the lax pairs for the holt system by using the known relation between the holt system and the henon heiles system the lax pairs for all the integrable cases of the holt system are constructed from the known lax representations for the henon heiles system

a theory of exact and approximate configurational invariants abstract a theory of integrable hamiltonian systems in two dimensions is formulated and applied the four dimensional phase space problem that is the vanishing of the poisson bracket between another invariant and the hamiltonian is here reduced to the solution of a series of two dimensional configuration space equations the theory is also applicable to more than two dimensions the constraints are found which admit to integrability of the orbits for magnetic or coriolis forces as well as for forces derivable from a potential when a system admits a given invariant the invariant is found for example by quadrature a number of examples including known and apparently previously unknown invariants are given the theory of exact integrals of the motion also can be extended to the derivation of approximate invariants the orbital structure of integrable or approximately integrable systems correlates with the degree maximum power of the velocity of a standardized invariant the theory admits a variational principle among other approximation techniques for the computation of a best approximate invariant the problem of the general cubic potential with one symmetric coordinate v 1 2 ax 2 1 2 by 2 cx 2 y 1 3 dy 3 of which the well studied henon heiles potential is the special case for a b and c d is examined in detail

periodic orbits and non integrability of henon heiles systems we apply the averaging theory of second order to study the periodic orbits for a generalized henon heiles system with two parameters which contains the classical henon heiles system two main results are shown the first result provides sufficient conditions on the two parameters of these generalized systems which guarantee that at any positive energy level the hamiltonian system has periodic orbits these periodic orbits form in the whole phase space a continuous family of periodic orbits parameterized by the energy the second result shows that for the non integrable henon heiles systems in the sense of liouville arnol d which have the periodic orbits analytically found with averaging theory cannot exist any second first integral of class in particular for any second first integral of class we prove that the classical henon heiles system and many generalizations of it are not integrable in the sense of liouville arnol d moreover the tools we use for studying the periodic orbits and the non liouville arnol d integrability can be applied to hamiltonian systems with an arbitrary number of degrees of freedom

nonisospectral scattering problems a key to integrable hierarchies we show that certain partial differential equations associated to nonisospectral scattering problems in 2 1 dimensions provide a key to associated integrable hierarchies of both ordinary and partial differential equations this is illustrated using an extension of a known second order and two new third order nonisospectral scattering problems these scattering problems allow us to derive new hierarchies of integrable partial differential equations in both 1 1 and 2 1 dimensions together with their underlying linear problems isospectral and nonisospectral and also new hierarchies of integrable ordinary differential equations again with their underlying linear problems

a generalized henon heiles system and related integrable newton equations a detailed description is given of integrable cases of the generalized henon heiles systems which differs from the standard h h ones by the term q22 their connection with fifth order one component soliton equations is discussed lax representations are constructed and the bi hamiltonian formulation of dynamics is given it is also shown that the gh h system can be mapped onto another system of newton equations with a nonstandard hamiltonian structure

non autonomous he non heiles systems abstract scaling similarity solutions of three integrable pdes namely the sawada kotera fifth order kdv and kaup kupershmidt equations are considered it is shown that the resulting odes may be written as non autonomous hamiltonian equations which are time dependent generalizations of the well known integrable henon heiles systems the time dependent hamiltonians are given by logarithmic derivatives of the tau functions inherited from the original pdes the odes for the similarity solutions also have inherited backlund transformations which may be used to generate sequences of rational solutions as well as other special solutions related to the first painleve transcendent

backlund transformations for many body systems related to kdv we present backlund transformations bts with parameter for certain classical integrable n body systems namely the many body generalized henon heiles garnier and neumann systems our construction makes use of the fact that all these systems may be obtained as particular reductions stationary or restricted flows of the kdv hierarchy alternatively they may be considered as examples of the reduced sl 2 gaudin magnet the bts provide exact time discretizations of the original continuous systems preserving the lax matrix and hence all integrals of motion and satisfy the spectrality property with respect to the backlund parameter

generalized lenard chains separation of variables and superintegrability we show that the notion of generalized lenard chains naturally allows formulation of the theory of multiseparable and superintegrable systems in the context of bi hamiltonian geometry we prove that the existence of generalized lenard chains generated by a hamiltonian function defined on a four dimensional n manifold guarantees the separation of variables as an application we construct such chains for the henon heiles systems and for the classical smorodinsky winternitz systems new bi hamiltonian structures for the kepler potential are found

quantization of chern simons gauge theory with complex gauge group the canonical quantization of chern simons gauge theory in 2 1 dimensions is generalized from the case in which the gauge group is a compact lie groupg to the case in which the gauge group is a complex lie groupg though the physical hilbert spaces become infinite dimensional in the latter case the quantization can be described as precisely as for compact gauge groups and using similar methods the special case in which the gauge group issl 2 gives a description of 2 1 dimensional quantum gravity with lorentz signature and positive cosmological constant or with euclidean signature and negative cosmological constant while it is not clear whether there is a 1 1 dimensional conformal field theory related to these 2 1 dimensional models there are natural computable candidates for the central charge and the conformal blocks of such a hypothetical theory

universality in the chaotic dynamics associated with saddle centers critical points abstract we describe a new statistical pattern characteristic of saddle center critical points in the dynamics of non integrable hamiltonian systems associated with the partition of the hamiltonian into rotational motion energy and hyperbolic motion energy pieces in a linear neighborhood of the saddle center the distribution of hyperbolic energy of orbits visiting a neighborhood of the saddle center defines a statistical law that for a large range of values of the hyperbolic energy has the form p z z in a regime of high nonintegrability we present numerical evidence to support the conjecture that this pattern is universal for any hamiltonian system with one or several saddle centers the parameter appears as a universal characteristic associated with the chaotic dynamics of saddle centers

on a new non isospectral variant of the boussinesq hierarchy we give a new non isospectral extension to 2 1 dimensions of the boussinesq hierarchy such a non isospectral extension of the third order scattering problem xxx u x v 0 has not been considered previously this extends our previous results on one component hierarchies in 2 1 dimensions associated to third order non isospectral scattering problems we characterize our entire 2 1 dimensional hierarchy and its linear problem using a single partial differential equation and its corresponding non isospectral scattering problem this then allows an alternative approach to the construction of linear problems for the entire 2 1 dimensional hierarchy reductions of this hierarchy yield new integrable hierarchies of systems of ordinary differential equations together with their underlying linear problems in particular we obtain a fourth painleve hierarchy i e a hierarchy of ordinary differential equations having the fourth painleveequation as its first member we also obtain a hierarchy having as its first member a generalization of an equation defining a new transcendent due to cosgrove

a connection between hh3 and korteweg de vries with one source in the system made of korteweg de vries with one source we first show by applying the painleve test that the two components of the source must have the same potential we then explain the natural introduction of an additional term in the potential of the source equations while preserving the existence of a lax pair this allows us to prove the identity between the traveling wave reduction and one of the three integrable cases of the cubic henon heiles hamiltonian system

non semisimple and complex gaugings of n 16 supergravity maximal and non maximal supergravities in three dimensions allow for a large variety of semisimple chern simons gauge groups in this paper we analyze non semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal n 16 gauged supergravity to exist we give a general procedure how to generate non semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within e8 8 examples include the theories with gauge group so 8 t28 that describe the reduction of iia iib supergravity on the seven sphere in addition we exhibit two strange embeddings of the complex gauge group open image in new window into real e8 8 and prove that both can be consistently gauged we discuss the structure of the associated scalar potentials as well as their relation to those of d 4 gauged supergravities

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

differential invariants of a class of lagrangian systems with two degrees of freedom abstract we consider systems of euler lagrange equations with two degrees of freedom and with lagrangian being quadratic in velocities for this class of equations the generic case of the equivalence problem is solved with respect to point transformations using lie s infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems we describe certain types of lagrangian systems in terms of their invariants the results are illustrated by several examples

lax representation for two particle dynamics splitting on two tori the lax representation in terms of matrices is constructed for separable multiply periodic systems splitting on two tori hyperelliptic kleinian functions and their reduction to elliptic functions are used

a note on integrable two degrees of freedom hamiltonian systems with a second integral quartic in the momenta two of the simplest integrable hamiltonians h x y px py px2 py2 2 v x y with a second integral quartic in the momenta are those with potentials v3 x y by 3x2 16y2 d x2 16y2 eta y and v4 x y a x4 6x2y2 8y4 c x2 4y2 vy 2 we show how v3 can be obtained from v4 in the process we obtain a new potential of the class vn that includes both v3 and v4 as particular cases for this potential we give the second integral of motion separating variables a lax representation and a bi hamiltonian structure thus synthesizing the corresponding results for potentials v3 and v4 the integrable extension vn mu x 2 is also discussed

elliptic solutions and blow up in an integrable henon heiles system we consider an integrable case of the henon heiles system and use an isomorphism with the two gap kdv flow to construct families of real elliptic trajectories which are associated with two gap elliptic solitons of the kdv equation some of these solutions exhibit blow up in finite time

quasi lagrangian systems of newton equations systems of newton equations of the form q 12a 1 q k with an integral of motion quadratic in velocities are studied these equations generalize the potential case when a i the identity matrix and they admit a curious quasi lagrangian formulation which differs from the standard lagrange equations by the plus sign between terms a theory of such quasi lagrangian newton qln systems having two functionally independent integrals of motion is developed with focus on two dimensional systems such systems admit a bi hamiltonian formulation and are proved to be completely integrable by embedding into five dimensional integrable systems they are characterized by a linear second order partial differential equation pde which we call the fundamental equation fundamental equations are classified through linear pencils of matrices associated with qln systems the theory is illustrated by two classes of systems separable potential systems and driven systems new separation variables for driven systems are foun

the he non heiles system revisited abstract the known integrable cases of the henon heiles system are shown to be closely related to the stationary flows of the known and only integrable fifth order single component and polynomial nonlinear evolution equations this is further evidence that these are the only integrable cases of the henon heiles system lax pairs are deduced for each of the integrable cases and used to construct the constants of motion a curious lax operator recently found by the painleve method is explained

integration of a generalized henon heiles hamiltonian the generalized henon heiles hamiltonian h 1 2 px2 py2 c1x2 c2y2 axy2 bx3 3 with an additional nonpolynomial term y 2 is known to be liouville integrable for three sets of values of b a c1 c2 it has been previously integrated by genus two theta functions only in one of these cases defining the separating variables of the hamilton jacobi equations we succeed here in the two other cases to integrate the equations of motion with hyperelliptic functions

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

exact solutions for a class of integrable henon heiles type systems we study the exact solutions of a class of integrable henon heiles type systems according to the analysis of bountis et al phys rev a 25 1257 1982 these solutions are expressed in terms of two dimensional kleinian functions special periodic solutions are expressed in terms of the well known weierstrass function we extend some of our results to a generalized henon heiles type system with n 1 degrees of freedom

properties of the canonical transformations of the time for the toda lattice and the henon heiles system for the toda lattice and the henon heiles system we consider properties of the canonical transformations of the extended phase space which preserve integrability at the special values of integrals of motion the integral trajectories separated variables and the action variables are invariant under change of the time on the other hand mapping of the time induces a shift of the generating function of the backlund transformation

nonlinear superposition formula for the kaup kupershmidt partial differential equation abstract the fifth order kaup kupershmidt kk equation is one of the solitonic equations related to the integrable cases of the henon heiles system as opposed to the sawada kotera equation which is its dual equation the construction of the n soliton solutions of kk is not an easy task in using a perturbation scheme or the hirota bilinear formalism from the backlund transformation obtained by singularity analysis considerations we here establish the permutability theorem for the kk equation this allows us to explain the anomalous structure of the n soliton solution empirically obtained by parker physica d 137 2000 25 33 34 48

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

study of chaos in hamiltonian systems via convergent normal forms abstract we use moser s normal forms to study chaotic motion in two degree hamiltonian systems near a saddle point besides being convergent they provide a suitable description of the cylindrical topology of the chaotic flow in that vicinity both aspects combined allowed a precise computation of the homoclinic interaction of stable and unstable manifolds in the full phase space rather than just the poincare section the formalism was applied to the henon heiles hamiltonian producing strong evidence that the region of convergence of these normal forms extends over that orginally established by moser

elliptic baker akhiezer functions and an application to an integrable dynamical system an ansatz for elliptic baker akhiezer functions is used to find new solutions for various spectral problems the sturm liouville problem with four and five gap lame potentials and two third order problems one of the third order problems is the classical halphen equation with a spectral parameter the eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized henon heiles system

stationary flows hamiltonian structures and canonical transformations abstract we consider the restriction of isospectral flows miura maps and darboux transformations to stationary manifolds specifically we present 1 i a systematic construction of hamiltonian structures on stationary manifolds 2 ii the reduction of darboux transformations to stationary manifolds using the relationship between the factorisation of the schrodinger operator and the mkdv hierarchy we derive interesting canonical transformations which preserve both hamiltonian structures and all commuting hamiltonians for the stationary flows 3 iii some interesting connections between stationary flows of integrable nonlinear evolution equations and integrable hamiltonian systems of natural type the corresponding miura maps give rise to canonical transformations between various of these integrable hamiltonian systems

newton representation of nonlinear ordinary differential equations abstract a new parametrization for higher order ode s is introduced it turns them into a set of newton equations when applied to stationary flows of soliton equations this parametrization leads to new integrable mechanical systems in the kdv case we find a new poisson bracket of 3rd order

coupled korteveg de vries hierarchy with sources and its newton decomposition a lax representation for the equations of the coupled korteveg de vries ckdv hierarchy with sources is derived from the energy dependent schrodinger spectral problem it is proved that each stationary flow of the ckdv hierarchy with sources can be reparametrized as a system of newton equations with velocity independent forces these newton systems have a lagrangian formulation and are completely integrable the developed decomposition techniques lead to construction of new infinite families of integrable classical mechanical systems

a multiple scales approach to maximal superintegrability in this paper we present a simple algorithmic test to establish if a hamiltonian system is maximally superintegrable or not this test is based on a very simple corollary of a theorem due to nekhoroshev and on a perturbative technique called multiple scales method if the outcome is positive this test can be used to suggest maximal superintegrability whereas when the outcome is negative it can be used to disprove it this method can be regarded as a finite dimensional analog of the multiple scales method as a way to produce soliton equations we use this technique to show that the real counterpart of a mechanical system found by jules drach in 1935 is in general not maximally superintegrable we give some hints on how this approach could be applied to classify maximally superintegrable systems by presenting a direct proof of the well known bertrand s theorem

extended soliton solutions for the kaup kupershmidt equation we extend the n soliton solutions of the kaup kupershmidt equation on a nonzero background decreasing as x 1 a 2 these new solutions describe the interaction of n solitary waves with a static bell shaped wave we give the conditions so that the backlund transformation relating those solutions and the n solitons of the sawada kotera equation will be satisfied

analysis of covariance and standardization as instances of prediction in this paper prediction provides the basis for unifying the procedures of covariances adjustment and standardization analysis of covariance is a method of forming predictions from a linear model it is used when qualitative effects are to be studied and the effects of continuous variables are to be adjusted for an essential feature is the division into effects of interest and effects for which adjustment is required covariates may also be qualitative as such they are used implicitly in experimental designs with blocks where treatment effects are adjusted for the effect of blocks the technique of standardization is well known in epidemiology and demography as a method of adjusting explicitly for qualitative effects the same division of effects applies when an analysis that uses generalized linear models is summarized two distinct types of prediction which give identical results in classical linear models are available prediction may be conditional on a fixed value of a covariate or marginal on a distribution of values such as the distribution in the set of data being analysed prediction methods are illustrated by the analysis of a table of proportions by use of a logit model

separability and lax pairs for henon heiles system the hamiltonian system corresponding to the generalized henon heiles hamiltonian h 1 2 px2 py2 1 2ax2 1 2by2 x2y ey3 is known to be integrable in the following three cases a b e 1 3 e 2 b 16a e 16 3 in the first two the system has been integrated by making use of genus one and genus two theta functions we show that in the third case the system can also be integrated by making use of elliptic functions finally using the fairbanks theorem we find lax pairs for each of the three integrable systems under consideration

two newton decompositions of staionary flows of kdv and harry dym hierarchies abstract we show that each stationary flow of the kdv and harry dym hierarchies of soliton equations which are 2 m 1 st order odes m 0 1 has two parametrisations as a set of newton equations with velocity independent forces forces are potential and these newton equations follow from a lagrangian function with an inefinite kinetic energy term these two parametrisations are canonically inequivalent and give rise to new bihamiltonian structures in classical mechanics lax representations for these newton equations are found

integrable maps for the garnier and for the neumann system we construct integrable maps for the garnier and for the neumann system they are related to the toda hierarchy exactly in the same way as the garnier and the neumann systems are related to the kdv hierarchy as restricted flows here we give lax representations for these maps and prove that they are completely integrable

on the integrability of stationary and restricted flows of the kdv hierarchy a bi hamiltonian formulation for stationary flows of the kdv hierarchy is derived in an extended phase space a map between stationary flows and restricted flows is constructed in one case it connects an integrable henon heiles system and the garnier system moreover a new integrability scheme for hamiltonian systems is proposed that holds in the standard phase space

coupled painleve systems and quartic potentials similarity reductions of the hirota satsuma system and another gauge related system yield non autonomous hamiltonian systems with quartic potentials we present classes of special solutions and backlund transformations which are interpreted in terms of the action of an affine weyl group on the space of parameters some other quartic oscillators related to coupled painleve type equations are briefly considered we show how separation of variables also has an application in this context

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series