Independent First Integrals In Involution Research Articles

A modified parameterization method for invariant Lagrangian tori for partially integrable Hamiltonian systems

In this paper we present an a-posteriori KAM theorem for the existence of an (n−d)-parameter family of d-dimensional isotropic invariant tori with Diophantine frequency vector ω∈Rd, of type (γ,τ), for n degrees of freedom Hamiltonian systems with (n−d) independent first integrals in involution. If the first integrals induce a Hamiltonian action of the (n−d)-dimensional torus, then we can produce n-dimensional Lagrangian tori with frequency vector of the form (ω,ωp), with ωp∈Rn−d. In the light of the parameterization method, we design a (modified) quasi-Newton method for the invariance equation of the parameterization of the torus, whose proof of convergence from an initial approximation, and under appropriate non-degeneracy conditions, is the object of this paper. We present the results in the analytic category, so the initial torus is real-analytic in a certain complex strip of width ρ, and the corresponding error in the functional equation is ɛ. We heavily use geometric properties and the so called automatic reducibility to deal directly with the functional equation and get convergence if γ−2ρ−2τ−1ɛ is small enough, in contrast with most of KAM results based on the parameterization method, that get convergence if γ−4ρ−4τɛ is small enough. The approach is suitable to perform computer assisted proofs.

Global dynamics of the Buckingham’s two-body problem

The equations of motion of the Buckingham system are the ones of a two-body problem defined by the Hamiltonian $$ H= \frac{1}{2} \bigl(p_{x}^{2}+p_{y}^{2} \bigr) + A e^{-B \sqrt{x ^{2}+y^{2}}}-\frac{M}{ (x^{2}+y^{2} )^{3}}, $$ where $A$ , $B$ and $M$ are positive constants. The angular momentum $p_{\theta }:= x p_{y}- y p_{x}$ and this Hamiltonian are two independent first integrals in involution. We denote by $I_{h}$ (respectively, $I_{c}$ ) the set of points of the phase space where $H$ (respectively, $p_{\theta }$ ) takes the value $h$ (respectively, $c$ ). Due to the fact that $H$ and $p_{\theta }$ are first integrals, the sets $I_{h}$ and $I_{hc}= I_{h} \cap I_{c}$ are invariant under the flow of the Buckingham systems. We describe the global dynamics of the Buckingham system describing the foliation of its phase space by the invariant sets $I_{h}$ , the foliation of the invariant set $I_{h}$ by its invariant subsets $I_{hc}$ , and the foliation of invariant set $I_{hc}$ by the orbits of the system.

Numerical discretization of completely integrable nonlinear Hamiltonian systems

Completely integrable nonlinear Hamiltonian systems offer attractive models for novel accelerator systems that are being developed to push the frontiers of high intensity and high power beam applications. However, all known numerical methods used to discretize and simulate these systems spoil the integrability property of the original, time-continuous, theoretically integrable system. Since the hallmark of integrability is the existence of sufficiently many functionally independent first integrals in involution, we investigated several methods' performance in terms of invariant preservation over longer timescales. We conclude that, while novel structure preserving integrators would be ideal, symplectic methods under mild restrictions offer adequate invariant preservation. We especially highlight the very good performance of the Stormer-Verlet method at low accuracy and a modified Lobatto method at high accuracy for the Integrable Optics Test Accelerator (IOTA). However, we show that symplectic integrators are limited by numerically induced resonances, which make interpretation of some simulation results challenging. These limitations and workarounds are discussed.

A-posteriori KAM theory with optimal estimates for partially integrable systems

In this paper we present a-posteriori KAM results for existence of d-dimensional isotropic invariant tori for n-DOF Hamiltonian systems with additional n−d independent first integrals in involution. We carry out a covariant formulation that does not require the use of action-angle variables nor symplectic reduction techniques. The main advantage is that we overcome the curse of dimensionality avoiding the practical shortcomings produced by the use of reduced coordinates, which may cause difficulties and underperformance when quantifying the hypotheses of the KAM theorem in such reduced coordinates. The results include ordinary and (generalized) iso-energetic KAM theorems. The approach is suitable to perform numerical computations and computer assisted proofs.

Stochastic averaging of quasi partially integrable and resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for quasi partially integrable and resonant Hamiltonian systems excited by the combined Gaussian and Poisson white noises is proposed. The averaged stochastic integro-differential equations (SIDEs) and generalized Fokker-Planck-Kolmogorov (GFPK) equation for the quasi partially integrable Hamiltonian system with r(1<r<n) independent first integrals in involution and β(1≤β≤r−2) resonant relations are derived. The dimension of averaged SIDEs and the GFPK equation is equal to the number of the independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. A 4-degree-of-freedom (DOF) quasi partially integrable and resonant Hamiltonian system under combined Gaussian and Poisson white noise excitations are calculated as an example to illustrate the application of the proposed method. Different resonant cases and non-resonant case are worked out for the purpose of comparison. The Monte Carlo simulations are also carried out to verify the effectiveness and accuracy of the proposed stochastic averaging method.

How many first integrals imply integrability in infinite-dimensional Hamilton system

The classical Liouville integrability means that there exist n independent first integrals in involution for 2 n -dimensional phase space. However, in the mfinite-dimensional case, an infinite number of independent first integrals in involution does not indicate that the system is solvable. How many first integrals do we need to make the system solvable? To answer the question, we obtain an infinite-dimensional Hamilton–Jacobi theory, and prove an infinite-dimensional Liouville theorem . Based on the theorem, we give a criterion of integrability of infinite-dimensional system. According to the criterion, we give a modified definition of the Liouville integrability in infinite dimension. This definition which dose not depend on the inverse scattering method is a natural generalization of the classical Liouville integrability. In general, an infinite number of first integrals is complete if all action variables of a Hamilton system such as KdV equation can be reconstmcted by the set of first integrals. Essentially, our theory only provides an equivalent representation to the statement that an infinite-dimensional dynamical system is integrable if it can be reformulated in terms of action-angle variables. However, our results answer the question on the relation between the first integrals and solvability of infinite-dimensional Hamilton system.

Stochastic stability of quasi-integrable Hamiltonian systems with time delay by using Lyapunov function method

The asymptotic Lyapunov stability of one quasi-integrable Hamiltonian system with time-delayed feedback control is studied by using Lyapunov functions and stochastic averaging method. First, a quasi-integrable Hamiltonian system with time-delayed feedback control subjected to Gaussian white noise excitation is approximated by a quasi-integrable Hamiltonian system without time delay. Then, stochastic averaging method for quasi-integrable Hamiltonian system is used to reduce the dimension of the original system, and after that the Lyapunov function of the averaged Ito equation is taken as the optimal linear combination of the corresponding independent first integrals in involution. Finally, the stability of the system is determined by using the largest eigenvalue of the linearized system. Two examples are used to illustrate the proposed procedure and the effects of delayed time on the Lyapunov stability are discussed as well.

The Global Phase Space for the 2- and 3-Dimensional Kepler Problems

We determine the foliations of the phase space of four particular integrable Hamiltonian systems obtained from the Kepler problem, namely the sidereal and the synodical Kepler Problem in the plane \({(\mathbb{R}^{2})}\) and in the space \({(\mathbb{R}^{3})}\). These problems differ in their formulation by the choice of the referentials and by the dimension of the phase space. These four Kepler problems have played a main role in Celestial Mechanics. Their importance is justified: First, the study of an integrable problem allow us to obtain information about a non-integrable problem sufficiently close to the integrable one. In fact this is the principle of perturbation theory. Second, from the point of view of the applications, the sidereal is basic for the computation of the planetary ephemerides and the synodical is the limit case of the non-integrable restricted circular 3-body problem when one of the masses of the two primaries tends to zero. We determine the foliations of the phase space of these four Kepler problems by the orbits (i.e. we characterize their global flow), and by fixing one, two or three independent first integrals in involution; of course, at most three for the two spatial problems, and at most two for the two planar problems.

Lyapunov functions for quasi-Hamiltonian systems

A procedure for constructing the Lyapunov functions and studying their asymptotic Lyapunov stability with probability one for quasi-Hamiltonian systems is proposed. For quasi-non-integrable Hamiltonian systems, the Hamiltonian (the total energy) is taken as the Lyapunov function. For quasi-integrable and quasi-partially-integrable Hamiltonian systems, the optimal linear combination of the independent first integrals in involution is taken as the Lyapunov function. The derivative of the Lyapunov function with respect to time is obtained by using the stochastic averaging method for quasi-Hamiltonian systems. The sufficient condition for the asymptotic Lyapunov stability with probability one of quasi-Hamiltonian systems is determined based on a theorem due to Khasminskii and compared with the corresponding necessary and sufficient condition obtained by using the largest Lyapunov exponent. Three examples are worked out to illustrate the proposed procedure and its effectiveness.

Remarks on a Lie Theorem on the Integrability of Differential Equations in Closed Form

is integrable by quadratures [1] (for details, see also [2, 3]). More precisely, all of its solutions can be found by “algebraic operations” (including inversion of functions) and “quadratures,” that is, computation of integrals of known functions of one variable. This definition has a local nature. The algebra g generates an n-dimensional solvable Lie group G that acts freely on R. Relations (1) imply that the transformations in G take the trajectories of system (2) to trajectories of the same system. In other words, the solvable group G “permutes” the trajectories of the differential equation (2). Lie treated his theorem as an analog of Galois theory for ordinary differential equations. However, there is a more natural analog of Galois theory, namely, Picard–Vessiot theory, which studies extensions of differential fields by solutions of linear differential equations (e.g., see [4]). Nevertheless, the Lie theorem is one of the key results in the problem of closed-form integrability of nonlinear differential equations. It suffices to note that, in the simplest case, this theorem readily implies the well-known Liouville theorem on the integrability of Hamiltonian differential equations with a complete set of independent first integrals in involution [3]. The following assertion is the main result of the present paper.

Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems

An n degree-of-freedom Hamiltonian system with r (1<r<n) independent first integrals which are in involution is called partially integrable Hamiltonian system and a partially integrable Hamiltonian system subject to light dampings and weak stochastic excitations is called quasi partially integrable Hamiltonian system. In the present paper, the averaged Itô and Fokker–Planck–Kolmogorov (FPK) equations for quasi partially integrable Hamiltonian systems in both cases of non-resonance and resonance are derived. It is shown that the number of averaged Itô equations and the dimension of the averaged FPK equation of a quasi partially integrable Hamiltonian system is equal to the number of independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. The technique to obtain the exact stationary solution of the averaged FPK equation is presented. The largest Lyapunov exponent of the averaged system is formulated, based on which the stochastic stability and bifurcation of original quasi partially integrable Hamiltonian systems can be determined. Examples are given to illustrate the applications of the proposed stochastic averaging method for quasi partially integrable Hamiltonian systems in response prediction and stability decision and the results are verified by using digital simulation.

The super-separability of the three-body inverse-square Calogero system

The geometrical theory of the variable separation for the Hamilton–Jacobi equation is applied to the classical three-body inverse-square Calogero system. It is proved that this system is separable in infinitely many inequivalent ways, related to five different kinds of separable webs in the Euclidean three-space, and the corresponding systems of independent first integrals in involution are computed.

Integrability of restricted multiple three-wave interactions. II. Coupling constants with ratios 1 and 2

The system of ordinary differential equations describing the multiple three-wave interactions with a common pump (daughter) wave is proved to be completely integrable by obtaining the necessary 2N+1 independent first integrals in involution for the case when the coupling constants and the frequency mismatches have ratio 1 and/or 2. This case was deemed integrable on the basis of a Painlevé analysis, but a direct proof has been lacking for some time. The first integrals are the N+1 quadratic Manley–Rowe relations, the cubic Hamiltonian, N−1 quartic integrals (analogous to the ones needed for complete integrability in the case of equal coupling constants and detunings in all wave triads), and a new sixth-order integral involving all wave quantities. The form of this last invariant was deduced from the recent result for the analogous interaction between five waves (N=2), and essentially made possible by the proper use of irreducible forms, elementary building blocks for polynomial first integrals in involution with the Manley–Rowe invariants.