On the accuracy of conservation of adiabatic invariants in slow-fast Hamiltonian systems

Let the adiabatic invariant of action variable in a slow-fast Hamiltonian system with two degrees of freedom have limits along the trajectories as time tends to plus and minus infinity. The difference of these two limits is exponentially small in analytic systems. An isoenergetic reduction and canonical transformations are applied to transform the slow-fast system to form of a system depending on a slowly varying parameter in a complexified phase space. On the basis of this method an estimate for the accuracy of conservation of adiabatic invariant is given.     

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable Hamiltonian systems. II

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence of adiabatic invariants associated with a slowly perturbed Hamiltonian system. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999.     

Hamiltonian systems and Vasil’ev invariants

It is proved that a dynamical system related to the first-order Vasil’ev invariants exactly coincides with the system of Cartesian vortices on the plane. Also, the general form of Hamiltonian systems corresponding to the second-order Vasil’ev invariant is written and some properties of such systems are indicated Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

Biasymptotic solutions of perturbed integrable Hamiltonian systems

We prove that small perturbations of a real analytic integrable Hamiltonian system ind degrees of freedom generically have biasymptotic orbits which are obtained as intersections of the stable and unstable manifolds of invariant hyperbolic tori of dimensiond–1. Hence, these solutions will be forward and backward asymptotic to such a torus and not to a periodic solution. The generic condition, which is open and dense, is given by an explicit condition on the averaged perturbation.     

Geometric and dynamical invariants of integrable Hamiltonian and dissipative systems

This paper presents results concerning the geometric invariant theory of completely integrable Hamiltonian systems and also the classification of integrable cases of low-dimensional and high-dimensional rigid body dynamics in a nonconservative force field. The latter problems are described by dynamical systems with variable dissipation. The first part of the work is the basis for the doctorial dissertation of V. V. Trofimov (1953–2003), which was already published in parts. However, in the present complete form, it has not appeared, and we decided to fill in this gap. The second part is a development of the results presented in the doctoral dissertation of M. V. Shamolin and has not appeared in the present variant. These two parts complement one another well, which initiated this work (its sketches already appeared in 1997).     

Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical -matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 264–273, 1982.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

---

Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

Geometric integration of Hamiltonian systems perturbed by Rayleigh damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter ε, and the schemes under study preserve the symplectic structure in the case ε=0. In the case 0<ε≪1 the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted.     

The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects

In this paper, we study the existence of multiple solutions for a class of second-order impulsive Hamiltonian systems. We give some new criteria for guaranteeing that the impulsive Hamiltonian systems with a perturbed term have at least three solutions by using a variational method and some critical points theorems of B. Ricceri. We extend and improve on some recent results. Finally, some examples are presented to illustrate our main results.     

Controllability of the perturbed adiabatic oscillator

In this paper, we consider the global controllability with bounded controls of the perturbed adiabatic oscillator. More explicitly, we study the equationÿ+(1+g(t))y=u, whereg(t) is a suitably small perturbation term andu is the control function, which is restricted either by norm or by range. We give sufficient conditions for global null controllability, under various conditions ong(t), whenu lies in a unit ball or is constrained to assume only positive values.     

Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I

By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. Academician. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal Vol. 51, No. 10, pp. 1379–1390, October 1999.     

Bundles and geometric structures associated with gyroscopic systems

The overview is devoted to topological and geometric structures associated with gyroscopic systems whose action functional S is multivalued. The expediency of constructing and studying them is, in particular, stipulated by the fact that the standard methods of the calculus of variations in the problem with fixed endpoints are not effective for such functionals. One of the methods for overcoming the difficulties arising here is the application of bundles, foliations, connections, and also Riemannian and Lorentz manifolds. In this way, it turns out to be possible to perform the reduction of the two-point problem for S to problems with fixed initial point and movable endpoint for the length functional {ie828-01} of a pseudo-Riemannian manifold foliated over the configurational space of the gyroscopic system considered. As the endpoint manifolds, the leaves of the Riemannian foliation are used, and the correspondence between the extremals of the functionals S and {ie828-02} is stated by using the Ehresmann connection of this bundle. The paper discusses the results on the motions of natural mechanical systems with gyroscopic forces and gyroscopic systems of relativistic type obtained by using the above reduction and also the topological and geometric constructions used in it. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

Applied aspects of the theory of adiabatic invariants

We consider the applicability of the theory of adiabatic invariants to determining the reaction of a linear oscillator that simulates the inverse problem of measurement technology taking account of the dynamics of a sensing element. An indication is given of the applicability of the ideas and methods developed in the papers of Ya. S. Podstrigach and his school to the solution of this circle of problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 163–167.     

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Hölder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Hölder continuous.