The classical KAM theorem for Hamiltonian systems via rational approximations

In this paper, we give a new proof of the classical KAM theorem on the persistence of an invariant quasi-periodic torus, whose frequency vector satisfies the Bruno-Rüssmann condition, in real-analytic non-degenerate Hamiltonian systems close to integrable. The proof, which uses rational approximations instead of small divisors estimates, is an adaptation to the Hamiltonian setting of the method we introduced in [4] for perturbations of constant vector fields on the torus.     

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, which can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that underlies the analytic part of Nekhoroshev’s theorem to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.     

On Diophantine Approximations of Dependent Quantities in the p-adic Case

In the present paper, we prove an analog of Khinchin's metric theorem in the case of linear Diophantine approximations of plane curves defined over the ring of $p$ -adic integers by means of (Mahler) normal functions. We also prove some general assertions needed to generalize this result to the case of spaces of higher dimension.     

Destruction Of Lagrangian Torus For Positive Definite Hamiltonian Systems

For an integrable Hamiltonian ${H_0=\frac{1}{2} \sum_{i=1}^dy_i^2}$ ${(d \geq 2)}$ , we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily ${C^{2d-\delta}}$ -small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under ${C^{2d+\delta}}$ -small perturbations by Pöschel (Comm Pure Appl Math 35:653–696, 1982).     

Generic perturbations of linear integrable Hamiltonian systems

In this paper, we investigate perturbations of linear integrable Hamiltonian systems, with the aim of establishing results in the spirit of the KAM theorem (preservation of invariant tori), the Nekhoroshev theorem (stability of the action variables for a finite but long interval of time) and Arnold diffusion (instability of the action variables). Whether the frequency of the integrable system is resonant or not, it is known that the KAM theorem does not hold true for all perturbations; when the frequency is resonant, it is the Nekhoroshev theorem that does not hold true for all perturbations. Our first result deals with the resonant case: we prove a result of instability for a generic perturbation, which implies that the KAM and the Nekhoroshev theorem do not hold true even for a generic perturbation. The case where the frequency is nonresonant is more subtle. Our second result shows that for a generic perturbation the KAM theorem holds true. Concerning the Nekhrosohev theorem, it is known that one has stability over an exponentially long (with respect to some function of ε ^?1) interval of time and that this cannot be improved for all perturbations. Our third result shows that for a generic perturbation one has stability for a doubly exponentially long interval of time. The only question left unanswered is whether one has instability for a generic perturbation (necessarily after this very long interval of time).     

Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach

In this paper, we present and illustrate a general methodology to apply KAM theory in particular problems, based on an a posteriori approach. We focus on the existence of real analytic quasi-periodic Lagrangian invariant tori for symplectic maps. The purpose is to verify the hypotheses of a KAM theorem in an a posteriori format: Given a parameterization of an approximately invariant torus, we have to check non-resonance (Diophantine) conditions, non-degeneracy conditions and certain inequalities to hold. To check such inequalities, we require to control the analytic norm of some functions that depend on the map, the ambient structure and the parameterization. To this end, we propose an efficient computer-assisted methodology, using fast Fourier transform, having the same asymptotic cost of using the parameterization method for obtaining numerical approximations of invariant tori. We illustrate our methodology by proving the existence of invariant curves for the standard map (up to \(\varepsilon =0.9716\)), meandering curves for the non-twist standard map and 2-dimensional tori for the Froeschlé map.     

Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$

How to find “best rational approximations” of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)? In this paper we specify this problem and make first steps in its study. It contains the classical problems of Diophantine and simultaneous approximation as particular subcases but in general is much wider. We prove estimates for n = 2 for both totally real and complex cases and give an algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.     

Wigner measures supported on weak KAM tori

In the setting of the Weyl quantization on the flat torus \(\mathbb{T}^n \) , we exhibit a class of wave functions with uniquely associated Wigner probability measure, invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in phase space. These sets are the graphs of Lipschitz-continuous weak KAM solutions of negative type of the stationary Hamilton-Jacobi equation. Such Wigner measures are, in fact, given by the Legendre transform of Mather’s minimal probability measures.     

Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

In classical PDE theory, Widder’s theorem gives a representation for non-negative solutions of the heat equation on \(\mathbb{R }^n\) . We show that an analogous theorem holds for local weak solutions of the canonical “heat equation” on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.     

Examples of incomplete normed barrelled spaces

In this note we investigate spaces of the type $ L_{\varepsilon}^{p}(\mu)=\lbrace f\in L^{p}(\mu);{\rm supp}f\in \varepsilon \rbrace $ where ε is an ideal of “small” measurable sets with certain properties. Typically, these spaces endowed with the p-norm are not complete and thus, classical Banach space theory cannot be used.However, we prove that for good ideals ε the normed space $L_\varepsilon ^{p}(\mu)$ is ultrabornological and hence barrelled and therefore many theorems of functional analysis like the closed graph theorem or the uniform boundedness principle are indeed applicable.     

Perturbations of Smooth Actions with Nontrivial Cohomology

We prove a general perturbation result for smooth Lie group actions with nontrivial finite‐dimensional cohomology. It describes sufficient conditions on cohomology over an action which imply that the action lies in a finite‐dimensional family of actions such that any small perturbation of the family intersects the smooth conjugacy class of the given action. We cast the classical KAM result on perturbations of Diophantine vector fields on tori into this general setup, and we address a few applications and potential applications of this result to homogeneous Lie group actions with finite‐dimensional first cohomology. © 2014 Wiley Periodicals, Inc.     

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular $L$ -functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on $M_{k}^{!}$ in terms of periods.     

On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes

Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an $O(\Delta t$ )) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.     

On the convergence of a smoothed penalty algorithm for semi-infinite programming

For semi-infinite programming (SIP), we consider a class of smoothed penalty functions, which approximate the exact $l_\rho (0<\rho \le 1)$ penalty functions. On base of the smoothed penalty function, we present a feasible penalty algorithm for solving SIP. Without any boundedness condition or coercive condition, we establish the global convergence theorem. Then we present a perturbation theorem for this algorithm and obtain a necessary and sufficient condition for the convergence to the optimal value of SIP. Under Mangasarian–Fromovitz constrained qualification condition, we further discuss the convergence properties for the algorithm based upon a subclass of smooth approximations to the exact $l_\rho $ penalty function. Finally, numerical results are given.     

On the representation of tight functionals as integrals

We consider a vector lattice $\mathcal L $ of bounded real continuous functions on a topological space $X$ that separates the points of $X$ and contains the constant functions. A notion of tightness for linear functionals is defined, and by an elementary argument we prove with the aid of the classical Riesz representation theorem that every tight continuous linear functional on $\mathcal L $ can be represented by integration with respect to a Radon measure. This result leads incidentally to an simple proof of Prokhorov’s existence theorem for the limit of a projective system of Radon measures.     

A point-wise criterion for quasi-periodic motions in the KAM theory

We consider initial value problems for nearly integrable Hamiltonian systems. We formulate a sufficient condition for each initial value to admit the quasi-periodic solution with a Diophantine frequency vector, without any nondegeneracy of the integrable part. We reconstruct the KAM theorem under Rüssmann’s nondegeneracy by the measure estimate for the set of initial values satisfying this sufficient condition. Our point-wise version is of the form analogous to the corresponding problems for the integrable case. We compare our framework with the standard KAM theorem through a brief review of the KAM theory.     

The Category of Ladders

In the first part of this paper we introduce the category of ladders and develop the underlying theory. A ladder consists of a sequence of vector spaces (V _n ) and linear operators ${(A^+_n), (A^-_n)}$ acting between these vector spaces in ascending and descending direction. Unlike as in classical quantum mechanics ladders are defined as objects of a category, the corresponding notion of ladder homomorphisms allows to perform a mathematically structural and rigorous analysis of ladder theory. We allow dependence on n for spaces and operators in the ladder. The job in ladder theory is to find SIE-subladders, on which the Intrinsic Endomorphisms ${A^-_n A^+_n}$ and ${A^+_n A^-_n}$ act as Scalars α _n . A fundamental ladder theorem will provide conditions on the (generalized) commutators or anticommutators assuring the existence of SIE-subladders. The second part contains examples of ladders from classical quantum mechanics, such as the Heisenberg ladder, the Dirac ladder the ladder for the Lie algebra ${{\bf sl}(2,\mathbb{C})}$ . Whereas these classical examples are distinguished by constance of operators and spaces, we then show how the generalized ladder theory allows to handle deformations: h-discretization, q-discretization and “periodization”. Other examples come from orthogonal polynomials. The Legendre, Laguerre and Bessel ladder are presented. Ladder theory allows a certain “anticommutator factorization” of the relevant second order differential operators. In a final section we apply ladder theory to ladders that are at the same time complexes. This enables us to give a transparent structural proof of Hodge’s theorem. The idea of ladder operators and factorization is well-known in quantum mechanics and quantum field theory. The book of Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007) contains a good survey of applications in physics, it provides historical background and links to sources in the physics literature. Note that our approach is independent—not only with respect to notation—of that in Shi-Hai Dong (Factorization method in quantum mechanics, Springer, Dordrecht, 2007).     

Almost Ricci solitons and K-contact geometry

We give a short Lie-derivative theoretic proof of the following recent result of Barros et al. “A compact non-trivial almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere”. Next, we obtain the result: a complete almost Ricci soliton whose metric \(g\) is \(K\) -contact and flow vector field \(X\) is contact, becomes a Ricci soliton with constant scalar curvature. In particular, for \(X\) strict, \(g\) becomes compact Sasakian Einstein.     

Approximation by radial basis functions with finitely many centers

Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasiinterpolants on a certain “native” space of functions $\mathcal{F}_\Phi $ . Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form $\mathcal{F}_{\Phi _0 } $ for less smooth functions Φ₀. It turns out that interpolation by translates of Φ to mollifications of functionsf from $\mathcal{F}_{\Phi _0 } $ yields approximations tof that attain the same asymptotic error bounds as (optimal) interpolation off by translates of Φ₀ on $\mathcal{F}_{\Phi _0 } $ .     

Mather Measures Associated with a Class of Bloch Wave Functions

In this paper we study the Wigner transform for a class of smooth Bloch wave functions on the flat torus ${\mathbb{T}^n = \mathbb{R}^n /2\pi \mathbb{Z}^n}$ : $$\psi_{\hbar,P} (x) = a (\hbar,P,x) {\rm e}^{ \frac{i}{\hbar} ( P\cdot x + \hat{v}(\hbar,P,x) )}.$$ On requiring that ${P \in \mathbb{Z}^n}$ and ${\hbar = 1/N}$ with ${N \in \mathbb{N}}$ , we select amplitudes and phase functions through a variational approach in the quantum states space based on a semiclassical version of the classical effective Hamiltonian ${{\bar H}(P)}$ which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform of ${\psi_{\hbar,P}}$ admits subsequences converging in the weak* sense to Mather probability measures on the phase space. These measures are invariant for the classical dynamics and Action minimizing.