Set Of First Integrals Research Articles

A generalization of the Lax equation defined by an arbitrary anti-automorphism

We consider a class of evolution equations on the Lie group GL(n,) or any of its closed subgroups, built by means of an arbitrary anti-automorphism of the associative algebra of all real n-dimensional matrices n × n. The set of first integrals and a method of construction for a Hamiltonian subclass is shown. This subclass has a connection with the factorization problem. A certain application of a matrix evolution equation built by means of transposition, related to the existence of (2,0)- and (0,2)-type tensor invariants in the theory of dynamical systems, is found.

A comparison of three methods of constructing Lyapunov functions

Chetayev's effective method [1] for constructing Lyapunov functions in the form of a set of first integrals of the equations of perturbed motion has been widely used since the 1950s in Russia. In the 1980s the energy-Casimir method [2] was developed in the U.S.A. as well as the energy-momentum method [3], employed for Hamiltonian systems. A comparison of these methods for systems with a finite number of degrees of freedom has shown that the energy-Casimir method is a more complicated version of Chetayev's method, while the energy-momentum method is essentially the Routh-Lyapunov method [4,5], stated in modern geometrical language. Some examples are considered.

Spinning particles in Schwarzschild spacetime

The motion of pseudo-classical spinning particles in a static, spherically symmetric spacetime (as described by the Schwarzschild coordinate system) is analysed. The constants of motion are derived and the full set of first integrals of motion is obtained. Various types of orbits are discussed. The exact solution for planar orbits is given.

AN INTEGRABLE SYSTEM EXTENDING THE KORTEWEG-DE VRIES EQUATION

Some exact solutions are found for a system of two nonlinear equations admitting a Lax representation. The dynamics of the scattering data of a fourth-order operator of a special form is indicated, and the system is shown to possess a countable set of first integrals.

SOME PROPERTIES OF THE COMPLEXIFICATION OF THE KdV EQUATION

An integrable complexification of the hierarchy of the KdV equation is constructed. A countable set of first integrals is found, along with the evolution of the scattering data for the complexification of the KdV equation obtained in [1]. Hirota's method is used to obtain some exact solutions of this equation. A representation of the complexification of the KdV equation as an equation of zero curvature is indicated. Bibliography: 10 titles.

BREAKING SOLITONS. IV

It is shown that a new nonlinear equation is Hamiltonian and has a countable set of first integrals. A construction is given of a two-dimensional modified equation as well as of a Lax representation for it. Bibliography: 9 titles.

BREAKING SOLITONS. III

It is shown that a new nonlinear equation is Hamiltonian and has a countable set of first integrals. A construction is given of a two-dimensional modified equation as well as of a Lax representation for it. Bibliography: 9 titles.

Integrable generalizations of non-linear multiple three-wave interaction models

Integrable generalizations of multiple three-wave interaction models in terms of r-matrix formulation are investigated. The Lax representations, complete sets of first integrals in involution are constructed, the quantization leading to Gaudin's models is discussed.

ALGEBRAIC CONSTRUCTIONS OF CERTAIN INTEGRABLE EQUATIONS

For differential equations in an arbitrary associative algebra, invariant constructions are presented that are connected with automorphisms of the algebra and admit a Lax representation. A Lax representation is found for a certain Hamiltonian integrodifferential equation, along with explicit formulas for a countable set of first integrals of it.

Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation

Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M n with measure μ. A construction of matrix equations having a set of attractors in the space of all matrix entries is given.

First Integrals and the Residual Solution for Orthotropic Plates in Plane Strain or Axisymmetric Deformations

Two classes of exact solutions are derived for the equations of three dimensional linear orthotropic elasticity theory governing flat (plate) bodies in plane strain or axisymmetric deformations. One of these is the analogue of the Lévy solution for plane strain deformations of isotropic plates and is designated as the interior solutions. The other complementary class correspond to the Papkovich‐Fadle Eigenfunction solutions for isotropic rectangular strips and is designated as the residual solutions. For sufficiently thin plates, the latter exhibits rapid exponential decay away from the plate edges. A set of first integrals of the elasticity equations is also derived. These first integrals are then transformed into a set of exact necessary conditions for the elastostatic state of the body to be a residual state. The results effectively remove the asymptoticity restriction of rapid exponential decay of the residual state inherent in the corresponding necessary conditions for isotropic plate problems. The requirement of rapid exponential decay effectively limits their applicability to thin plates. The result of the present paper extend the known results to thick plate problems and to orthotropic plate problems. They enable us to formulate the correct edge conditions for two‐dimensional orthotropic thick plate theories with stress or mixed edge data.

SOME CONSTRUCTIONS OF INTEGRABLE DYNAMICAL SYSTEMS

New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg-de Vries equation. For an arbitrary space with finite measure and measure-preserving mapping differential equations are constructed on the space of measurable functions on . Here differentiation is with respect to time and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals. Bibliography: 18 titles.

On symmetries and first integrals of dynamical systems

A previous result according to which, given a symmetry vector of a dynamical system, under certain co-ordinate-dependent conditions, a first integral could be expressed as the divergence of the symmetry vector has been generalized. By introducing convenient connections it is found that not only one but a set of first integrals can, in general, be interpreted as divergences of a single symmetry vector. It is also shown that any symmetry vector is a linear combination of a maximal independent set of symmetry vectors of the same dynamical system with first integrals as coefficients. This leads to other relationships between first integrals andsets of symmetry vectors. Finally it is seen how a type-(r, s) tensor invariant of the one-parameter group generated by a dynamical system yieldsr generally different type-(r−1),s) tensor invariants. This is, in a sense, a further generalization of the former result. The type-(r, s) tensor invariant also leads to a type-(r, s+1) tensor invariant of the same dynamical system.

Symmetry of equations of motion for a free particle in riemann space

Spaces are classified in which the equations of motion of a free test particle admit a partial separation of variables aided by a set of first integrals of the second order.

Integration of dynamic equations with constraint multipliers: PMM vol. 36, n≗1, 1972, pp. 163–171

By generalizing the unfinished investigation of Jacobi [1], Suslov showed in [2] and [3] that, knowing the complete integral W of a first order partial differential equation we can construct a set of first integrals of the equations of motion of a mechanical system with constraint multipliers. Suslov confined himself to the case when the constraints imposed on the system are specified in a finite form. For such systems the set of the first integrals mentioned above defines the general solution of the equations of motion, thus providing a method of integrating these equations. A problem of extending this method to cover nonholonomic systems engaged the efforts of the authors of [4–10], [9–12], [13, 14]. The author of [6] established that in the case of nonholonomic constraints the integral W must satisfy an additional system of equations, therefore the set of first integrals indicated by Suslov, generally speaking, defines a particular solution of the equations of motion. The authors of [12, 13] (see also the dissertation of E. Kh. Naziev “Certain Problems of Analytical Dynamics”, MGU, 1969) obtained the necessary and sufficient conditions which must be imposed on the equations of constraints and on the integral W, in an explicit form. When these conditions hold, the Suslov method can be used to obtain a particular solution of equations with multipliers of nonholonomic constraints. The conditions however are all different. In Sect. 1 of the present paper we show that the most general of these conditions [12] are not necessary and we obtain the necessary conditions. In Sect. 2 we prove the sufficiency of our conditions and show that the sufficiency of [10, 12] conditions follows as a particular case. Compatibility of the equations determining the integral W was studied in [10, 12] by the usual method of constructing all possible Poisson brackets. In Sects. 3 and 4 we show that the integral W needs not, in fact, satisfy the equations obtained by making the Poisson brackets all equal to zero. The case of a homogeneous sphere rolling on a plane without slipping is used as an example.

Remarks on the Nature of Relativistic Particle Orbits

Instantaneous action-at-a-distance relativistic particle dynamics, embraced in Newtonian-like equations of motion χ̈i = Fi(χi, χ̇i) with suitable F's, is examined in once-integrated or ``kinematical'' form χ̇i = fi(χi, Vi) with Vi a set of first integrals transforming as velocities. The Lorentz covariance requirements on fi are worked out and illustrative examples are given, including a family of many-valued ones. A general meaning for integrals of χ̈i = Fi being in involution is adduced, and general counterparts to some well-known theorems in Hamiltonian dynamics are obtained accordingly. A novel elementary proof of the zero-interaction theorem is appended.