Maciej Blaszak

Multi-Hamiltonian theory of dynamical systems

Preliminary considerations elements of differential calculus for tensor fields the theory of Hamiltonian and bi-Hamiltonian systems lax representations of multi-Hamiltonian systems multi-Hamiltonian soliton particles multi-Hamiltonian finite dimensional systems multi-Hamiltonian lax dynamics. (Part contents)

Classical R-matrices on Poisson algebras and related dispersionless systems

Abstract An R -matrix theory on algebras of formal Laurent series is considered. On this basis a multi-Hamiltonian Lax dynamics are constructed, related to dispersionless mKP class of systems and dispersionless (1+2)-Harry Dym class of systems.

On separability of bi-Hamiltonian chain with degenerated Poisson structures

Separability of bi-Hamiltonian finite-dimensional chains with two degenerated Poisson tensors, which have Pfaffian quasi-bi-Hamiltonian representation, is proved.

Lie algebraic approach to the construction of (2+1)-dimensional lattice-field and field integrable Hamiltonian equations

Two different methods for the construction of (2+1)-dimensional integrable lattice-field and field Hamiltonian dynamical systems are presented. The first method is based on the so-called central extension procedure applied to the Lie algebra of shift operators and the Lie algebra of pseudodifferential operators. The second method is the so-called operand formalism. Both methods allow a construction of some new integrable nonlinear Hamiltonian lattice-field and field equations in (2+1)-dimensional space.

Generalized Stäckel transform and reciprocal transformations for finite-dimensional integrable systems

We present a multiparameter generalization of the Stackel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stackel transform preserves Liouville integrability, noncommutative integrability and superintegrability. The corresponding transformation for the equations of motion proves to be nothing but a reciprocal transformation of a special form, and we investigate the properties of this reciprocal transformation. Finally, we show that the Hamiltonians of the systems possessing separation curves of apparently very different form can be related through a suitably chosen generalized Stackel transform.

Theory of separability of multi-Hamiltonian chains

The theory of separability of one-Casimir bi-Hamiltonian chains is extended onto unsplit multi-Casimir bi-Hamiltonian chains. Multi-Casimir extensions of the known one-Casimir chains are constructed.

Bi-Hamiltonian separable chains on Riemannian manifolds

Abstract Bi-Hamiltonian separable chains in so-called Nijenhuis coordinates, which are quadratic in the momentum variables, are related with a large classes of Stackel systems.

Separation of variables in quasi-potential systems of bi-cofactor form

We perform variable separation in the quasi-potential systems of equations of the form ¨ q =− A −1 ∇k =− ˜ A −1 ∇ ˜ k ,w hereA and ˜ A are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux–Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions.

From Bi-Hamiltonian Geometry to Separation of Variables: Stationary Harry-Dym and the KdV Dressing Chain

Abstract Separability theory of one-Casimir Poisson pencils, written down in arbitrary coordinates, is presented. Separation of variables for stationary Harry-Dym and the KdV dressing chain illustrates the theory.

Maximal superintegrability of Benenti systems

For a class of Hamiltonian systems, naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.