Integrals of Motion

The Lagrangian formalism is used to derive covariant equations that are suitable for use in continuously distributed matter in curved spacetime. Special attention is given to theoretical representation, in which the Lagrangian and its derivatives are directly involved. The obtained results, including equation for metric, equation of motion, equations for fields, are applied to purely vector fields. As a consequence, formulas are determined for calculating the basic quantities necessary to describe physical systems. In this case, not only the pressure field and the acceleration field are taken into account, but also the electromagnetic and gravitational fields outside the matter, which contribute to the four-momentum and to the fourdimensional angular momentum pseudotensor of each system. It is shown that the canonical representation of the angular momentum pseudotensor is its representation with covariant indices. The radius-vector of the center of momentum of a physical system is determined in covariant form.

We study nonlocal reductions of coupled equations in 1 + 1 dimensions of the Heisenberg ferromagnet type. The equations under consideration are completely integrable and have a Lax pair related to a linear bundle in pole gauge. We describe the integrable hierarchy of nonlinear equations related to our system in terms of generating operators. We present some special solutions associated with four distinct discrete eigenvalues of scattering operator. Using the Lax pair diagonalization method, we derive recurrence formulas for the conserved densities and find the first two simplest conserved densities. Contents 1 Introduction 1 2 Lax Representation 3 3 Integrable Hierarchies 4 4 Special Solutions 6 5 Integrals of Motion 11 6 Conclusion 13

The hierarchy of equations belonging to two different but related integrable systems, the nonlinear Schrödinger (NLS) and its derivative variant the derivative nonlinear Schrödinger (DNLS), is subjected to two distinct deformation procedures, viz., quasi-integrable deformation that generally does not preserve the integrability, with the system only asymptotically integrable, and non-holonomic deformation that does. Quasi-integrable deformation is carried out generically for the NLS hierarchy, while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No quasi-integrable anomaly is observed at the level of equations of motion which suggests that at that level, the quasi-integrable deformation may be identified as some integrable deformation. Non-holonomic deformation is applied to the NLS hierarchy generally, with the specific focus on the NLS equation itself and on the coupled Korteweg-de Vries type NLS equation. For the DNLS hi...

In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

We explore properties and representations of first and second order spectral shift functions evaluated for pairs of self-adjoint matrices guided by examples. We examine the conditions sufficient for the first and second order spectral shift functions to be identically 0 in 3.11, 4.10, and 4.13.

Jacobean integral of motion of a system of two artificial cable connected satellites in elliptical orbit under several influences of general nature is obtained. Cable connecting the two satellites is light, flexible, non-conducting and elastic in nature. The influences of general nature are earth’s magnetic field

Abstract: We consider the St\" ackel transform, also known as the coupling-constant metamorphosis, which under certain conditions turns a Hamiltonian dynamical system into another such system and preserves the Liouville integrability. We show that the corresponding transformation for the equations of motion is nothing but the reciprocal transformation of a special form and we investigate the properties of this transformation. This result is further applied for the study of the $ k $-hole deformations of the Benenti systems ...

A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU (m + n)/S(U (m) × U (n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.

We study derivative nonlinear Schrödinger equations related to symmetric spaces of the type A.III. We discuss the spectral properties of the corresponding Lax operator and develop the direct scattering problem connected to it. By applying an appropriately chosen dressing factor we derive soliton solutions to the nonlinear equation. We find the integrals of motion by using the method of diagonalization of Lax pair.

We consider quadratic bundles related to Hermitian symmetric spaces of the type SU (m + n)/S(U (m) × U (n)). We discuss the spectral properties of scattering operator, develop the direct scattering problem associated with it and stress on the effect of reduction on these. By applying a modification of Zakharov-Shabat's dressing procedure we demonstrate how one can obtain reflectionless potentials. That way one is able to generate soliton solutions to the nonlinear evolution equations belonging to the integrable hierarchy associated with quadratic bundles under study.

We present a multiparameter generalization of the Stäckel transform (the latter is also known as the coupling-constant metamorphosis) and show that under certain conditions this generalized Stäckel 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 Stäckel transform.

The covariant expressions are derived for the energy, momentum, and angular momentum of an arbitrary physical system of particles and vector fields acting on them. These expressions are based on the Lagrange function of the system, are the additive integrals of motion, and are conserved over time in closed systems. The angular momentum pseudotensor and the radius-vector of the system's center of momentum are determined in a covariant form. By integrating the motion equation over the volume the integral vector is calculated and the impossibility of treatment of the integral vector as the system's four-momentum is proved as opposed to how it is done in the general theory of relativity. In contrast to the system's four-momentum, which collectively characterizes the motion of the system's particles in the surrounding fields, the physical meaning of the integral vector consists in the taking account of all the energies and energy fluxes of the fields generated by the particles. The difference between the four-momentum and the integral vector is associated not only with the duality of particles and fields, but also with different transformation laws for four-vectors and four-tensors of second order. As a result, the integral vector turns out to be a pseudovector of a special kind.

Preprint version of the paper (no subscription required): http://arxiv.org/abs/nlin/0701015

We establish quantum and classical exact solvability for two large classes of maximally superintegrable Benenti systems in n dimensions with arbitrarily large n. Namely, we solve the Hamilton-Jacobi and Schrödinger equations for the systems in question. The results obtained are illustrated for a model with the cubic potential.

We consider the Klein-Gordon equation in generalized higher-dimensional Kerr-NUT-(A)dS spacetime without imposing any restrictions on the functional parameters characterizing the metric. We establish commutativity of the second-order operators constructed from the Killing tensors found in [J. High Energy Phys. 02 (2007) 004] and show that these operators, along with the first-order operators originating from the Killing vectors, form a complete set of commuting symmetry operators (i.e., integrals of motion) for the Klein-Gordon equation. Moreover, we demonstrate that the separated solutions of the Klein-Gordon equation obtained in [J. High Energy Phys. 02 (2007) 005] are joint eigenfunctions for all of these operators. We also present an explicit form of the zero mode for the Klein-Gordon equation with zero mass. In the semiclassical approximation we find that the separated solutions of the Hamilton-Jacobi equation for geodesic motion are also solutions for a set of Hamilton-Jacobi-type equations which correspond to the quadratic conserved quantities arising from the above Killing tensors.