On the Hamiltonian formalism for Korteweg—de Vries type hierarchies

Vladimir State Pedagogical Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 2, pp. 79–82, April–June, 1992.     

A new construction of recursion operators for systems of the hydrodynamic type

We consider a certain class of two-dimensional systems of the hydrodynamic type that contains all examples known to us and can be associated with a class of linear wave equations possessing an algebra of ladder operators. We use this to give a simple construction of recursion operators for these systems, not always coinciding with those of Sheftel and Teshukov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 37–49, January, 1999.     

Hamiltonian formalism for particles with a generalized rigidity

The Hamiltonian formalism is developed for mechanical systems described by reparametrization-invariant Lagrangians dependent on the external curvatures of a world line. The complete sets of constraints are constructed for the Lagrangians quadratic in the external curvatures, the Lagrangians proportional to an arbitrary external curvature, and the Lagrangians linear in the first two curvatures. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 1, pp. 130–139, October, 1998.     

Iterated Hamiltonian type systems and applications

We discuss, in arbitrary dimension, certain Hamiltonian type systems and prove existence, uniqueness and regularity properties, under the independence condition. We also investigate the critical case, define a class of generalized solutions and prove existence and basic properties. Relevant examples and counterexamples are also indicated. The applications concern representations of implicitly defined manifolds and their perturbations, motivated by differential systems involving unknown geometries.     

A partial Hamiltonian approach for current value Hamiltonian systems

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.     

Hamiltonian formalism in the presence of higher derivatives

We briefly review basic formulas of the Hamiltonian formalism in classical mechanics in the case where the Lagrangian contains N time derivatives of n coordinate variables. For nonlocal models, N = ∞. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 208–216, November, 2008.     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Existence and multiplicity of solutions for nonlocal systems with Kirchhoff type

Firstly, we use Nehari manifold and Mountain Pass Lemma to prove an existence result of positive solutions for a class of nonlocal elliptic system with Kirchhoff type. Then a multiplicity result is established by cohomological index of Fadell and Rabinowitz. We also consider the critical case and prove existence of positive least energy solution when the parameter β is sufficiently large.     

Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type,which do not possess Riemann invariants

We formulate several conjectures concerning the structure and properties of thenxn integrable nondiagonalizable hamiltonian systems of hydrodynamic type. Forn=3 our results are, in fact, complete: a 3×3 nondiagonalizable hamiltonian system is integrable if and only if it is weakly nonlinear (linearly degenerate).Institute for Mathematical Modelling, Academy of Sciences of Russia, 125047, Miusskaya, 4, Moscow, Russia. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 257–262, May, 1994.