arXiv research

Researchers find a method to represent bi-Hamiltonian systems using Lagrangian representations.

We consider the two body problem with central interaction on two point homogeneous spaces from point of view of the invariant differential operators theory. The representation of the two particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connect…

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(Λ^2V)$ that lie in the kernel of the natural map $S^2(Λ^2V)\to Λ^4V$. Non-equivalent operators corres…

Study non-homogeneous operators in 1+0 systems, classifying and analyzing their geometric properties.

The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.

Motivated by various results on homogeneous geodesics of Riemannian spaces, we study homogeneous trajectories, i.e. trajectories which are orbits of a one-parameter symmetry group, of Lagrangian and Hamiltonian systems. We present criteria under which an orbit of a one-parameter subgroup of a symmetry group G is a solu…

Develops integrators for Hamiltonian systems in Jacobi manifolds.

New integrators preserve geometric structure in Hamiltonian systems.

Representations of coherent state Lie algebras on coherent state manifolds as first order differential operators are presented. The explicit expressions of the differential action of the generators of semisimple Lie groups determine for linear Hamiltonians in the generators of the groups first order differential equati…

A new method simplifies contact Hamiltonian mechanics.

We apply the concept of castling transform of prehomogeneous vector spaces to produce new examples of minimal homogeneous Lagrangian submanifolds in the complex projective space. Furthermore we verify the Hamiltonian stability of a low dimensional example that can be obtained in this way.

The two-body problem with a central interaction on simply connected constant curvature spaces of an arbitrary dimension is considered. The explicit expression for the quantum two-body Hamiltonian via a radial differential operator and generators of the isometry group is found. We construct a self-adjoint extension of t…

The paper defines unimodularity for coisotropic Poisson spaces and discusses invariant volume forms.

The application of the Legendre transformation to a hyperregular Lagrangian system results in a Hamiltonian vector field generated by a Hamiltonian defined on the phase space of the mechanical system. The Legendre transformation in its usual interpretation can not be applied to homogeneous Lagrangians found in relativi…

The Clifford torus is a torus in a three-dimensional sphere. Homogeneous tori are simple generalization of the Clifford torus which still in a three-dimensional sphere. There is a way to construct tori in a three-dimensional sphere using the Hopf fibration. In this paper, all Hamiltonian stationary Lagrangian tori whic…

The paper extends Hamiltonian Monte Carlo to Lie groups and constrained mechanics.

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is giv…

Study classifies Hamiltonian operators with skew-symmetric constraints.

Paper extends Poisson-Lichnerowicz cohomology to scalar difference Hamiltonian operators.

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropr…

Study geometric quantization of Hamiltonian flows using Berezin-Toeplitz operators.

We develop notions of twisted spinor bundle and twisted pre-quantum bundle on quasi-Hamiltonian G-spaces. The main result of this paper is that we construct a Dirac operator with index given by positive energy representation of loop group. This generalizes the quantization of Hamiltonian $G$-spaces to quasi-Hamiltonian…

New Haantjes operators extend bi-hamiltonian systems.

We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result …

We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated as a zero curvature on a loop group. As an application, we show in details why such tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.

The image of the Gauss map of any oriented isoparametric hypersurface of the unit standard sphere $S^{n+1}(1)$ is a minimal Lagrangian submanifold in the complex hyperquadric $Q_n({\mathbf C})$. In this paper we show that the Gauss image of a compact oriented isoparametric hypersurface with $g$ distinct constant princi…

Characterizes symplectic and variational operators for scalar evolution equations.

We study the family of $α$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_μ(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold correspondi…

Study first BGG operators on homogeneous geometries.

The author calculates curvatures for homogeneous sub-Riemannian manifolds using specific riggings.

This paper extends the evolution operator to contact mechanics, linking Lagrangian and Hamiltonian formulations.

An invariant description of Bianchi Homogeneous (B.H.) 3-spaces is presented, by considering the action of the Automorphism Group on the configuration space of the real, symmetric, positive definite, $3\times 3$ matrices. Thus, the gauge degrees of freedom are removed and the remaining (gauge invariant) degrees, are th…

Starting from a Lie algebroid ${\cal A}$ over a space V we lift its action to the canonical transformations on the principle affine bundle ${\cal R}$ over the cotangent bundle $T^*V$. Such lifts are classified by the first cohomology $H^1({\cal A})$. The resulting object is the Hamiltonian algebroid ${\cal A}^H$ over $…

Solves Nekhoroshev's problem on invariant tori for Hamiltonian systems with cyclic variables.

This paper shows that Hamiltonians and operators can also be put to good use even in contexts which are not purely physics based. Consider the world of finance. The work presented here {models a two traders system with information exchange with the help of four fundamental operators: cash and share operators; a portfol…

Given a Poisson structure (or, equivalently, a Hamiltonian operator) $P$, we show that its Lie derivative $L_τ(P)$ along a vector field $τ$ defines another Poisson structure, which is automatically compatible with $P$, if and only if $[L_τ^2(P),P]=0$, where $[\cdot,\cdot]$ is the Schouten bracket. We further prove that…

In this paper, the method of approximate transformation groups which was proposed by Baikov, Gazizov and Ibragimov, is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and ap…

A Lie group G in a group pair (D,G), integrating a Lie algebra g in a Manin pair (d,g) has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-h…

Develops global pseudo-differential calculus on homogeneous vector bundles.

We consider infinite dimensional port-Hamiltonian systems. Based on a power balance relation we introduce the port-Hamiltonian system representation where we pay attention to two different scenarios, namely the non-differential operator case and the differential operator case regarding the structural mapping, the dissi…

We find flat band Hamiltonians and Ginsparg-Wilson relations for symmetry classes.

Study BGG operators on homogeneous conformal geometries.

Quantizes Stäckel integrable systems into self-adjoint operators.

We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and Kupershmidt's deformation of a bi-Hamiltonian system.

We present infinitely many nonlocal conservation laws, a pair of compatible local Hamiltonian structures and a recursion operator for the equations describing surfaces in three-dimensional space that admit nontrivial deformations which preserve both principal directions and principal curvatures (or, equivalently, the s…

The study confirms essential self-adjointness for certain differential operators on manifolds.

The paper identifies curvature patterns in 3D homogeneous spaces.