Roman G. Smirnov

Group invariant classification of separable Hamiltonian systems in the Euclidean plane and the O(4)-symmetric Yang–Mills theories of Yatsun

We present a new and effective method of determining separable coordinate systems for natural Hamiltonians with two degrees of freedom in flat Riemannian space. The method is based on intrinsic properties of the associated Killing tensors and their invariants under the group of rigid motions E(2). Applications to the Hamiltonian systems derived by the late V. A. Yatsun from O(4)-symmetric Yang–Mills theories are presented. In addition, an equivalence between separability of two-dimensional Hamiltonian systems and the existence of Pfaffian quasi-bi-Hamiltonian representations is specified.

A geometrical approach to the problem of integrability of Hamiltonian systems by separation of variables

Abstract We propose a geometrical approach to the problem of integrability of Hamiltonian systems of low dimensions using the Hamilton–Jacobi method of separation of variables, based on the method of moving frames. As an illustration we present a complete classification of all separable Hamiltonian systems defined in two-dimensional Riemannian manifolds of arbitrary curvature and a criterion for separability. Connections to bi-Hamiltonian theory are also found.

Intrinsic Characterizations of Orthogonal Separability for Natural Hamiltonians with Scalar Potentials on Pseudo-Riemannian Spaces

Abstract Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijenhuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined in two-dimensional locally flat Lorenzian spaces is presented.

A class of Liouville-integrable Hamiltonian systems with two degrees of freedom

A class of two-dimensional Liouville-integrable Hamiltonian systems is studied. Separability of the corresponding Hamilton–Jacobi equation for these systems is shown to be equivalent to other fundamental properties of Hamiltonian systems, such as the existence of the Lax and bi-Hamiltonian representations of certain fixed types. Applications to physical models, including the Calogero–Moser model, an integrable case of the Henon-Heiles potential and the nonperiodic Toda lattice are presented.

Benenti's theorem and the method of moving frames

Abstract We apply the classical method of moving frames to the problem of classification of Hamiltonian systems with separable potentials defined in a pseudo-Riemannian space of arbitrary curvature. The approach is based on Benentis criterion of orthogonal separability.

Best uniform restricted ranges approximation. II

Abstract A new theory of best uniform restricted approximation of complex valued functions is established.

A systematic study of the Toda lattice in the context of the Hamilton-Jacobi theory

Abstract. Integrability of the Toda lattice is studied in the framework of the Hamilton-Jacobi theory of separation of variables. It is shown based on the Benenti theory that only the two dimensional, non-periodic Toda lattice is separable in the sense of the existence of a point transformation to a separable system of coordinates. The separation of variables is exhibited in this case explicitly.

The action-angle coordinates revisited: bi-Hamiltonian systems

Abstract We present a constructive scheme based on the action-angle coordinates to the problem of generating compatible bi-Hamiltonian structures related to a given completely integrable Hamiltonian systems. The approach is applied to the Kepler problem, for which a plethora of invariant quantities is derived.