Related papers: Geometry of jet spaces and integrable systems
We discover a fundamental exterior differential system of Riemannian geometry; indeed, an intrinsic and invariant global system of differential forms of degree $n$ associated to any given oriented Riemannian manifold $M$ of dimension $n+1$.…
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects, for which the infinitesimal configuration…
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beitr\"{a}ge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in…
Main ideas of the differential geometry on affine bundles are presented. Affine counterparts of Lie algebroid and Poisson structures are introduced and discussed. The developed concepts are applied in a frame-independent formulation of the…
We define a jet-space analog of the BV-Laplacian, avoiding delta-functions and infinite constants; instead we show that the main properties of the BV-Laplacian and its relation to the Schouten bracket originate from the underlying jet-space…
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…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for…
The real plane with its set of orientations or angles in $[0,\pi)$ is the simplest non trivial example of a (projective) Hilbert space and provides nice illustrations of quantum formalism. We present some of them, namely covariant integral…
The paper describes the algebraic structure of the graded algebra of differentially homogeneous polynomials of fixed finite order. We show that it is a finitely generated algebra, and we exhibit a minimal set of generators. Along the way,…
This survey examines separation of variables for algebraically integrable Hamiltonian systems whose tori are Jacobians of Riemann surfaces. For these cases there is a natural class of systems which admit separations in a nice geometric…
A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a bivector field over the space of solutions of the Euler-Lagrange equations of a…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
The Eisenhart geometric formalism, which transforms an Euclidean natural Hamiltonian $H=T+V$ into a geodesic Hamiltonian ${\cal T}$ with one additional degree of freedom, is applied to the four families of quadratically superintegrable…
This talk introduces a Cartan-geometric framework for generalised geometries governed by a differential graded Lie algebra. In contrast to ordinary Cartan geometry, the tangent bundle is extended and qu both a global duality group and a…
This work is devoted to a systematic exposition of the dynamics of a rigid body, considered as a system with kinematic constraints. Having accepted the variational problem in accordance with this, we no longer need any additional postulates…
A systematic construction of St\"{a}ckel systems in separated coordinates and its relation to bi-Hamiltonian formalism are considered. A general form of related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is derived. One…
The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex…
We present examples of Lax-integrable multi-dimensional systems of partial differential equations with higher local symmetries. We also consider Lagrangian deformations of these equations and construct variational bivectors on them.
Basic concepts and definitions in differential geometry and topology which are important in the theory of solitons and instantons are reviewed. Many examples from soliton theory are discussed briefly, in order to highlight the application…