Integrals Of Motion In Involution Research Articles

Second order Killing tensors related to symmetric spaces

We discuss the pairs of quadratic integrals of motion belonging to the n-dimensional space of independent integrals of motion in involution, that provide integrability of the corresponding Hamiltonian equations of motion by quadratures. In contrast to the Eisenhart theory, additional integrals of motion are polynomials of the fourth, sixth and other orders in momenta. The main focus is on the second-order Killing tensors corresponding to quadratic integrals of motion and relating to the special combinations of rotations and translations in Euclidean space.

On rotationally invariant integrable and superintegrable classical systems in magnetic fields with non-subgroup type integrals

The aim of the present article is to construct quadratically integrable three dimensional systems in non-vanishing magnetic fields which possess so-called non-subgroup type integrals. The presence of such integrals means that the system possesses a pair of integrals of motion in involution which are (at most) quadratic in momenta and whose leading order terms, that are necessarily elements of the enveloping algebra of the Euclidean algebra, are not quadratic Casimir operators of a chain of its subalgebras. By imposing in addition that one of the integrals has the leading order term we can consider three such commuting pairs: circular parabolic, oblate spheroidal and prolate spheroidal. We find all possible integrable systems possessing such structure of commuting integrals and describe their Hamiltonians and their integrals.We show that our assumptions imply the existence of a first order integral Lz, i.e. rotational invariance, of all such systems. As a consequence, the Hamilton–Jacobi equation of each of these systems with magnetic field separates in the corresponding coordinate system, as it is known to be the case for all quadratically integrable systems without magnetic field, and in contrast with the subgroup type, i.e. Cartesian, spherical and cylindrical, cases, with magnetic fields.We also look for superintegrable systems within the circular parabolic integrable class. Assuming the additional integral to be first order we demonstrate that only previously known systems exist. However, for a particular second order ansatz for the sought integral () we find a minimally quadratically superintegrable system. It is not quadratically maximally superintegrable but appears to possess bounded closed trajectories, hinting at hypothetical higher order superintegrability.

Spherical type integrable classical systems in a magnetic field

We show that four classes of second order spherical type integrable classical systems in a magnetic field exist in the Euclidean space , and construct the Hamiltonian and two second order integrals of motion in involution for each of them. For one of the classes the Hamiltonian depends on four arbitrary functions of one variable. This class contains the magnetic monopole as a special case. Two further classes have Hamiltonians depending on one arbitrary function of one variable and four or six constants, respectively. The magnetic field in these cases is radial. The remaining system corresponds to a constant magnetic field and the Hamiltonian depends on two constants. Questions of superintegrability—i.e. the existence of further integrals—are discussed.

An exactly solvable quantum four-body problem associated with the symmetries of an octacube

In this article, we show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6m, 2m, m, and 3m in a hard-wall box can be found exactly using Bethe Ansatz. The Ansatz is based on the exceptional affine reflection group associated with the symmetries and tiling properties of an octacube—a Platonic solid unique to four-dimensions, with no three-dimensional analogues. We also uncover the Liouville integrability structure of our problem: the four integrals of motion in involution are identified as invariant polynomials of the finite reflection group F4, taken as functions of the components of momenta.

Cubic polynomials on Lie groups: reduction of the Hamiltonian system

This paper analyzes the optimal control problem of cubic polynomials on compact Lie groups from a Hamiltonian point of view and its symmetries. The dynamics of the problem is described by a presymplectic formalism associated with the canonical symplectic form on the cotangent bundle of the semidirect product of the Lie group and its Lie algebra. Using these control geometric tools, the relation between the Hamiltonian approach developed here and the known variational one is analyzed. After making explicit the left trivialized system, we use the technique of Marsden–Weinstein reduction to remove the symmetries of the Hamiltonian system. In view of the reduced dynamics, we are able to guarantee, by means of the Lie–Cartan theorem, the existence of a considerable number of independent integrals of motion in involution.

Poisson vertex algebras in the theory of Hamiltonian equations

We lay down the foundations of the theory of Poisson vertex algebras aimed at its applications to integrability of Hamiltonian partial differential equations. Such an equation is called integrable if it can be included in an infinite hierarchy of compatible Hamiltonian equations, which admit an infinite sequence of linearly independent integrals of motion in involution. The construction of a hierarchy and its integrals of motion is achieved by making use of the so called Lenard scheme. We find simple conditions which guarantee that the scheme produces an infinite sequence of closed 1-forms $$\omega_j, j \in {\mathbb {Z}}_+$$ , of the variational complex Ω. If these forms are exact, i.e., ω j are variational derivatives of some local functionals ∫ h j , then the latter are integrals of motion in involution of the hierarchy formed by the corresponding Hamiltonian vector fields. We show that the complex Ω is exact, provided that the algebra of functions is $$\fancyscript {V}$$ is “normal”; in particular, for arbitrary $$\fancyscript {V}$$ , any closed form in Ω becomes exact if we add to $$\fancyscript {V}$$ a finite number of antiderivatives. We demonstrate on the examples of the KdV, HD and CNW hierarchies how the Lenard scheme works. We also discover a new integrable hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and demonstrate its applicability on the examples of the NLS, pKdV and KN hierarchies.

N-dimensional integrability from two-photon coalgebra symmetry

A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N − 2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie–Poisson coalgebra (h6, Δ). The set of (N − 2) constants of the motion is shown to be a universal one for all these Hamiltonians, irrespective of the dependence of the latter on several arbitrary functions and N free parameters. Within this large class of quasi-integrable N-dimensional Hamiltonians, new families of completely integrable systems are identified by finding explicitly a new independent integral through the analysis of the sub-coalgebra structure of h6. In particular, new completely integrable N-dimensional Hamiltonians describing natural systems, geodesic flows and static electromagnetic Hamiltonians are presented.

Integrable almost-symplectic Hamiltonian systems

We extend the notion of Liouville integrability, which is peculiar to Hamiltonian systems on symplectic manifolds, to Hamiltonian systems on almost-symplectic manifolds, namely, manifolds equipped with a nondegenerate (but not closed) 2-form. The key ingredient is to require that the Hamiltonian vector fields of the integrals of motion in involution (or equivalently, the generators of the invariant tori) are symmetries of the almost-symplectic form. We show that, under this hypothesis, essentially all of the structure of the symplectic case (from quasiperiodicity of motions to an analog of the action-angle coordinates and of the isotropic-coisotropic dual pair structure characteristic of the fibration by the invariant tori) carries over to the almost-symplectic case.

Exact stationary solutions of stochastically excited and dissipated partially integrable Hamiltonian systems

The concept of partially integrable Hamiltonian system is introduced. It is shown that the functional form of the exact stationary solution of a stochastically excited and dissipated, partially integrable Hamiltonian system depends upon the number of independent integrals of motion in involution and the number of resonant relations in the integrable part of the associated Hamiltonian system. Two forms of exact stationary solutions of stochastically excited and dissipated, partially integrable Hamiltonian systems with and without resonant relations in the associated Hamiltonian system are proposed and the procedures to obtain them are given. The general results are illustrated with three examples.

A finite-dimensional integrable system and the r-matrix method

The nonlinearization method is used on the soliton hierarchy associated with the 4×4 AKNS eigenvalue problem, from which a new finite-dimensional Hamiltonian system is obtained by nonlinearization of the eigenvalue problem and its ajoint one. A Lax representation is deduced for the system. The Lax operator admits the representation of an r-matrix. The 4N functionally independent and involutive integrals of motion are obtained via the r-matrix, which shows that the system is completely integrable in the Liouville sense.

Binary nonlinearization of AKNS spectral problem under higher-order symmetry constraints

Binary nonlinearization of AKNS spectral problem is extended to the cases of higher-order symmetry constraints. The Hamiltonian structures, Lax representations, r-matrices and integrals of motion in involution are explicitly proposed for the resulting constrained systems in the cases of the first four orders.The obtained integrals of motion are proved to be functionally independent and thus the constrained systems are completely integrable in the Liouville sense.

Quasi-integrable and superintegrable systems from a `deformed-mass' two-photon algebra

A quantum deformation of the two-photon (or Schrodinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n−2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n−1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.

New integrable systems: hydrogen atom in external fields

We report the discovery of two new three-dimensional conservative Hamiltonian systems which are completely integrable. In both cases, two additional integrals of motion in involution are generators of purely dynamical symmetries, i.e., they do not follow from any apparent geometrical symmetry of the system and are fairly complicated functions of canonical coordinates. Physically, the obtained Hamiltonian functions are good approximations for the description of the hydrogen atom in a circularly polarized microwave field (for low frequencies) or in crossed magnetic and electric fields (for low magnetic field). More formally, they correspond to the hydrogen atom in the presence of both a circularly polarized laser field and a generalized Van der Waals potential.

On the complete integrability of nonlinearized lax systems for the classical Boussinesq hierarchy

TheN involutive integrals of motion with linearly independent gradients for the nonlinearized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that whenn=1, 2, 3, the nonlinearized time parts of Lax systems for the CB hierarchy are transformed into three finite-dimensional integrable Hamiltonian systems under the constraint of the nonlinearized spatial part.

Direct integration of generalized Lie symmetries of nonlinear Hamiltonian systems with two degrees of freedom: integrability and separability

Many of the integrable coupled nonlinear oscillator systems are associated with generalized Lie symmetries involving velocity dependent terms. For a class of systems with two degrees of freedom, the authors show that by integrating the characteristic equation associated with the generalized symmetries, the required involutive integrals of motion can be obtained explicitly in a straightforward manner, almost by inspection and without recourse to Noether's theorem. Further, all the separable coordinates can be obtained by integrating a subset of the characteristic equation associated with the coordinate variables alone. The explicit examples include the two coupled generalized Henon-Heiles, quartic, sextic and other polynomial oscillator systems as well as the perturbed Kepler system.

Integrable Hamiltonian systems related to the polynomial eigenvalue problem

The independent integrals of motion in involution for the Hamiltonian system related to the second-order polynomial eigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.

A hierarchy of integrable Hamiltonian systems associated with φxx=(λ3−u0−λu1−λ2u2)φ

A hierarchy of completely integrable Hamiltonian systems is obtained by restricting a hierarchy of integrable evolution equations associated with φxx=(λ3−u0−λu1−λ2u2)φ to an invariant subspace of the recursion operator of these ev olution equations. The independent integrals of motion in involution and Hamiltonian functions for these systems are constructed by using the recursion formula related to the eigenvalue problem.

On the nonintegrability of Yang-Mils potentials

The nonintegrability of three Hamiltonian systems obtained by simplifications of the Yang-Mills classical field theory is proved by application of two theorems of Yoshida (1987, 1988). It is also shown that for the two four-degree of freedom systems there exists a set of three independent integrals of motion in involution.

Abelian integrals and the reduction method for an integrable Hamiltonian system

A classical finite-dimensional integrable Hamiltonian system, corresponding to the motion of a particle constrained to an n-dimensional sphere ∑nμ=0 x2μ=1, with the Hamiltonian H=∑μ( (1)/(2) y2μ+u2μ/x2μ +εαμx2μ) (where uμ, αμ, and ε are constants and yμ are the momenta conjugate to xμ), is integrated using several different methods. These are the following: (1) The projection of geodesic (free) flow on a larger space, namely the sphere S2n+1 (for ε=0). The flow is obtained in terms of elementary functions. (2) Separation of variables in the Hamilton–Jacobi equation in elliptic coordinates or, alternatively, the use of a complete set of integrals of motion in involution to reduce Hamilton’s equations to quadratures. The flow is obtained in terms of Abelian integrals which are then inverted in terms of generalized θ functions. The relation between the different methods and results is clarified using methods of algebraic geometry, in particular the geometry of quadrics.

Integrability of One Particle in a Perturbed Central Quartic Potential

We show here that the Hamiltonian (1.1) has n functionally independent integrals of motion in involution which are rational both in phase space variables and in parameters. Moreover these integrals are quadratic in momenta and the Hamilton-Jacobi equation of the system (1.1) is separable in generalized elliptic coordinates. A Lax representation for (1.1) and for higher flows is found. The system (1.1) constrained to an ellipsoid remains integrable.