Time-Dependent First Integrals,Nonlinear Dynamical Systems,and Numerical Integration

Many nonlinear dynamical systems expressed as autonomous systems of first-order ordinary differential equations admit first integrals and explicitly time-dependent first integrals. Under numerical integration these first integrals should be preserved. We discuss this case for explicitly time-dependent first integrals.     

A note on the integrability conditions for the existence of rational first integrals of the geodesic equations in a Riemannian space

Rational first integrals of the equation determining the geodesies of a given Riemannian space are investigated. It is shown that integrability conditions can be obtained for the existence of such integrals, the conditions being analogous to those associated with linear first integrals but, unfortunately, of a vastly more complicated form.     

关于线性阻尼振子第一积分的研究

通过引入一维线性阻尼振子基本积分来构造其他第一积分, 包括不含时的积分. 将这种方法推广到多维情形, 构造二维和n维线性阻尼振子不同形式的第一积分; 证明不同类型的二维线性阻尼振子都存在三个独立的不含时的第一积分, n维线性阻尼振子存在2n-1个独立的不含时的第一积分. 利用变量变换将线性阻尼振子的第一积分变换成简谐振子形式的第一积分. 关键词： 线性阻尼振子 第一积分 基本积分 简谐振子     

Nonlinear dynamical systems,first integrals,Bose operators,and Lie algebras

A connection between nonlinear autonomous systems of ordinary differential equations, first integrals, Bose operators and Lie algebras is described. An extension to nonlinear partial differential equations is given.     

First integrals versus configurational invariants and a weak form of complete integrability

A confusion over the concept of first integrals, which has been created in a recent paper by Hall [13] is clarified. The clear distinction between first integrals and functions which are first integrals only on a specific, fixed hypersurface is discussed. Hall's terminology of configurational invariants is adopted for the latter case. The possible relevance of knowing configurational invariants for a Hamiltonian system is illustrated by results concerning a weak form of the theory on complete integrability.     

Dielectric constant of polarizable,nonpolar fluids and suspensions

We study the dielectric constant of a polarizable, nonpolar fluid or suspension of spherical particles by use of a renormalized cluster expansion. The particles may have induced multipole moments of any order. We show that the Clausius-Mossotti formula results from a virtual overlap contribution. The corrections to the Clausius-Mossotti formula are expressed with the aid of a cluster expansion. The integrands of the cluster integrals are expressed in terms of two-body nodal connectors which incorporate all reflections between a pair of particles. We study the two- and three-body cluster integrals in some detail and show how these are related to the dielectric virial expansion and to the first term of the Kirkwood-Yvon expansion.     

转动相对论系统动力学的积分理论

建立转动相对论系统动力学方程的积分理论.给出系统运动的第一积分,分别利用系统的循环积分和能量积分降阶运动方程,得到推广的Routh方程和推广的Whittaker方程,建立系统运动的正则方程和变分方程,并由第一积分构造系统的积分不变量.给出系统的Poincaré-Cartan型积分变量关系和积分不变量. 关键词： 转动相对论 运动方程 积分方法     

Lie and Noether symmetries of geodesic equations and collineations

The Lie symmetries of the geodesic equations in a Riemannian space are computed in terms of the special projective group and its degenerates (affine vectors, homothetic vector and Killing vectors) of the metric. The Noether symmetries of the same equations are given in terms of the homothetic and the Killing vectors of the metric. It is shown that the geodesic equations in a Riemannian space admit three linear first integrals and two quadratic first integrals. We apply the results in the case of Einstein spaces, the Schwarzschild spacetime and the Friedman Robertson Walker spacetime. In each case the Lie and the Noether symmetries are computed explicitly together with the corresponding linear and quadratic first integrals.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoflan system, the first integrals and the construction of integral invariants are studied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfect differential method. Secondly, the equations of nonsimultaneous variation of the system are established by using the relation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the first integral and the integral invariant of the system is studied, and it is proved that, using a t~rst integral, we can construct an integral invarlant of the system. Finally, the relation between the relativistic Birkhoflan dynamics and the relativistic Hamilton;an dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltonian system are obtained. Two examples are given to illustrate the application of the results.     

First Integrals and Integral Invariants of Relativistic Birkhoffian Systems

For a relativistic Birkhoffian system, the first integrals and the construction of integral invariants arestudied. Firstly, the cyclic integrals and the generalized energy integral of the system are found by using the perfectdifferential method. Secondly, the equations of nonsimultaneous variation of the system are established by using therelation between the simultaneous variation and the nonsimultaneous variation. Thirdly, the relation between the firstintegral and the integral invariant of the system is studied, and it is proved that, using a first integral, we can construct anintegral invariant of the system. Finally, the relation between the relativistic Birkhoffian dynamics and the relativisticHamiltonian dynamics is discussed, and the first integrals and the integral invariants of the relativistic Hamiltoniansystem are obtained. Two examples are given to illustrate the application of the results.     

First Integrals,Liouville Theorem,and Dirac Brackets

In this paper, we discuss the conditions for the existence of first integrals of movement and the Liouville theorem on integrable systems. We revise the core results of the Hamilton-Jacobi theory and discuss the extension of the formalism to encompass constrained systems using Dirac brackets, originally developed in the context of the canonical quantization of constrained systems. As an application, we analyze a Hamiltonian that represents the classical limit of a Fermionic system of oscillators.     

Radial integrals, oscillator strengths, and polarizabilities of the thallium, lead and bismuth atoms

The radial integrals for transitions of 6p-electrons to the excited states in the Tl, Pb and Bi atoms have been found by numerical integration of the Dirac equation with an effective potential. The radial integrals for excitation of 6s- and 5d-electrons from the closed subshells have been obtained from analysis of the available experimental data on the oscillator strengths in Tl, Hg, Au and the polarizability of Hg. Using these radial integrals, we determine the polarizabilities of the Tl, Pb and Bi atoms by a simple method that is similar to Sternheimer's method.     

Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

We revisit the work of the first named author and using simpler algebraic arguments we calculate integrals of polynomial functions with respect to the Haar measure on the unitary group U(d). The previous result provided exact formulas only for 2d bigger than the degree of the integrated polynomial and we show that these formulas remain valid for all values of d. Also, we consider the integrals of polynomial functions on the orthogonal group O(d) and the symplectic group Sp(d). We obtain an exact character expansion and the asymptotic behavior for large d. Thus we can show the asymptotic freeness of Haar-distributed orthogonal and symplectic random matrices, as well as the convergence of integrals of the Itzykson–Zuber type. B.C. is supported by a JSPS postdoctoral fellowship. P.Ś. was supported by State Committee for Scientific Research (KBN) grant 2 P03A 007 23.     

Integrable nonlinear equations and Liouville's theorem,I

A symplectic structure for stationary Lax equations of the type [L, P]=0 is constructed, whereL is a matrix differential operator of the first order. It is shown that the equation has a sufficient for the complete integrability amount of first integrals in involution. The well-known linearization of the equation by the Abelian mapping is obtained in a natural manner in consequent exercising of Liouville's procedure.     

Proper rational and analytic first integrals for asymmetric 3-dimensional Lotka-Volterra systems

We go beyond in the study of the integrability of the classical model of competition between three species studied by May and Leonard [19], by considering a more realistic asymmetric model. Our results show that there are no global analytic first integrals and we provide all proper rational first integrals of this extended model by classifying its invariant algebraic surfaces.     

On algebraic construction of certain integrable and super-integrable systems

We propose a new construction of two-dimensional natural bi-Hamiltonian systems associated with a very simple Lie algebra. The presented construction allows us to distinguish three families of super-integrable monomial potentials for which one additional first integral is quadratic, and the second one can be of arbitrarily high degree with respect to the momenta. Many integrable systems with additional integrals of degree greater than two in momenta are given. Moreover, an example of a super-integrable system with first integrals of degree two, four and six in the momenta is found.     

Darboux integrability of a generalized Friedmann-Robertson-Walker Hamiltonian system

We study the Darboux first integrals of a generalized Friedmann-Robertson-Walker Hamiltonian system.