arXiv subjects
Publications and source records attributed to Maria Przybylska.
We show that the geodesic motion in the Zipoy-Voorhees space-time is not Liouville integrable, in that there does not exist an additional first integral meromorphic in the phase space variables.
This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. The non-integrability of this system is proven using differential Galois theory.
The classical Euler--Poinsot case of the rigid body dynamics admits a class of simple but non-trivial integrable generalizations, which modify the Poisson equations describing the motion of the body in space. These generalizations possess first integrals which are polynomial in the angular momenta. We consider the modified Poisson equations as a system of linear equations with elliptic coefficients and show that all the solutions of it are single-valued. By using the vector generalization of the Picard theorem, we derive the solutions explicitly in terms of sigma functions of the corresponding elliptic curve. The solutions are accompanied with a numerical example. We also compare the generalized Poisson equations with % some generalizations of the classical 3rd order Halphen equation.
The one dimensional Dirac equation with a rational potential is reducible to an ordinary differential equation with a Riccati-like coefficient. Its integrability can be studied with the help of differential Galois theory, although the results have to be stated with recursive relations, because in general the equation is of Heun type. The inverse problem of finding integrable rational potentials based on the properties of the singular points is also presented; in particular, a general class of integrable potentials leading to the Whittaker equation is found.
We consider a special type of triple pendulum with two pendula attached to end mass of another one. Although we consider this system in the absence of the gravity, a quick analysis of of Poincaré cross sections shows that it is not integrable. We give an analytic proof of this fact analysing properties the of differential Galois group of variational equation along certain particular solutions of the system.
We make several comments on "Note on the Analytical Solution of the Rabi Model" (arXiv:1210.4946).
We show how to use properly the Bargmann space of entire functions in the analysis of the Rabi model. We are able to correct a serious error in recent papers on the topic and develop a corrected method of finding the spectrum applicable also to other, more general systems.
We study the integrability in the Liouville sense of natural Hamiltonian systems with a homogeneous rational potential $V(\vq)$. Strong necessary conditions for the integrability of such systems were obtained by an analysis of differential Galois group of variational equations along certain particular solutions. These conditions have the form of arithmetic restrictions putted on eigenvalues of Hessian $V"(\vd)$ calculated at a non-zero solution $\vd$ of equation $\grad V(\vd)=\vd$. Such solutions are called proper Darboux points. It was recently proved that for generic polynomial homogeneous potentials there exist universal relations between eigenvalues of Hessians of the potential taken at all proper Darboux points. The question about the existence of such relations for rational potentials seems to be hard. One of the reason of this fact is the presence of indeterminacy points of the potential and its gradient. Nevertheless, for two degrees of freedom we prove that such relation exists. This result is important because it allows to show that the set of admissible values for eigenvalues of Hessian at a proper Darboux point for potentials satisfying necessary conditions for the integrability is finite. In turn, it gives a tool for classification of integrable rational potentials. Also, quite recently, it was shoved that for polynomial homogeneous potentials additional necessary conditions for the integrability can be deduced from the existence of improper Darboux points, i.e., points $\vd$ which are non-zero solution of equation $\grad V(\vd)=\vzero$. These new conditions have also the form of arithmetic restrictions imposed on eigenvalues of $V"(\vd)$. In this paper we prove that for rational potentials improper Darboux points give the same necessary conditions for the integrability.
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.
We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$ as well as their equivalents on $S^2$ and $H^2$ for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which always can be chosen polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta.
This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realise the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.
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.
We consider natural complex Hamiltonian systems with $n$ degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each $k$ there exists an explicitly known infinite set $\scM_k\subset\Q$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$, belong to $\scM_k$. The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning $V$ we prove the following fact. For each $k$ and $n$ there exists a finite set $\scI_{n,k}\subset\scM_k$ such that if the system is integrable, then all eigenvalues of the Hessian matrix $V''(\vd)$ belong to $\scI_{n,k}$. We give an algorithm which allows to find sets $\scI_{n,k}$. We applied this results for the case $n=k=3$ and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.
In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$ is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic, if it admits a nonzero solution of equation $V'(\vd)=0$. The existence of such solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory gives also additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(λ_1,...,λ_n)$ of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$. Furthermore, we show that similarly to the generic case also for nongeneric potentials some universal relations between $(λ_1,...,λ_{n})$ calculated at various solutions of $V'(\vd)=\vd$ exist. We derive them for case $n=k=3$ applying the multivariable residue calculus. We demonstrate the strength of the obtained results analysing in details the nongeneric cases for $n=k=3$. Our analysis cover all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n=k=3$ thanks to this analysis a three-parameter family of potentials integrable or super-integrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.
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.
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.
This is the second part of integrability analysis of cosmological models with scalar fields. Here, we study systems with conformal coupling, and show that apart from four cases, where explicit first integrals are known, the generic system is not integrable. We also comment on some chaotic properties of the system, and the issues of integrability restricted to the real domain.
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 $k$.