First integrals of local analytic differential systems

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré–Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as “conserved” by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac–Frommer center problem.     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

与一族(1+1)维孤子方程相联系的有限维可积系（英文）

In this paper, a new spectral problem is proposed and the corresponding soliton equations hierarchy are also obtained. Under a constraint between the potentials and the eigenfunctions, the eigenvalue problem is nonlinearized so as to be a new finite-dimensional Hamiltonian system. By resotring to the generating function approach, we obtain conserved integrals and the involutivity of the conserved integrals. The finite-dimensional Hamiltonian system is further proved to be completely integrable in the Liouville sense. Finally, we show the decomposition of the soliton equations.     

A partial Hamiltonian approach for current value Hamiltonian systems

We develop a partial Hamiltonian framework to obtain reductions and closed-form solutions via first integrals of current value Hamiltonian systems of ordinary differential equations (ODEs). The approach is algorithmic and applies to many state and costate variables of the current value Hamiltonian. However, we apply the method to models with one control, one state and one costate variable to illustrate its effectiveness. The current value Hamiltonian systems arise in economic growth theory and other economic models. We explain our approach with the help of a simple illustrative example and then apply it to two widely used economic growth models: the Ramsey model with a constant relative risk aversion (CRRA) utility function and Cobb Douglas technology and a one-sector AK model of endogenous growth are considered. We show that our newly developed systematic approach can be used to deduce results given in the literature and also to find new solutions.     

Area-Preserving Normalizations for Centers of Planar Hamiltonian Systems

It is well known that a nondegenerate center of an analytic Hamiltonian planar system can be brought to normal form by means of an analytic canonical change of coordinates. This normal form, that we denote by CNF, does not depend on the coordinate transformation. In this paper we give an elementary proof of these facts and we show some interesting applications of the machinery that we develop in order to prove them. For instance, we describe the space of coordinate transformations that bring a Hamiltonian nondegenerate center to its CNF, and we prove that they are all canonical when the center is non-isochronous. We also show that two Hamiltonian systems with a nondegenerate center are canonically conjugated if and only if both centers have the same period function.     

On action-angle coordinates and the Poincaré coordinates

This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants “∮_γ p dq”. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did.     

Symplectic invariants near hyperbolic-hyperbolic points

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system.We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.      

KAM theory near multiplicity one resonant surfaces in perturbations of a-priori stable hamiltonian systems

Summary We consider a near-integrable Hamiltonian system in the action-angle variables with analytic Hamiltonian. For a given resonant surface of multiplicity one we show that near a Cantor set of points on this surface, whose remaining frequencies enjoy the usual diophantine condition, the Hamiltonian may be written in a simple normal form which, under certain assumptions, may be related to the class which, following Chierchia and Gallavotti [1994], we calla-priori unstable. For the a-priori unstable Hamiltonian we prove a KAM-type result for the survival of whiskered tori under the perturbation as an infinitely differentiable family, in the sense of Whitney, which can then be applied to the above normal form in the neighborhood of the resonant surface. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan C. Simo.     

On the nesting of Painlevé hierarchies: A Hamiltonian approach

We consider the phenomenon whereby two different Painlevé hierarchies, related to the same hierarchy of completely integrable equations, are such that solutions of one member of one of the Painlevé hierarchies are also solutions of a higher-order member of the other Painlevé hierarchy. An explanation is given in terms of the Hamiltonian structures of the related underlying completely integrable hierarchies, and is sufficiently generally formulated so as to be applicable equally to both continuous and discrete Painlevé hierarchies. Special integrals of a further Painlevé hierarchy related by Bäcklund transformation to the other Painlevé hierarchy mentioned above can also be constructed. Examples of the application of this approach to Painlevé hierarchies related to the Korteweg–de Vries, dispersive water wave, Toda and Volterra integrable hierarchies are considered. Our results provide further evidence of the importance of the underlying structures of related completely integrable hierarchies in understanding the properties of Painlevé hierarchies.     

Generalized affine lie algebras and the solution of a class of flows associated with the QR eigenvalue algorithm

In [3] the authors show that the QR algorithm to compute the eigenvalues of a matrix is the integer time evaluation of a completely integrable Hamiltonian system. Here the authors show that all the associated commuting integrals generate flows that can be solved explicitly via a factorization procedure on a suitable finite, or infinite-dimensional, Lie algebra.     

Direction-Reversing Traveling Waves in the Even Fermi-Pasta-Ulam Lattice

Summary. This paper considers the famous Fermi-Pasta-Ulam (FPU) lattice with periodic boundary conditions and quartic nonlinearities. Due to special resonances and discrete symmetries, the Birkhoff normal form of this Hamiltonian system is Liouville integrable. The normal form equations can easily be solved if the number of particles in the lattice is odd, but if the number of particles is even, several nontrivial phenomena occur. In the latter case we observe that the phase space of the normal form is decomposed in invariant subspaces that describe the interaction between the Fourier modes with wave number j and the Fourier modes with wave number n / 2-j . We study how the level sets of the integrals of the normal form foliate these invariant subspaces. The integrable foliations turn out to be singular and the method of singular reduction shows that the normal form has invariant pinched tori and monodromy. Monodromy is an obstruction to the existence of global action-angle variables. The pinched tori are interpreted as homoclinic and heteroclinic connections between traveling waves. Thus we discover a class of solutions of the normal form which can be described as direction-reversing traveling waves. The relation between the FPU lattice and its Birkhoff normal form can be understood from KAM theory and approximation theory. This explains why we observe the impact of the direction-reversing traveling waves numerically as a relaxation oscillation in the original FPU system.     

Normal form of a quantum Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point

According to classical result of Moser [1] a real-analytic Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point can be reduced to the normal form by a real-analytic symplectic change of variables. In this paper the result is extended to the case of the non-commutative algebra of quantum observables.We use an algebraic approach in quantum mechanics presented in [2] and develop it to the non-autonomous case. We introduce the notion of quantum non-autonomous canonical transformations and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a non-commutative normal form and prove that a time-periodic quantum observable with one degree of freedom near a hyperbolic fixed point can be reduced to a normal form by a canonical transformation. Unlike traditional results, where only formal theory of normal forms is constructed, we prove a convergence of the normalizing procedure.      

Zero bifurcation diagrams for Abelian integrals: A study on higher-order hyperelliptic Hamiltonian systems with three perturbation parameters

In this paper, we present the zero bifurcation diagrams for the Abelian integrals of two hyperelliptic Hamiltonian systems with three perturbation parameters using an algebraic-geometric approach. The method can be used to study the bifurcation diagrams for higher-order Hamiltonian systems with polynomial perturbations of any degree.     

The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch's case

This paper deals with a geometric approach to the integration of Clebsch's case of equations describing the motion of a solid body in an ideal fluid. This problem is defined by a nonlinear system of 6 differential equations admitting 4 polynomial first integrals. We show that the intersection of surface levels of these integrals can be completed to an abelian surface, i.e., a 2-dimensional algebraic torus. Also, we prove that the problem can be linearized, i.e., can be written in terms of abelian integrals, on a Prym variety of a genus 3 curve obtained naturally. Received August 1998     

Quadratic integrals of linear Hamiltonian systems

Momentum mapping of an autonomous, real linear Hamiltonian system is determined by its set of quadratic integrals. Such a system can be identified with an element of the real symplectic algebra and its quadratic integrals correspond to the centralizer of this element inside the symplectic algebra. In this paper, using a new set of normal forms for the elements of the real symplectic algebra, we compute their centralizers explicitly.Research supported in part by the National Science Foundation under NSF-MCS 8205355.     

On a Certain Sub-Riemannian Geodesic Flow on the Heisenberg Group

Under study is an integrable geodesic flow of a left-invariant sub-Riemannian metric for a right-invariant distribution on the Heisenberg group. We obtain the classification of the trajectories of this flow. There are a few examples of trajectories in the paper which correspond to various values of the first integrals. These trajectories are obtained by numerical integration of the Hamiltonian equations. It is shown that for some values of the first integrals we can obtain explicit formulae for geodesics by inverting the corresponding Legendre elliptic integrals.     

Linear Hamiltonian Systems: Quadratic Integrals,Singular Subspaces and Stability

A chain of quadratic first integrals of general linear Hamiltonian systems that have not been represented in canonical form is found. Their involutiveness is established and the problem of their functional independence is studied. The key role in the study of a Hamiltonian system is played by an integral cone which is obtained by setting known quadratic first integrals equal to zero. A singular invariant isotropic subspace is shown to pass through each point of the integral cone, and its dimension is found. The maximal dimension of such subspaces estimates from above the degree of instability of the Hamiltonian system. The stability of typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an equilibrium point. General results are applied to the investigation of linear mechanical systems with gyroscopic forces and finite-dimensional quantum systems.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

An isoperimetric inequality for Hamiltonian stationary Lagrangian tori in {\mathbb C}^2 related to Oh's conjecture

We compute loops integrals on Hamiltonian stationary Lagrangian tori in which are symplectic invariants, then we show an isoperimetric inequality involving these invariants and the area. Finally, we show that the flat torus has least area among Hamiltonian stationary Lagrangian tori of its isotopy class. Received: 4 December 2000; in final form: 18 January 2002 / Published online: 5 September 2002     

Isomorphism and Hamilton representation of some nonholonomic systems

We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.