Third and fourth order invariants for one-dimensional time-dependent classical systems

The construction of invariants up to fourth order in velocities has been carried out for one-dimensional, time-dependent classical dynamical systems. While the exact results are recovered for the first and second order integrable systems, the results for the third and fourth order invariants are expressed in terms of nonlinearpotential equations. Noticing the separability of the potential in space and time variables these nonlinear equations are reduced to a tractable form. A possible solution for the third order case suggests a new integrable systemV(q, t) ∼t ^−4/3 q ^1/2. Alexander von Humboldt-Stiftung Fellow, on leave from the Department of Physics, Ramjas College (University of Delhi), Delhi 110 007, India.     

Construction of exact dynamical invariants of two-dimensional classical system

A general method is used for the construction of second constant of motion of fourth order in momenta using the complex coordinates (z, _z ^- ). A fourth-order potential equation is obtained whose solutions directly provide a large class of integrable systems. The potential equation is tested with an interesting example which admits second constants of motion.     

Further examples of integrable systems in two dimensions

The construction of the second constant of motion of second order for two-dimensional classical systems is carried out in terms ofz=q ₁ +iq ₂ andq=q ₁ −iq ₂. As a result a class of Toda-type potentials admitting second order invariants is explored.     

Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant

In this paper we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be isotropic and the magnetic field is applied along the z-axis. The canonical transformation is invoked to remove the interaction term and the system is reduced to a model containing the second harmonic generation. Two classes of the real and complex quadratic invariants (constants of motion) are obtained. We have employed the Lie algebraic technique to find the most general solution for the wave function for both real and complex invariants. Some discussions related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself are also given.     

Quantum Invariants for Solving the Time-Dependent Schrödinger Equation in One Dimension

Extending the work of Lewis and Leach on classical invariants for solving the classical equation of motion in one-dimensional system, the quantum invariants in polynomial form of momentum are obtained. The involved Hamiltonian is time-dependent and quadratic in momentum.     

Integrable classical systems in higher dimensions

A general method for the construction of the second constant of motion (up to second order) for higher-dimensional classical systems is carried out. Correspondingly, the first- and the second-order potential equations are obtained whose solutions can directly provide the integrable systems.     

Spin dynamics in orthogonal fields

The technique of Lorentz transformations is used to obtain exact solutions of the classical vector and tensor equations of motion of a spin in constant homogeneous and orthogonal fields (E < H). It is shown that the vector and tensor variants of the solutions go over into one another uniquely. The invariants of the motion are written down explicitly. The results of the paper agree with the quantum theory of the motion of a spin.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 44–48, January, 1981.     

A new perturbative approach to the classical anharmonic oscillator

The periodic motion of the classical anharmonic oscillator characterized by the potentialV(x)=1/2x ²+λ/2k x ^2k is considered. The period is first determined to all orders inλ in a perturbative series. Making use of this, the solution of the nonlinear equation of motion is then expressed in the form of a Fourier series. The Fourier coefficients are obtained by solving simple algebraic relations. Secular terms are inherently absent in this perturbative scheme. Explicit solution is presented for generalk up to the second order, from which the Duffing and the sextic oscillator results follow as special cases.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

A relativistic analogue of the Kepler problem

The Poincaré invariant system of two point particles with an instantaneous interaction-at-a-distance originally proposed by Fokker is studied in the Hamiltonian formalism. The interaction, which agrees to first order in the coupling constant with the electromagnetic one obtained from the Liénard-Wiechert fields, is described in an advanced-retarded state space. The first particle moves in the advanced field of the second which in turn is subject to the retarded field of the first. The acceleration terms in the Liénard-Wiechert fields are neglected. In this theory the state space of the system is a twelve-dimensional manifold Σ and the motions are described as integral curves of a vector field that is obtained as the projection of the generator of time translations in space-time. The Poincaré group acts on this manifold Σ in a well-defined way and leaves a symplectic form ω invariant. Thus the set of all possible motions of this system can be studied by the methods of modern symplectic mechanics. In this paper the general method is explained and the set of all bounded motions for two equal rest masses and an attractive force is studied qualitatively and numerically. In the limit (binding energy)/(sum of rest masses) · (speed of light)² → 0 all the features of the classical Kepler motion are obtained.     

Differentiability of invariants and Liapunov equations in Hamiltonian systems

In this communication, we investigate the behavior of the derivatives of invariants for Hamiltonian systems, using information derived from an analysis of the Liapunov exponents of the system. We show that under certain conditions on the analyticity properties of the solutions of the equations of motion, it is possible to construct 2n invariants of motion which are guaranteed to beC ^∞ as functions of phase space and time in a suitably defined domainD.     

The ground state properties of the He4-nucleus

Conclusions Table VIII summarizes the computed binding energy of He⁴-nucleus which includes the zeroth order contribution and the correction up to the third order for three different potentials. The binding energy does not contain the C.M. energy, which has been calculated up to the second order. The r.m.s. radii corrected for the C.M. motion and for not-point-like nucleons are calculated up to the second order for potential RHEL 1 and up to the first order for Reid and RHEL 2 potentials. The binding energy and r.m.s. radii are computed for two different self-consistent conditions, the first of which is the usual classical condition (2.16), the second reads E ⁽¹⁾ = 0.In all cases the absolute value of the binding energy of the He⁴-nucleus is lower than the experimental value.It has been shown that the perturbation series built up on the Goldstone reaction matrix diverges, when a self-consistent technique is not used. This represents certain danger also for the self-consistent formulation, although it gives plausible results up to the third order. The result obtained seems to indicate that the discrepancy between the experimental and theoretical values for the binding energy may be caused by neglect of some fundamental facts (relativistic effects, many-body forces etc.) in the present many-body theory.The authors would like to thank the Rutherford High Energy Laboratory for the help in performing necessary computations and for the encouraging interest in this problem.     

Motion of extended bodies in general relativity and gravitational radiation

In the framework of general relativity, we consider the motion of extended spherically symmetric bodies coupled together only by gravitational interactions. The self-gravitational radiation of the bodies is taken into account. The problem is solved by the Fock method of successive approximations in harmonic coordinates, assuming slow motion and weak fields (v²/c² U/c^{2 2} 1) with terms up to and including the fifth order in taken into account. The equations of motion of the two bodies are derived, including the gravitational radiation reaction terms. It is shown that the system is conservative up to fifth order but to second order in e the six classical integrals of the motion are replaced by only two (the total energy and total angular momentum). An induced rotation effect in a system of initially nonrotating bodies is obtained. It is shown that in the fifth order approximation the system is nonconservative because of gravitational radition. An expression is obtained for the rate of loss of energy from the system directly from the equations of motion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 12–16, January, 1985.     

Localized and periodic wave patterns for a nonic nonlinear Schrödinger equation

Propagating modes in a class of ‘nonic’ derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by application of two key invariants of motion. A nonlinear equation for the squared wave amplitude is derived thereby which allows the exact representation of periodic patterns as well as localized bright and dark pulses in terms of elliptic and their classical hyperbolic limits. These modes represent a balance among cubic, quintic and nonic nonlinearities.     

Kauffman knot polynomials in classical abelian Chern-Simons field theory

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant t^I(L) is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, t^I satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t^I satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.     

NMR line broadening in solids by slowing down of spin fluctuations

The¹⁰⁹Ag nuclear magnetic resonance line in a sample of polycrystalline AgF is observed to broaden substantially when the¹⁹F spins are irradiated near the magic angle in their rotating frame. This is due to the reduction of¹⁹F—¹⁹F dipolar coupling, which normally causes fluctuations in the¹⁹F—¹⁰⁹Ag interactions (Abragam and Winter), inducing an exchange narrowing analogous to classical motional narrowing. The¹⁰⁹Ag linewidths obtained over the entire motional range at different¹⁹F frequencies are compared with those calculated exactly from the ratio of second to fourth moment.     

微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量

把极角θ视为独立变量,得到Kepler系统的轨道微分方程.首先讨论Kepler系统轨道微分方程的Lie对称性和不变量,微扰Kepler系统轨道微分方程的精确Lie对称性和精确不变量,其次讨论微扰Kepler系统轨道微分方程的近似Lie对称性和近似不变量,并得到了微扰Kepler系统的9个一阶近似Lie对称性和6个一阶近似不变量,其中1个实为精确不变量,而其余5个分别等于微扰系数ε乘以Kepler系统相应的5个不变量。     

Integrable Substructure in a Korteweg Capillarity Model. A Karman-Tsien Type Constitutive Relation

A classical Korteweg capillarity system with a Karman-Tsien type (κ, ρ) constitutive relation is shown, via a Madelung transformation and use of invariants of motion, to admit integrable Hamiltonian subsystems.     

Nuclear rotational-vibrational collective motion with nonvanishing vortex-spin

The theory of linear collective motion is developed by the method of canonical transformations, recovering, as special cases, the earlier results of Casimir, Bohr-Mottelson, and Villars. In the approximation of constant Eckart-frame vectors, the kinetic energy Hamiltonian is shown to commute with the invariant operators of the collective motion symmetry group CM(3). The collective motion approach of Tomonaga, and the symmetry approach of Gell-Mann, are discussed and shown to be essentially equivalent, and to be properly contained in the CM(3) structure. The invariant operators of CM(3) are determined and shown to imply two invariants for nuclear collective motion: the volume Λ and the vortex-spin v. Representations of CM(3) are obtained and related to wavefunctions of the generator-coordinate form.     

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 -?² plus a quantum deformation of order ?⁴ and ?⁶. The systems inside the classes are transformed using Stäckel transforms in the quantum case as in the classical case. The general form of the Stäckel transform between superintegrable systems is discussed.