Symmetry and Perturbation Theory

FOREWORD

ACKNOWLEDGMENTS

CONTENTS

In this note, we give a description of the invariant bi-Lagrangian structures on a homogeneous symplectic manifold (M = G/K, ω) of a semisimple Lie group G. We then use this description in order to obtain invariant generalized Monge-Ampére equations, in the sense of V. Lychagin, on certain homogeneous contact manifolds, associated with homogeneous bi-Lagrangian manifolds.

This article is an introduction to Darboux integrability in terms of a recently developed theory of superposition.

In this article, we show that if one writes down the structure equations for the evolution of a curve embedded in an 4n–dimensional symplectic manifold with zero curvature, this leads to a Nijenhuis operator for an integrable scalar-vector evolution equation generalizing the known cases of the vmKdV equation and the noncommutative scalar KdV. The procedure also gives us the symplectic and Hamiltonian operators.

This expository paper gathers the contents of talks given by the authors in the conference SPT2007. It is focused on the recent results on the structure of the singular set of generalized solutions to n–body type problems extending the classical Von Zeipel Theorem and Sundman asymptotic estimates.

The Riemann tensor of a given metric, of any dimension and signature, can be computed ‘by hand calculation’, avoiding the explicit calculation of the (1/2) n²(n+1) Christoffel symbols. The algorithm presented here works with n quadratic form Qⁱ in the velocity-variables coming from the Lagrange geodesic equations, and with 2n cubic forms and generated by them. An example of this method is illustrated: it concerns the application of Geodesic Equivalence theory to General Relativity.

Recently, semi-global results have been reported by Wagener⁶ for the k : ℓ resonance where ℓ = 1, 2. In this work we add the ℓ = 3 strong resonance case and give an overview for ℓ = 1,2,3. For an introduction to the topic, as well as results on the non-resonant and weakly-resonant cases, see Refs. 1,2,6.

Small dissipation limits for nearly–integrable systems are considered; theorems concerning the existence of quasi–periodic attractors smoothly approaching KAM tori are presented and an application to the capture in 3:2 resonance of Mercury is discussed.

In this short note we review the role of the Darboux matrix transformation in soliton theory; in particular we present a Darboux construction of solutions of integrable PDEs with nonvanishing boundary values.

No abstract received.

We outline some recent results for convergent and divergent normal form transformations (NFT) for families of commuting analytic vector fields having a common fixed point. First we treat vector fields having linear parts with non-trivial Jordan blocks. We propose geometrically invariant conditions leading to the existence of convergent NFT. Divergent solutions of some overdetermined systems of linear homological equations are constructed and sharp Gevrey estimates for formal divergent solutions are derived. Next, we investigate some classes of resonant vector fields and show that they do not admit convergent NFT and we compute the Gevery index of their formal NFT and describe completely the centralizers in the framework of the smooth vector fields.

A rigid body with three equal moments of inertia is moving in a nonlinear force field with potential z³. Next to the S¹–symmetry about the vertical axis and a further S¹–symmetry introduced by normalization, there is a discrete symmetry due to a special choice of the mass distribution. The continuous symmetries allow to reduce to a one-degree-of-freedom problem, which exhibits bifurcations related to the elliptic umbilic catastrophe. This bifurcation carries over from the integrable approximation to the original system and further to perturbations that break the S¹–symmetry of the potential.

I present here recent results mostly described Ref. [4–6] on relations between integrability properties of nonlinear PDEs and nonlocal geometry of differential equations, see Ref. [11], obtained in a long collaboration with Paul Kersten (University of Twente) and Alik Verbovetsky (Independent University of Moscow).

In this note we discuss some formal properties of universal linearization operator, relate this to brackets of non-linear differential operators and discuss application to the calculus of auxiliary integrals, used in compatibility reductions of PDEs.

We study global variational principles on jet prolongations of fibred manifolds, whose Lagrangians are differential invariants. In particular, a relationship between solutions of the Euler-Lagrange equations and conservation laws, related with vector fields on the basis of underlying fibred manifolds, is established.

We introduce exterior differential systems associated with Euler-Lagrange equations on fibred manifolds and investigate their symmetries and conservation laws.

A construction of Appell polynomials obtained from formal group exponential laws is reviewed. Applications to the theory of hyperfunctions are illustrated.

We introduce, in the spirit of Witten's gauging of exterior differential, a deformed Lie derivative that allows a geometrical interpretation of variational λ-symmetries, in complete analogy with standard variational symmetries.

We derive a covering for the modified Khokhlov–Zabolotskaya equation from Maurer–Cartan forms of its contact symmetry pseudo-group. Then we use the covering to find multi-valued solutions of the Khokhlov–Zabolotskaya equation.

A theorem about the matrix of fractional monodromy will be formulated. The monodromy corresponds to going around a fiber with a singular point of oscillator type with arbitrary resonance. The reason of fractional monodromy and fuzziness of such a monodromy is explained.

Within the framework of inverse Lie problems we give some non–trivial examples of Lie remarkable equations, i.e., classes of partial differential equations that are in correspondence with their Lie point symmetries.

We survey a recent extension of the moving frames method for infinite-dimensional Lie pseudo-groups. Applications include a new, direct approach to the construction of Maurer–Cartan forms and their structure equations for pseudo-groups, and new algorithms, based on constructive commutative algebra, for uncovering the structure of the algebra of differential invariants for pseudo-group actions.

In this report I will outline some recent results — obtained in joint work with G. Gentile — on the existence of periodic solutions for a class of non-linear equations modelling waves, with Dirichlet boundary conditions on the D > 1 dimensional square. The main idea is to combine a Lyapunov-Schmidt reduction and a renormalisation technique, in order to deal with the small divisors. In the massless case this implies finding a non-degenerate solution of the infinite dimensional bifurcation equation.

In this article some qualitative aspects of non-smooth systems on ℝⁿ are studied through methods of Geometric Singular Perturbation Theory (GSP-Theory). We present some results that generalize some settings in low dimension, that bridge the space between such systems and singularly perturbed smooth systems. We analyze the local behavior around typical singularities and prove that the dynamics of the so called Sliding Vector Field is determined by the reduced problem on the center manifold.

In this note we show how very general continuation results can be used to describe (1) families of periodic orbits in conservative systems, and (2) families of symmetric and doubly-symmetric orbits in reversible conservative systems. We describe a general approach which repairs the lack of submersivity for such problems due to the presence of first integrals. Our results can in particular be applied in the Hamiltonian context; we very briefly describe how they can be used for the continuation of choreographies in the N-body problem.

Averaging-normalization, applied to weakly nonlinear wave equations provides a tool for identification of slow manifolds in these infinite-dimensional systems. After discussing the general procedure we demonstrate its effectiveness for a Rayleigh wave equation to find low-dimensional invariant manifolds.

The concept of superintegrability in quantum mechanics is extended to the case of a particle with spin s = 1/2 interacting with one of spin s = 0. Non-trivial superintegrable systems with 8- and 9-dimensional Lie algebras of first-order integrals of motion are constructed in two- and three-dimensional spaces, respectively.

Within the qualitative approach to the study of finite particle quantum systems different possible ways of the generalization of Hamiltonian monodromy are discussed. It is demonstrated how several simple integrable models like non-linearly coupled resonant oscillators, or coupled rotators, lead to physically natural generalizations of the monodromy concept. Fractional monodromy, bidromy, and the monodromy in the case of multi-valued energy-momentum maps are briefly reviewed.

Flow subjected to an axial variation of the external heat transfer coefficient and nonlinear boundary conditions due to radiation, through a conducting thin duct subjected to variable heat transfer coefficient…

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Let (x, y, z) be coordinates on the trivial bundle τ = ℝ² × ℝ. A parabolic Monge-Ampère equation (PMAE) is a second order PDE on τ of the form

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Krause's concept of a complete symmetry group (csg) of a differential equation (de) is the group underlying the algebra associated with the set of symmetries, be they point, contact, generalised or nonlocal, required to specify the equation or system completely [4]. Subsequent work has added a third requirement for a set of symmetries to represent the csg of a de [1]: the group must be of minimal dimension…

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The usual way to write the wave equations of relativistic quantum mechanics in a curved spacetime is by covariantization: the searched equation in curved spacetime should coincide with the flat-spacetime version in coordinates where the connection cancels at the event X considered. This is connected with the equivalence principle. For the Dirac equation with standard (spinor) transformation, this procedure leads to the Dirac-Fock-Weyl (DFW) eqn, which does not obey the equivalence principle. Alternatively, in this work we want to apply directly the classical-quantum correspondence…

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A reduction method of ODEs not possessing Lie point symmetries makes use of the so called λ-symmetries. These are not standard symmetries, nevertheless for any given λ-symmetry of an ODE one can always reconstruct nonlocal symmetries of . As a consequence, using these nonlocal symmetries, the λ-symmetry reduction method reduce to a standard method of symmetry-reduction.

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The classical Laplace equation in Cartesian coordinates on ℝⁿ, , is usually extended to a general n-dimensional Riemannian manifold (M, g) as Δψ = 0, where is the Laplace-Beltrami operator…

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The simplest and most widely used model of a bouncing ball (or grains of a granular fluid) assumes that the ball is a rigid body, and that an impact with the floor is an instantaneous event, which reverses the vertical component of the speed of the ball: to model energy dissipation caused by an impact, it is customary to introduce a positive coefficient of restitution r < 1, which is the ratio between the absolute values of the vertical speeds immediately after and before an impact…

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Consider the Degasperis-Procesi equation

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Variational Poisson–Nijenhuis structures on nonlinear PDEs are introduced and investigated. Relations between Schouten and Nijenhuis brackets with the Lie bracket of shadows of symmetries are established. This approach allows to construct a framework for the theory of nonlocal Poisson–Nijenhuis structures.

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We show that finitely differentiable diffeomorphisms that are either symplectic, volume-preserving, or contact can be approximated with analytic diffeomorphisms that are, respectively, symplectic, volume-preserving, or contact…

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The purpose of this paper is to see how to construct asymptotic solutions using Renormalizaton group method by means of Lie symmetry group. Due to limitation of space, we see only the application of the method to an example. More detailed and generalized procedure and the validity of this method are shown in another paper [1]…

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Let G be a Lie group (a gauge group) and F₁ and F₂ be gauge-natural bundle functors in the sense of Eck [1] defined on the category of principal G-bundles. An operator D : F₁ → F₂ is said to be a gauge invariant (natural) differential operator (NDO) if it is invariant w.r.t. the principal bundle authomorphisms φ : P → P. Moreover, D is a natural operator iff it commutes with the Lie derivatives in the sense that L_ΞD(P)(σ) = TD(P)(L_Ξσ) for any right G-invariant vector field Ξ on the principal bundle P. Let us consider the affine bundle π : PCon P → M of principal connections on any principal G-bundle P which is a gauge-natural bundle (GNB) of order (1,1) and consider an r-th order gauge-invariant (natural) Lagrangian …

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A free fly of two material points M₀ and M₁ representing an orbital station and a subsatellite correspondingly under the action of gravity of a fixed attracting point P supposed as a planet is under consideration. Denote masses of the station and the subsatellite by m₀ and m₁. The points supposed to be connected by an ideal flexible, massless, and inextensible tether of constant length. For that reason the free fly is interrupted from time to time by the impacts with the unilateral constraint of the tether. The problem is considered in its unrestricted form: material endpoints, the orbital station and the sub-satellite, composing the tethered satellite system (TSS), move independently in field of gravity of the fixed attracting center P each thus performing a piecewise Keplerian motion…

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This note is a short version of [3]. We consider N positive masses in a euclidean space ℝ^d, subject to a gravitational interaction, and we find some interesting solutions having a given asymptotic behaviour. The equation of motion of the N-body problem can be written…

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We find a sufficient condition for a J-tensor J to generate from any given flat bicofactor system a multiparameter family of geodesically equivalent flat bi-cofactor systems.

Position coordinates of “exotic” particles, associated with the 2-parameter central extension of the planar Galilei group, do not commute. Realization of the model in solid state Physics, via Berry phase effects, and the relation to anyons are discussed.

We summarise some properties of the dynamics of two interacting particles moving under the action of a uniform magnetic field.

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Consider n = 2l ≥ 4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group D_l, where D_l is the group of order 2l generated by two rotations of angle π around two secant lines in space meeting at an angle of π/l. By adding a homogeneous gravitational (Newtonian) potential one finds a special n-body problem with three degrees of freedom, in which all orbits have zero angular momentum.

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We derive a Lax-representation for integrable maps (or OΔEs) obtained by travelling wave reductions from integrable PΔEs with Lax pairs.

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We consider the problem of calculating the pseudorotational energy levels associated with the molecular Jahn-Teller effect for electronic triplets. We show that the energy levels are related to the spectra of Laplace-Beltrami operators on line bundles over the leaves of Cartan's isoparametric foliation of S⁴. These spectra are computed explicitly using the Peter-Weyl theorem and harmonic analysis on line bundles.

These pages are an account of a research project aiming to classify hyperbolic PDE in the plane of generic type which are Darboux integrable. Here we focus on some symbolic computation tools, the Five Variables package the author developed, which work in the environment Differential Geometry of Maple11 and that can be used to produce a variety of examples.

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An open problem is the global classification, up to linear transformations, of polynomial two-component integrable equations (u_t, v_t) = K,

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The goal of the present investigation is to understand the typical bifurcation patterns organized around a Hopf-saddle-node (HSN) bifurcation of fixed points, defined as follows: a C^∞-family of diffeomorphisms F_α : ℝ³ → ℝ³, where α ∈ ℝ^P is a multi-parameter, is an HSN-family if

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We discuss construction of relative differential invariants for Lie algebras that is relevant e.g. for description of equations invariant under these algebras.

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SPT2007 – List of participants

List of communications presented at SPT2007

Previous SPT conferences and proceedings