Natural Hamiltonian Systems Research Articles

DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

This paper is an overview of our works that 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 realize 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.

From Jacobi problem of separation of variables to theory of quasipotential Newton equations

Our solution to the Jacobi problem of finding separation variables for natural Hamiltonian systems H = ½p2 + V(q) is explained in the first part of this review. It has a form of an effective criterion that for any given potential V(q) tells whether there exist suitable separation coordinates x(q) and how to find these coordinates, so that the Hamilton-Jacobi equation of the transformed Hamiltonian is separable. The main reason for existence of such criterion is the fact that for separable potentials V(q) all integrals of motion depend quadratically on momenta and that all orthogonal separation coordinates stem from the generalized elliptic coordinates. This criterion is directly applicable to the problem of separating multidimensional stationary Schrodinger equation of quantum mechanics.

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

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 $(\lambda_1,...,\lambda_n)$ 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 $(\lambda_1,...,\lambda_{n})$ 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.

Hamiltonian Systems Admitting a Runge–Lenz Vector and an Optimal Extension of Bertrand’s Theorem to Curved Manifolds

Bertrand’s theorem asserts that any spherically symmetric natural Hamiltonian system in Euclidean 3-space which possesses stable circular orbits and whose bounded trajectories are all periodic is either a harmonic oscillator or a Kepler system. In this paper we extend this classical result to curved spaces by proving that any Hamiltonian on a spherically symmetric Riemannian 3-manifold which satisfies the same conditions as in Bertrand’s theorem is superintegrable and given by an intrinsic oscillator or Kepler system. As a byproduct we obtain a wide panoply of new superintegrable Hamiltonian systems. The demonstration relies on Perlick’s classification of Bertrand spacetimes and on the construction of a suitable, globally defined generalization of the Runge–Lenz vector.

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

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.

Existence of multiple periodic solutions for a natural Hamiltonian system in a potential well

We prove, under a pinching hypothesis, a theorem of existence of at least n periodic orbits for a closed regular energy hypersurface Σ in R 2 n which is the level set of a natural (i.e. of the type “potential + kinetic energy”) Hamiltonian function and projects onto a potential well in the configuration space.

Universal localizing bounds for compact invariant sets of natural polynomial Hamiltonian systems

In this Letter we study the localization problem of compact invariant sets of natural Hamiltonian systems with a polynomial Hamiltonian. Our results are based on applying the first order extremum conditions. We compute universal localizing bounds for some domain containing all compact invariant sets of a Hamiltonian system by using one quadratic function of a simple form. These bounds depend on the value of the total energy of the system, degree and some coefficients of a potential and, in addition, some positive number got as a result of a solution of one maximization problem. Besides, under some quasihomogeneity condition(s) we generalize our construction of the localization set.

Necessary conditions for super-integrability of Hamiltonian systems

We formulate a general theorem which gives a necessary condition for the maximal super-integrability of a Hamiltonian system. This condition is expressed in terms of properties of the differential Galois group of the variational equations along a particular solution of the considered system. An application of this general theorem to natural Hamiltonian systems of n degrees of freedom with a homogeneous potential gives easily computable and effective necessary conditions for the super-integrability. To illustrate an application of the formulated theorems, we investigate: three known families of integrable potentials, and the three body problem on a line.

Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems

We consider a 2 d.o.f. natural Hamiltonian system with one degree of freedomcorresponding to fast motion and the other one corresponding to slow motion. The Hamiltonianfunction is the sum of potential and kinetic energies, the kinetic energy being a weighted sumof squared momenta. The ratio of time derivatives of slow and fast variables is of order$\epsilon $«$ 1$. At frozen values of the slow variables there is a separatrix on the phaseplane of the fast variables and there is a region in the phase space (the domain ofseparatrix crossings) where the projections of phase points onto the plane of the fastvariables repeatedly cross the separatrix in the process of evolution of the slowvariables. Under a certain symmetry condition we prove the existence of many, of order$1/\epsilon$, stable periodic trajectories in the domain of the separatrix crossings. Each ofthese trajectories is surrounded by a stability island whose measure is estimated frombelow by a value of order $\epsilon$. Thus, the total measure of the stability islands isestimated from below by a value independent of $\epsilon$. We find the location of stableperiodic trajectories and an asymptotic formula for the number of these trajectories. Asan example, we consider the problem of motion of a charged particle in the parabolicmodel of magnetic field in the Earth magnetotail.

( 1 + 1 ) -dimensional separation of variables

In this paper we explore general conditions which guarantee that the geodesic flow on a two-dimensional manifold with indefinite signature is locally separable. This is equivalent to showing that a two-dimensional natural Hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a secind rank Killing tensor. We examine the possibility that the integral is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary values of the Hamiltonian (strong integrability). Using null coordinates, we show that the leading-order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability. These functions are quadratic polynomials in the coordinates in the case of strong integrability. We show that for (1+1)-dimensional systems, there are three possible types of conformal Killing tensors and, therefore, three distinct separability structures in contrast to the single standard Hamilton-Jacobi-type separation in the positive definite case. One of the new separability structures is the complex∕harmonic type which is characterized by complex separation variables. The other new type is the linear∕null separation which occurs when the conformal Killing tensor has a null eigenvector.

Higher order first integrals in classical mechanics

We present a practical algorithm for computing first integrals of motion which are polynomial in the momenta for natural Hamiltonian systems defined in a flat pseudo-Riemannian space of arbitrary dimension and signature. We then apply our algorithm to explore the integrability of two physical systems. First, we study the Holt potential in two dimensions and derive analogous potentials which admit an additional first integral quartic in the momenta. Second, we analyze a class of cylindrically symmetric potentials in three-dimensional Euclidean space and recover known families of second-order maximally superintegrable systems.

Finiteness of integrable n-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 $k$.

An inverse problem of Hamiltonian dynamics

We study the question of whether for a natural Hamiltonian system on a two-dimensional compact configuration manifold, a single trajectory of sufficiently high energy is almost surely enough to reconstruct a real analytic potential.

Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential

In this paper we study the integrability of natural Hamiltonian systems with a homogeneous polynomial potential. The strongest necessary conditions for their integrability in the Liouville sense have been obtained by a study of the differential Galois group of variational equations along straight line solutions. These particular solutions can be viewed as points of a projective space of dimension smaller by one than the number of degrees of freedom. We call them Darboux points. We analyze in detail the case of two degrees of freedom. We show that, except for a radial potential, the number of Darboux points is finite and it is not greater than the degree of the potential. Moreover, we analyze cases when the number of Darboux points is smaller than maximal. For two degrees of freedom the above-mentioned necessary condition for integrability can be expressed in terms of one nontrivial eigenvalue of the Hessian of potential calculated at a Darboux point. We prove that for a given potential these nontrivial eigenvalues calculated for all Darboux points cannot be arbitrary because they satisfy a certain relation which we give in an explicit form. We use this fact to strengthen maximally the necessary conditions for integrability and we show that in a generic case, for a given degree of the potential, there is only a finite number of potentials which satisfy these conditions. We also describe the nongeneric cases. As an example we give a full list of potentials of degree four satisfying these conditions. Then, investigating the differential Galois group of higher order variational equations, we prove that, except for one discrete family, among these potentials only those which are already known to be integrable are integrable. We check that a finite number of potentials from the exceptional discrete family are not integrable, and we conjecture that all of them are not integrable.

Configurational invariants of Hamiltonian systems

In this paper we explore the general conditions in order that a two-dimensional natural Hamiltonian system possess a second invariant which is a polynomial in the momenta and is therefore Liouville integrable. We examine the possibility that the invariant is preserved by the Hamiltonian flow on a given energy hypersurface only (weak integrability) and derive the additional requirement necessary to have conservation at arbitrary energy (strong integrability). Using null complex coordinates, we show that the leading order coefficient of the polynomial is an arbitrary holomorphic function in the case of weak integrability and a polynomial in the coordinates in the strongly integrable one. We review the results obtained so far with strong invariants up to degree four and provide some new examples of weakly integrable systems with linear and quadratic invariants.

Darboux polynomials and first integrals of natural polynomial Hamiltonian systems

We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals.

Darboux polynomials and first integrals of natural polynomial Hamiltonian systems

We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function. Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals.

A note on topology of isoenergy manifold of n-degree natural Hamiltonian systems and existence of global Poincaré section

The fundamental group of compact isoenergy manifolds of natural n-degree of freedom Hamiltonian systems is discussed. It is demonstrated that a certain topology of corresponding accessible region in configuration space gives rise to topological obstruction to the existence of global Poincaré sections.

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation $${\frac{1}{2}}$$ ∑(∂u/∂q i )2+V(q)=E consists of finding new curvilinear coordinates x i (q) in which the transformed equation admits a complete separated solution u(x)=∑u (i)(x i ;α). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrodinger) equation as well.

Geometric approach to Lyapunov analysis in Hamiltonian dynamics.

As is widely recognized in Lyapunov analysis, linearized Hamilton's equations of motion have two marginal directions for which the Lyapunov exponents vanish. Those directions are the tangent one to a Hamiltonian flow and the gradient one of the Hamiltonian function. To separate out these two directions and to apply Lyapunov analysis effectively in directions for which Lyapunov exponents are not trivial, a geometric method is proposed for natural Hamiltonian systems, in particular. In this geometric method, Hamiltonian flows of a natural Hamiltonian system are regarded as geodesic flows on the cotangent bundle of a Riemannian manifold with a suitable metric. Stability/instability of the geodesic flows is then analyzed by linearized equations of motion that are related to the Jacobi equations on the Riemannian manifold. On some geometric settings on the cotangent bundle, it is shown that along a geodesic flow in question, there exist Lyapunov vectors such that two of them are in the two marginal directions and the others orthogonal to the marginal directions. It is also pointed out that Lyapunov vectors with such properties cannot be obtained in general by the usual method that uses linearized Hamilton's equations of motion. Furthermore, it is observed from numerical calculation for a model system that Lyapunov exponents calculated in both methods, geometric and usual, coincide with each other, independently of the choice of the methods.