The Poisson equations in the nonholonomic Suslov problem: Integrability, meromorphic and hypergeometric solutions

We consider the problem of integrability of the Poisson equations describing spatial motion of a rigid body in the classical nonholonomic Suslov problem. We obtain necessary conditions for their solutions to be meromorphic and show that under some further restrictions these conditions are also sufficient. The latter lead to a family of explicit meromorphic solutions, which correspond to rather special motions of the body in space. We also give explicit extra polynomial integrals in this case. In the more general case (but under one restriction), the Poisson equations are transformed into a generalized third order hypergeometric equation. A study of its monodromy group allows us also to calculate the "scattering" angle: the angle between the axes of limit permanent rotations of the body in space

Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields

In this paper we analyse the integrability of a dynamical system describing the rotational motion of a rigid satellite under the influence of gravitational and magnetic fields. In our investigations we apply an extension of the Ziglin theory developed by Morales-Ruiz and Ramis. We prove that for a symmetric satellite the system does not admit an additional real meromorphic first integral except for one case when the value of the induced magnetic moment along the symmetry axis is related to the principal moments of inertia in a special way.Comment: 39 pages, 4 figures, missing bibliography was adde

Non-integrability of the generalised spring-pendulum problem

We investigate a generalisation of the three dimensional spring-pendulum system. The problem depends on two real parameters $(k,a)$, where $k$ is the Young modulus of the spring and $a$ describes the nonlinearity of elastic forces. We show that this system is not integrable when $k\neq -a$. We carefully investigated the case $k= -a$ when the necessary condition for integrability given by the Morales-Ramis theory is satisfied. We discuss an application of the higher order variational equations for proving the non-integrability in this case.Comment: 20 pages, 1 figur

Global integrability of cosmological scalar fields

We investigate the Liouvillian integrability of Hamiltonian systems describing a universe filled with a scalar field (possibly complex). The tool used is the differential Galois group approach, as introduced by Morales-Ruiz and Ramis. The main result is that the generic systems with minimal coupling are non-integrable, although there still exist some values of parameters for which integrability remains undecided; the conformally coupled systems are only integrable in four known cases. We also draw a connection with chaos present in such cosmological models, and the issues of integrability restricted to the real domain.Comment: This is a conflated version of arXiv:gr-qc/0612087 and arXiv:gr-qc/0703031 with a new theory sectio

Integrability of planar polynomial differential systems through linear differential equations

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constructed by using this method. We also present a new example of this kind of families. We give an analogous method for constructing rational equations but by means of a linear differential equation of first order.Comment: 24 pages, no figure