Integrability of natural Hamiltonian systems with homogeneous potentials of degree zero

We derive necessary conditions for integrability in the Liouville sense of natural Hamiltonian systems with homogeneous potential of degree zero. We derive these conditions through an analysis of the differential Galois group of variational equations along a particular solution generated by a non-zero solution \vd\in\C^n of nonlinear equations \grad V(\vd)=\vd. We proved that if the system integrable then the Hessian matrix V''(\vd) has only integer eigenvalues and is semi-simple.Comment: 13 page

Non-perturbative non-integrability of non-homogeneous nonlinear lattices induced by non-resonance hypothesis

We prove the non-integrability (non-existence of additional analytic conserved quantities other than Hamiltonian) for Fermi-Pasta-Ulam (FPU) lattices by virtue of Lyapunov-Kovalevskaya- -Ziglin-Yoshida's monodromy method about the variational equations. The key to this analysis is that the normal variational equations along a certain solution happen to be in a type of Lam\'e equations. We also introduce the classification problem towards non-homogeneous nonlinear lattices including FPU lattices using non-integrability preserving transformation.Comment: Latex, 21 pages, to appear in Physica D (1996), ps.Z file available at http://www.bip.riken.go.jp/irl/chaosken/reulam.ps.

The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on $S^3$ and $L^3$. An analogue of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of the Newtonian center moving along a geodesic on $S^2$ and $L^2$ (the restricted two-body problem). When the curvature is small, the pericenter shift is computed using the perturbation theory. We also present the results of the numerical analysis based on the analogy with the motion of rigid body.Comment: 29 pages, 7 figure

Classical nonintegrability of a quantum chaotic SU(3) Hamiltonian system

We prove nonintegrability of a model Hamiltonian system defined on the Lie algebra $\mathfrak{su}_3$ suitable for investigation of connections between classical and quantum characteristics of chaos.Comment: 17 page

Finiteness of integrable nn-dimensional homogeneous polynomial potentials

We consider natural Hamiltonian systems of $n>1$ degrees of freedom with polynomial homogeneous potentials of degree $k$. We show that under a genericity assumption, for a fixed $k$, at most only a finite number of such systems is integrable. We also explain how to find explicit forms of these integrable potentials for small $k$

Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve μ2=νn−1,n∈Z\mu^2=\nu^n-1, n\in{\Bbb Z}: ergodicity, isochrony, periodicity and fractals

We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically periodic with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behaviour. We suggest a possible theoretical explanation of these different behaviours. We also introduce a one-parameter family of 2-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such mapping the center map. Computer experiments for the center map show a typical multi-fractal behaviour with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.Comment: LaTex, 28 pages, 10 figure

On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $\sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.Comment: 7 page

A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta

We searched integrable 2D homogeneous polynomial potential with a polynomial first integral by using the so-called direct method of searching for first integrals. We proved that there exist no polynomial first integrals which are genuinely cubic or quartic in the momenta if the degree of homogeneous polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge

Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial Differential Equations

The automation of the traditional Painleve test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of ordinary and partial differential equations which may be parameterized by arbitrary functions (or constants). Except where limited by memory, there is no restriction on the number of independent or dependent variables. The package is quite robust in determining all the possible dominant behaviors of the Laurent series solutions of the differential equation. The omission of valid dominant behaviors is a common problem in many implementations of the Painleve test, and these omissions often lead to erroneous results. Finally, our package is compared with the other available implementations of the Painleve test.Comment: Published in the Journal of Nonlinear Mathematical Physics (http://www.sm.luth.se/math/JNMP/), vol. 13(1), pp. 90-110 (Feb. 2006). The software can be downloaded at either http://www.douglasbaldwin.com or http://www.mines.edu/fs_home/wherema

Chaos around Holographic Regge Trajectories

Using methods of Hamiltonian dynamical systems, we show analytically that a dynamical system connected to the classical spinning string solution holographically dual to the principal Regge trajectory is non-integrable. The Regge trajectories themselves form an integrable island in the total phase space of the dynamical system. Our argument applies to any gravity background dual to confining field theories and we verify it explicitly in various supergravity backgrounds: Klebanov-Strassler, Maldacena-Nunez, Witten QCD and the AdS soliton. Having established non-integrability for this general class of supergravity backgrounds, we show explicitly by direct computation of the Poincare sections and the largest Lyapunov exponent, that such strings have chaotic motion.Comment: 28 pages, 5 figures. V3: Minor changes complying to referee's suggestions. Typos correcte