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A Renormalization Proof of the KAM Theorem for Non-Analytic Perturbations  [PDF]
Emiliano De Simone
Physics , 2006, DOI: 10.1142/S0129055X07003085
Abstract: We shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non-analytic perturbation (the latter will be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which the perturbations are analytic approximations of the original one. We shall finally show that the sequence of the approximate solutions will converge to a differentiable solution of the original problem.
Weak KAM theorem on non compact manifolds  [PDF]
Albert Fathi,Ezequiel Maderna
Mathematics , 2015, DOI: 10.1007/s00030-007-2047-6
Abstract: In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.
The classical KAM theorem for Hamiltonian systems via rational approximations  [PDF]
Abed Bounemoura,Stephane Fischler
Mathematics , 2014, DOI: 10.1134/S1560354714020087
Abstract: 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-R\"ussmann 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 a previous work for perturbations of constant vector fields on the torus.
KAM Theorem and Renormalization Group  [PDF]
E. De Simone,A. Kupiainen
Physics , 2007,
Abstract: We give an elementary proof of the analytic KAM theorem by reducing it to a Picard iteration of a PDE with quadratic nonlinearity, the so called Polchinski renormalization group equation studied in quantum field theory.
Twistless KAM tori  [PDF]
Giovanni Gallavotti
Physics , 1993, DOI: 10.1007/BF02108809
Abstract: A selfcontained proof of the KAM theorem in the Thirring model is discussed.
A lecture on the classical KAM theorem  [PDF]
Jürgen P?schel
Mathematics , 2009,
Abstract: The purpose of this lecture is to describe the KAM theorem in its most basic form and to give a complete and detailed proof. This proof essentially follows the traditional lines laid out by the inventors of this theory, and the emphasis is more on the underlying ideas than on the sharpness of the arguments.
KAM for the non-linear Beam equation 2: A normal form theorem  [PDF]
L. Hakan Eliasson,Beno?t Grèbert,Serge? B. Kuksin
Mathematics , 2015,
Abstract: We prove an abstract KAM theorem adapted to space-multidimensional hamiltonian PDEs with regularizing nonlinearities. It applies in particular to the singular perturbation problem studied in the first part of this work.
An alternative proof of the non-Archimedean Montel theorem for polynomial dynamics  [PDF]
Junghun Lee
Mathematics , 2015,
Abstract: We will see an alternative proof of the non-Archimedean Montel theorem, which is also called Hsia's criterion, for polynomial dynamics.
An abstract KAM theorem  [PDF]
Mauricio Garay
Mathematics , 2013,
Abstract: The KAM iterative scheme turns out to be effective in many problems arising in perturbation theory. I propose an abstract version of the KAM theorem to gather these different results.
KAM Theorem and Quantum Field Theory  [PDF]
J. Bricmont,K. Gawedzki,A. Kupiainen
Physics , 1998, DOI: 10.1007/s002200050573
Abstract: We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are shown to follow from the Ward identities expressing the translation invariance of the corresponding field theory.
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