applications such as computer vision. Here we describe what we call a discrete group based moving frame, which is essentially a sequence of moving frames with overlapping domains. We demonstrate a small set of generators of the algebra of invariants, which we call the discrete Maurer--Cartan invariants, for which there are recursion formulae. We show that this offers significant computational advantages over a single moving frame for our study of discrete…
Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ 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 ot…
The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separations for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. It allows us to say about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.
Integrable spinning extension of a free particle on 2-sphere is constructed in which spin degrees of freedom are represented by a 3-vector obeying the Bianchi type-V algebra. Generalizations involving a scalar potential giving rise to two quadratic constants of the motion, or external field of the Dirac monopole, or the motion on the group manifold of SU(2) are built. A link to the model of a relativistic spinning particle propagating on the near horizon 7d Myers-Perry black hole background is considered. Implications of the construction in this work for the D(2,1;a) superconformal mechanics a…
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