Quadratic Integrals Of Motion Research Articles

Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces

Abstract Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces.

Three-Component Stäckel Model of the Galaxy Based on the Rotation Curve from Maser Data

A three-component Stackel model of the Galaxy, including the bulge, disk, and halo, is constructed. Parameter estimates of the potential are obtained as a result of fitting the model rotation curve to azimuthal velocities found from data on trigonometric parallaxes and spatial velocities of masers. The fitting method takes into account the measurement and natural dispersions of azimuthal velocities and uses an algorithm for excluding objects with excessive residuals. In order to obtain more uniform samples, the objects were divided into two groups: masers associated with high-mass star forming regions and masers of other types. A significant kinematic inhomogeneity of these groups was identified and taken into account: the azimuthal velocity dispersion is $\sigma_{0,1}=4.3\pm 0.4$~km\,s$^{-1}$, in the first group and $\sigma_{0,2}=15.2\pm1.3$~km\,s$^{-1}$ in the second. After constructing the model of the Galactic-plane potential, it was generalized to the entire space under the assumption of the existence of a third quadratic integral of motion. When reconstructing the Galactic rotation curve in detail, the used algorithm gives an analytical expression for the Stackel potential, which significantly simplifies the task of constructing the Galaxy's phase density model in the Stackel approximation. In order to make the Stackel model more realistic, one needs to develop methods of direct account of data on the vertical distribution of density in the Galaxy.

On the properties of numerical solutions of dynamical systems obtained using the midpoint method

The article considers the midpoint scheme as a finite-difference scheme for a dynamical system of the form ̇ = (). This scheme is remarkable because according to Cooper’s theorem, it preserves all quadratic integrals of motion, moreover, it is the simplest scheme among symplectic Runge-Kutta schemes possessing this property. The properties of approximate solutions were studied in the framework of numerical experiments with linear and nonlinear oscillators, as well as with a system of several coupled oscillators. It is shown that in addition to the conservation of all integrals of motion, approximate solutions inherit the periodicity of motion. At the same time, attention is paid to the discussion of introducing the concept of periodicity of an approximate solution found by the difference scheme. In the case of a nonlinear oscillator, each step requires solving a system of nonlinear algebraic equations. The issues of organizing computations using such schemes are discussed. Comparison with other schemes, including those symmetric with respect to permutation of and .̂

Quantum cosmology of Bianchi VIII, IX LRS geometries

In the present work we revisit the axisymmetric Bianchi VIII and IX models. At the classical level we reproduce the known analytic solution, in a novel way making use of two quadratic integrals of motion, the constraint equation, as well as a linear non-local integral of motion. These quantities correspond to two second rank Killing tensors and a homothetic vector field existing on the relevant configuration space. On the corresponding phase space the two quadratic charges commute with the Hamiltonian constraint but not among themselves. Thus, after turning these charges into operators we obtain two different solutions to the Wheeler DeWitt equation utilizing each of the quadratic operators. The homothetic vector is then used, as a natural guide line, to define a normalizable conditional probability which assigns zero to the classically collapsed configurations.

Hamiltonian description of nonlinear curl forces from cofactor systems

Recently, Berry and Shukla presented (J Phys A 45:305201, 2012; J Phys A 46:422001, 2013; Proc R Soc A 471:20150002, 2015) a fundamental new dynamics concerning forces (accelerations) depending only on position, i.e. without velocity-dependent dissipation, which was partly anticipated in the papers of cofactor systems introduced by the Linkoping school (Rauch-Wojciechowski et al. in J Math Phys 40:6366–6398, 1999; Lundmark in Stud Appl Math 110(3):257–296, 2003; Lundmark in Integrable nonconservative Newton systems with quadratic integrals of motion. Linkoping Studies in Science and Technology. Thesis No. 756, Linkoping Univ., Linkoping, 1999). In this paper, we extend their results to nonlinear curl forces, where the nonlinearity is with respect to the coordinate dependence of the forces, and study the Hamiltonians for homogeneous quadratic and cubic cases presenting the conditions for existence of Hamiltonian curl forces. In particular, we examine the existence and expressions of the Hamiltonian curl forces for planar systems when the accelerations are given by general (both homogeneous and inhomogeneous) second-order and homogeneous cubic polynomials, and also associate the cubic case with an example from optics.

On Superintegrable Systems with a Cubic Integral of Motion**Supported by the National Natural Science Foundation of China under Grant Nos. 11771352 and 11371293

We present a variety of superintegrable systems in polar coordinates possessing a cubic and a quadratic integral of motion, where Noether integrals of kinetic energy are used to build the integrals. In addition, the associated polynomial Poisson algebras with their algebraic dependence relations are exhibited.

A Gyrostatic Low-Order Model for the El Ñino-Southern Oscillation

The paper demonstrates that, similar to other important models of atmospheric phenomena, beginning with the celebrated Lorenz model of the Rayleigh-Bénard convection, the Vallis low-order model (LOM) of the El Ñino-Southern Oscillation admits a gyrostatic form and discusses how gyrostatic LOMs may offer a general framework for deriving effective physically sound models for atmospheric dynamics and time series analysis. Any such model has a quadratic integral of motion (interpreted as some form of energy), which eliminates unphysical behaviors that have often plagued atmospheric LOMs and paves way for developing Hamiltonian LOMs. Restricting LOMs to a gyrostatic form also helps to design LOMs of optimum size and provides a modular construction of LOMs using gyrostats as elementary building blocks.

Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems

The superintegrability of two-dimensional Hamiltonians with a position dependent mass (pdm) is studied (the kinetic term contains a factor m that depends of the radial coordinate). First, the properties of Killing vectors are studied and the associated Noether momenta are obtained. Then the existence of several families of superintegrable Hamiltonians is proved and the quadratic integrals of motion are explicitly obtained. These families include, as particular cases, some systems previously obtained making use of different approaches. We also relate the superintegrability of some of these pdm systems with the existence of complex functions endowed with interesting Poisson bracket properties. Finally the relation of these pdm Hamiltonians with the Euclidean Kepler problem and with the Euclidean harmonic oscillator is analyzed.

Two integrable systems with integrals of motion of degree four

We discuss the possibility of using second-order Killing tensors to construct Liouville-integrable Hamiltonian systems that are not Nijenhuis integrable. As an example, we consider two Killing tensors with a nonzero Haantjes torsion that satisfy weaker geometric conditions and also three-dimensional systems corresponding to them that are integrable in Euclidean space and have two quadratic integrals of motion and one fourth-order integral in momenta.

Stäackel-type Dynamic Model of the Galaxy Based on Maser Kinematic Data

Abstract A dynamic model of the Galaxy is constructed based on kinematic data for masers with trigonometric parallaxes. Maser data is used to compute the model potential in the Galactic plane. The potential is then generalized to three dimensions assuming the existence of a third quadratic integral of motion. The resulting Galactic model potential is of Stäckel’s type. The corresponding space density function is determined from Poisson’s equation.

Polynomial integrals of motion in dilaton gravity theories

Investigating modern gravity and cosmology models involves a stage of analyzing associated nonlinear dynamical systems. In general, such systems are not integrable, but they often admit additional integrals of motion. Based on features of dynamical systems appearing in the theory of dilaton gravity, we formulate a universal algorithm seeking the integrals of motion polynomial in momentum. Using this algorithm, we investigate the dilaton gravity theories with a single scalar field for the presence of linear and quadratic integrals of motion.

Canonical quantization of static spherically symmetric geometries

The conditional symmetries of the reduced Einstein–Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

Quantization of the Schwarzschild geometry

The conditional symmetries of the reduced Einstein-Hilbert action emerging from a static, spherically symmetric geometry are used as supplementary conditions on the wave function. Based on their integrability conditions, only one of the three existing symmetries can be consistently imposed, while the unique Casimir invariant, being the product of the remaining two symmetries, is calculated as the only possible second condition on the wave function. This quadratic integral of motion is identified with the reparametrization generator, as an implication of the uniqueness of the dynamical evolution, by fixing a suitable parametrization of the r-lapse function. In this parametrization, the determinant of the supermetric plays the role of the mesure. The combined Wheeler – DeWitt and linear conditional symmetry equations are analytically solved. The solutions obtained depend on the product of the two "scale factors".

On the two-dimensional classical motion of a charged particle in an electromagnetic field with an additional quadratic integral of motion

The problem of quadratic Hamiltonians with an electromagnetic field commuting in the sense of the standard Poisson brackets has been considered. It has been shown that, as in the quantum case, any such pair can be reduced to the canonical form, which makes it possible to construct the complete classification of the solutions in the class of meromorphic solutions for the main function of one variable. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables, which is similar to that in the theory of integrable tops. This transformation has been considered for the two-dimensional Hamiltonian of a charged particle with an additional quadratic integral of motion.

On a two-dimensional Schrödinger equation with a magnetic field with an additional quadratic integral of motion

The problem of commuting quadratic quantum operators with a magnetic field has been considered. It has been shown that any such pair can be reduced to the canonical form, which makes it possible to construct an almost complete classification of the solutions of equations that are necessary and sufficient for a pair of operators to commute with each other. The transformation to the canonical form is performed through the change of variables to the Kovalevskaya-type variables; this change is similar to that in the theory of integrable tops. As an example, this procedure has been considered for the two-dimensional Schrodinger equation with the magnetic field; this equation has an additional quantum integral of motion.

Quadratic algebras for three-dimensional superintegrable systems

The three-dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress, and Miller have proved that in the case of nondegenerate potentials with quadratic integrals of motion there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral implies that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. In this contribution we investigate the structure of this algebra. We show that in all the nondegenerate cases there is at least one subalgebra of three integrals having a Poisson quadratic algebra structure, which is similar to the two-dimensional case.

Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Stäckel transforms

The two-dimensional quantum superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -ħ2 plus a quantum deformation of order ħ4 and ħ6. The systems inside the classes are transformed using Stackel transforms in the quantum case as in the classical case. The general form of the Stackel transform between superintegrable systems is discussed.

Quantum superintegrable systems with quadratic integrals on a two dimensional manifold

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems, as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schrödinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but can be solved. Therefore, there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by −ℏ2 plus quantum corrections of orders ℏ4 and ℏ6.

Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold

In this paper we prove that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes.

The equation [formula omitted], multiplication of cofactor pair systems, and the Levi-Civita equivalence problem

Cofactor pair systems generalize the separable potential Hamiltonian systems. They admit n quadratic integrals of motion, they have a bi-Hamilton formulation, they are completely integrable and they are equivalent to separable Lagrangian systems. Cofactor pair systems can be constructed through a peculiar multiplicative structure of the so-called quasi-Cauchy–Riemann equations ( cof J ) − 1 ∇ V = ( cof J ̃ ) − 1 ∇ V ̃ , where J and J ̃ are special conformal Killing tensors. In this work we have isolated the properties that are responsible for the multiplication, allowing us to give an elegant characterization of systems that admit multiplication. In this characterization the equation X ∇ det X = det X ∇ tr X plays a central role. We describe how multiplication of quasi-Cauchy–Riemann equations can be considered as a special case of a more general kind of multiplication, defined on the solution space of certain systems of partial differential equations. We investigate algebraic properties of this multiplication and provide several methods for constructing new systems with multiplicative structure. We also discuss the role of the multiplication in the theory of equivalent dynamical systems on Riemannian manifolds, developed by Levi-Civita.