periodic solutions with prescribed energy for singular conservative systems involving strong forces this paper establishes under appropriate conditions which are stated below the existence of periodic solutions of the equation ii t vv q t 0 1 1 where q t e rn n 1 2 and v i is the gradient of a real function i which has a singularity at the origin in the literature the problem of the existence of periodic solutions of 1 1 has been addressed following two different points of view i given t 0 find periodic solutions of 1 1 having t as period fixed period problem ii given h e r find periodic solutions of 1 1 such that for all t e r

multiple brake orbits for some classes of singular hamiltonian systems from which one can easily construct a periodic solution of period 2t by reflecting q around t 0 t if q x e e 1 v x hj is homeomorphic to a ball then the existence of one brake orbit is a classical result of seifert 11 generalizations of that result as well as alternative proofs were given in 2 6 141 in the case in which the topology of sz is richer then the existence of multiple solutions has been proved by bolotin and kozlov in 7 see also 8 in this paper we are mainly concerned with the case in which the potential v has singularities inside 2 in the sense that v x co as x x e ai and the topology of q is nontrivial more precisely in section 2 we shall assume that sz has a certain number of holes k and that v behaves like lxi o1 with 0 cx 2 near x 0 and we shall show that 1 3 has at least k classical brake orbits let us point out that the main interest of this result lies in the fact that we are able to prove that the solutions we find are classical ones that is they do not collide with the singularity while most of the papers dealing with existence of fixed energy 9 or fixed period lo solutions for singular potentials only prove existence of generalized solutions that is solutions which can collide with the singularity existence of a noncollision periodic solution for a singular potential

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

computationally efficient stepup and stepdown procedures for real predictor coefficients algorithms are presented for the stepup and stepdown problem for the real case it is shown that the split levinson algorithm and similar algorithms can be used to reduce the multiplicative complexity of the problem considerably

toda lattices and positive entropy integrable systems this paper studies completely integrable hamiltonian systems on t where is a mathbb t n 1 bundle over mathbb t n with an split free abelian monodromy group for each periodic toda lattice there is an integrable hamiltonian system on t with positive topological entropy bolsinov and taimanov s example of an integrable geodesic flow with positive topological entropy fits into this general construction with the a 1 1 toda lattice topological entropy is used to show that the flows associated to non dual toda lattices are typically topologically non conjugate via an energy preserving homeomorphism the remaining cases are approached via the homology spectrum an energy preserving conjugacy implies the congruence of two rational quadratic forms over the unit group of a number field f when f is normal a classification of flows is obtained in degree 3 this results from a well known result of gelfond in higher degrees the result is conditional on the conjecture that a rationally independent set of logarithms of algebraic numbers is algebraically independent over

two dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties abstract a multi layered model for frictionless contact analysis of functionally graded materials fgms with arbitrarily varying elastic modulus under plane strain state deformation has been developed based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves the fgm is divided into several sub layers and in each sub layers the shear modulus is assumed to be linear function while the poisson s ratio is assumed to be a constant with the model the frictionless contact problem of a functionally graded coated half space is investigated by using the transfer matrix method and fourier integral transform technique the problem is reduced to a cauchy singular integral equation the contact pressure contact region and indentation are calculated for various indenters by solving the equations numerically

sliding frictional contact between a rigid punch and a laterally graded elastic medium abstract analytical and computational methods are developed for contact mechanics analysis of functionally graded materials fgms that possess elastic gradation in the lateral direction in the analytical formulation the problem of a laterally graded half plane in sliding frictional contact with a rigid punch of an arbitrary profile is considered the governing partial differential equations and the boundary conditions of the problem are satisfied through the use of fourier transformation the problem is then reduced to a singular integral equation of the second kind which is solved numerically by using an expansion collocation technique computational studies of the sliding contact problems of laterally graded materials are conducted by means of the finite element method in the finite element analyses the laterally graded half plane is discretized by quadratic finite elements for which the material parameters are specified at the centroids flat and triangular punch profiles are considered in the parametric analyses the comparisons of the results generated by the analytical technique to those computed by the finite element method demonstrate the high level of accuracy attained by both methods the presented numerical results illustrate the influences of the lateral nonhomogeneity and the coefficient of friction on the contact stresses

the problem of two fixed centers bifurcations actions monodromy abstract a comprehensive analysis of the euler jacobi problem of motion in the field of two fixed attracting centers is given first classically and then quantum mechanically in semiclassical approximation the system was originally studied in the context of celestial mechanics but starting with pauli s dissertation became a model for one electron molecules such as h 2 symmetric case of equal centers or hhe2 asymmetric case of different centers the present paper deals with arbitrary relative strength of the two centers and considers separately the planar and the three dimensional problems all versions represent non trivial examples of integrable dynamics and are studied here from the unifying point of view of the energy momentum mapping from phase space to the space of integration constants the interesting objects are the critical values of this mapping i e its bifurcation diagram and their pre images which organize the foliation of phase space into liouville arnold tori the classical analysis culminates in the explicit derivation of the action variable representation of iso energetic surfaces the attempt to identify a system of global actions smoothly dependent on the integration constants wherever these are non critical leads to the detection of monodromy of a special kind which is here described for the first time the classical monodromy has its counterpart in the quantum version of the two center problem where it prevents the assignments of unique quantum numbers even though the system is separable

two dimensional thermoelastic contact problem of functionally graded materials involving frictional heating the two dimensional thermoelastic sliding frictional contact of functionally graded material fgm coated half plane under the plane strain deformation is investigated in this paper a rigid punch is sliding over the surface of the fgm coating with a constant velocity frictional heating with its value proportional to contact pressure friction coefficient and sliding velocity is generated at the interface between the punch and the fgm coating the material properties of the coating vary exponentially along the thickness direction in order to solve the heat conduction equation analytically the homogeneous multi layered model is adopted for treating the graded thermal diffusivity coefficient with other thermomechanical properties being kept as the given exponential forms the transfer matrix method and fourier integral transform technique are employed to convert the problem into a cauchy singular integral equation which is then solved numerically to obtain the unknown contact pressure and the in plane component of the surface stresses the effects of the gradient index peclet number and friction coefficient on the thermoelastic contact characteristics are discussed in detail numerical results show that the distribution of the contact stress can be altered and therefore the thermoelastic contact damage can be modified by adjusting the gradient index peclet number and friction coefficient

spectra of sol manifolds arithmetic and quantum monodromy the spectral problem of three dimensional manifolds m 3 a admitting sol geometry in thurston s sense is investigated topologically m 3 a are torus bundles over a circle with a unimodular hyperbolic gluing map a the eigenfunctions of the corresponding laplace beltrami operators are described in terms of modified mathieu functions it is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form as a result the spectral statistics is shown to disagree with the berry tabor conjecture the topological nature of the monodromy for both classical and quantum systems on sol manifolds is demonstrated

on local description of two dimensional geodesic flows with a polynomial first integral in this paper we present a construction of multiparametric families of two dimensional metrics with a polynomial first integral of arbitrary degree in momenta such integrable geodesic flows are described by solutions of some semi hamiltonian hydrodynamic type system we give a constructive algorithm for the solution of the derived hydrodynamic type system i e we found infinitely many conservation laws and commuting flows thus we were able to find infinitely many particular solutions of this hydrodynamic type system by the generalized hodograph method therefore infinitely many particular two dimensional metrics equipped with first integrals polynomial in momenta were constructed

integrability of hamiltonian systems and transseries expansions this paper studies analytic liouville nonintegrable and c liouville integrable hamiltonian systems with two degrees of freedom we prove the property for a class of hamiltonians more general than the one studied in gorni and zampieri differ geom appl 22 287 296 2005 we also show that a certain monodromy property of an ordinary differential equation obtained as a subsystem of a given hamiltonian and the transseries expansion of a first integral play an important role in the analysis cf 4 in the former half the analytic liouville nonintegrability for a class of hamiltonians satisfying the above condition is shown for these analytic nonintegrable hamiltonians one cannot construct nonanalytic first integrals concretely as in gorni and zampieri differ geom appl 22 287 296 2005 in the latter half the nonanalytic integrability from the viewpoint of a transseries expansion of a first integral is discussed more precisely we construct a first integral in a formal transseries expansion in a general situation then we show convergence of transseries or existence of the first integral being asymptotically equal to a given formal transseries solution

splitting of some s domain stability test algorithms some classic s domain algorithms are first expressed in a unified form from this all possible split forms are then derived and classified in particular attention is focused on two families of algorithms which include the standard routh algorithm and an s domain levinson like algorithm respectively the related split forms are analysed and compared their complexity can be reduced by suitably choosing some scaling factors it turns out that the usual form of the routh algorithm which is intrinsically split and the simplified version of the split levinson like algorithm are substantially equivalent and superior to any other possible split algorithm

the remainder in weyl s law for heisenberg manifolds we prove that the error term r in weyl s law is o 5 6 for certain three dimensional heisenberg manifolds we also show that the l 2 norm of the weyl error term integrated over the moduli space of left invariant heisenberg metrics is 3 4 we conjecture that r o 3 4 i s a sharp deterministic upper bound for heisenberg three manifolds

topologically distinct collision free periodic solutions for the n center problem this work concerns the planar n center problem with homogeneous potential of degree alpha alpha in 1 2 the existence of infinitely many topologically distinct non collision periodic solutions with a prescribed energy is proved a notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the n center problem with alpha in 1 2 is proven in case alpha 1 a slightly different result is shown it is the case that there is either a non collision periodic solution or a collision reflection solution the results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions the proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments

a minimax method for a class of hamiltonian systems with singular potentials abstract this paper presents a minimax method which gives existence and multiplicity results for time periodic solutions of a class of hamiltonian systems when a singular potential is present the singularity satisfies the strong force condition of gordon when milder singularities are permitted a notion of generalized t periodic solution is introduced and we get existence and multiplicity results for such solutions

periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials abstract we study the existence of periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials we found periodic solutions lying between two prescribed spheres

complete involutive algebras of functions on cotangent bundles of homogeneous spaces homogeneous spaces of all compact lie groups admit riemannian metrics with completely integrable geodesic flows by means of c smooth integrals 9 10 the purpose of this paper is to give some constructions of complete involutive algebras of analytic functions polynomial in velocities on the co tangent bundles of homogeneous spaces of compact lie groups this allows us to obtain new integrable riemannian and sub riemannian geodesic flows on various homogeneous spaces such as stiefel manifolds flag manifolds and orbits of the adjoint actions of compact lie groups

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

chaotic scattering by steep repelling potentials consider a classical two dimensional scattering problem a ray is scattered by a potential composed of several tall repelling steep mountains of arbitrary shape we study when the traditional approximation of this nonlinear far from integrable problem by the corresponding simpler billiard problem of scattering by hard wall obstacles of similar shape is justified for one class of chaotic scatterers named here regular sinai scatterers the scattering properties of the smooth system indeed limit to those of the billiards for another class the singular sinai scatterers these two scattering problems have essential differences though the invariant set of such singular scatterers is hyperbolic possibly with singularities that of the smooth flow may have stable periodic orbits even when the potential is arbitrarily steep it follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards deviation from a singular scatterer thus even in this singular case the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow finally we see that corners have nontrivial and significant impact on the scattering functions

axisymmetric frictionless contact of a functionally graded piezoelectric layered half space under a conducting punch abstract this paper studies the axisymmetric frictionless contact of a functionally graded piezoelectric layered half space subjected to three typical rigid punches i e flat circular punch spherical punch and conical punch the punch is assumed as a perfect conductor with a constant electric potential the electro mechanical properties of the functionally graded piezoelectric materials fgpms vary exponentially along the thickness direction by employing the hankel integral transform technique the problem is reduced to a set of singular integral equations of the first kind these equations are then discretized and solved numerically to determine the unknown normal contact pressure and electric charge the effects of the gradient index and punch geometry on the surface electro mechanical contact behaviors are discussed in detail

closed orbits of fixed energy for singular hamiltonian systems this paper deals with the existence of periodic solutions of q v q 0 such that 1 2 q 2 v q h where q r n h is a given number v c 2 r n 0 r has a singularity at x 0 and v denotes the gradient of v

tonelli hamiltonians without conjugate points and c 0 integrability we prove that all the tonelli hamiltonians defined on the cotangent bundle t t n of the n dimensional torus that have no conjugate points are c 0 integrable i e t t n is c 0 foliated by a family fc of invariant c 0 lagrangian graphs assuming that the hamiltonian is c infty we prove that there exists a g delta subset gc of fc such that the dynamics restricted to every element of gc is strictly ergodic moreover we prove that the lyapunov exponents of every c 0 integrable tonelli hamiltonian are zero and deduce that the metric and topological entropies vanish

on the integrability of the n centre problem it is known that for n 3 centres and positive energies the n centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy here we consider the energies above some threshold and show whereas for arbitrary g 1 independent integrals of gevrey class g exist no real analytic that is gevrey class 1 independent integral exists

comparison between some stability criteria of discrete time filters routh s algorithm is written in the form of polynomial recursion and transposed for testing stability in the wide sense of discrete time linear systems the algorithm obtained is a three term recursion of symmetric polynomials this recursion easily gives both the coefficients appearing in the continued fraction expansion associated to a given polynomial in the z domain and a computation scheme for the bilinear transformation of polynomials finally this recursion is compared to other algorithms already known for testing stability

a stability test for continuous systems abstract a test for determining the zero distribution of a real polynomial with respect to the imaginary axis is presented it is based on the construction of a sequence of polynomials of descending degree which draws on the levinson algorithm the proof of the stability criterion which is based on intuitive geometrical considerations is very simple and the computational complexity is not greater than that for the routh test the critical situations that may arise are also examined and some examples are given

a note on periodic solutions of prescribed energy for singular hamiltonian systems abstract in this note we show under a less restrictive setting than previously that there exists a collision brake orbit of prescribed energy to a certain second order hamiltonian system

a generalised test for discrete system stability in this correspondence we show that by using the continued fraction expansion 8 twice we can determine whether the discrete system is stable or not and if yes the number of real and complex roots appearing inside the unit circle

a prescribed energy problem for a singular hamiltonian system with a weak force we consider the existence of periodic solutions of a hamiltonian system q v q 0 hs such that 12 q t 2 v q t h for all t where q rn n 3 h 0 we prove the existence of a generalized solution that may enter the singularity 0 moreover under the assumption v q 1 q w q where 0 2 and q w q q 1 w q q 2 2w q 0 as q 0 we estimate the number of collisions of generalized solutions in particular we get the existence of a classical non collision solution of hs for l 2 when n 4 and for 4 3 2 when n 3

zero entropy non integrable geodesic flows and a non commutative rotation vector let g be a 2 step nilpotent lie algebra we say g is non integrable if for a generic pair of points p p g the isotropy algebras do not commute g p g p 0 theorem if g is a simply connected 2 step nilpotent lie group g lie g is non integrable d g is a cocompact subgroup and g is a left invariant riemannian metric then the geodesic flow of g on t d g is neither liouville nor non commutatively integrable with c first integrals the proof uses a generalization of the rotation vector pioneered by benardete and mitchell