Recent zbMATH articles in MSC 70Hhttps://www.zbmath.org/atom/cc/70H2021-04-16T16:22:00+00:00WerkzeugTopology change of level sets in Morse theory.https://www.zbmath.org/1456.370612021-04-16T16:22:00+00:00"Knauf, Andreas"https://www.zbmath.org/authors/?q=ai:knauf.andreas"Martynchuk, Nikolay"https://www.zbmath.org/authors/?q=ai:martynchuk.nikolayThis paper considers Morse functions \(f \in C^2(M, \mathbb{R})\) on a smooth \(m\)-dimensional manifold without boundary. For such functions and for every critical point \(x \in M\) of \(f\), the Hessian of \(f\) at \(x\) is nondegenerate.
The authors are interested in the topology of the level sets \(f^{-1}(a) = \partial{M^a}\) where \(M^a = \{ x \in M : f(x) \le a\}\) is a so-called sublevel set. The authors address the question of under what conditions the topology of \(\partial{M^a}\) can change when the function \(f\) passes a critical level with one or more critical points. ``Change in topology'' here means roughly that if \(a\) and \(b\) are regular values with \(a < b\), then \(H_k(\partial{M^a};G) \ne H_k(\partial{M^b};G)\) where \(k\) is an integer and the \(H_k\) are homology groups over some abelian group \(G\) when the function \(f\) passes through a level with one or more critical points. A motivation for this question derives from the \(n\)-body problem: does the topology of the integral manifolds always change when passing through a bifurcation level?
The first part of the paper considers level sets of abstract Morse functions that satisfy the Palais-Smale condition and whose level sets have finitely generated homology groups. They show that for such functions the topology of \(f^{-1}(a)\) changes when passing a single critical point if the index of the critical point is different from \(m/2\) where \(m\) is the dimension of the manifold \(M\). Then they move on to consider the case where \(M\) is a vector bundle of rank \(n\) over a manifold \(N\) of dimension \(n\), and where (up to a translation) \(f\) is a Morse function that is the sum of a positive definite quadratic form on the fibers and a potential function that is constant on the fibers. Here the authors have in mind Hamiltonian functions \(H\) on the cotangent bundle of the base manifold \(N\). In this case they show that the topology of \(H^{-1}(h)\) always changes when passing a single critical point if the Euler characteristic of \(N\) is not \(\pm 1\).
Finally the authors apply their results to examples from Hamiltonian and celestial mechanics, with emphasis on the planar three-body problem. There they show that the topology always changes for the planar three-body problem provided that the reduced Hamiltonian is a Morse function with at most two critical points on each level set.
Reviewer: William J. Satzer Jr. (St. Paul)Double obstacle problems and fully nonlinear PDE with non-strictly convex gradient constraints.https://www.zbmath.org/1456.352362021-04-16T16:22:00+00:00"Safdari, Mohammad"https://www.zbmath.org/authors/?q=ai:safdari.mohammadSummary: We prove the optimal \(W^{2, \infty}\) regularity for fully nonlinear elliptic equations with convex gradient constraints. We do not assume any regularity about the constraints; so the constraints need not be \(C^1\) or strictly convex. We also show that the optimal regularity holds up to the boundary. Our approach is to show that these elliptic equations with gradient constraints are related to some fully nonlinear double obstacle problems. Then we prove the optimal \(W^{2, \infty}\) regularity for the double obstacle problems. In this process, we also employ the monotonicity property for the second derivative of obstacles, which we have obtained in a previous work.The Hamilton-Jacobi equation and holographic renormalization group flows on sphere.https://www.zbmath.org/1456.830822021-04-16T16:22:00+00:00"Kim, Nakwoo"https://www.zbmath.org/authors/?q=ai:kim.nakwoo"Kim, Se-Jin"https://www.zbmath.org/authors/?q=ai:kim.se-jinSummary: We study the Hamilton-Jacobi formulation of effective mechanical actions associated with holographic renormalization group flows when the field theory is put on the sphere and mass terms are turned on. Although the system is supersymmetric and it is described by a superpotential, Hamilton's characteristic function is not readily given by the superpotential when the boundary of AdS is curved. We propose a method to construct the solution as a series expansion in scalar field degrees of freedom. The coefficients are functions of the warp factor to be determined by a differential equation one obtains when the ansatz is substituted into the Hamilton-Jacobi equation. We also show how the solution can be derived from the BPS equations without having to solve differential equations. The characteristic function readily provides information on holographic counterterms which cancel divergences of the on-shell action near the boundary of AdS.Classical motions of infinitesimal rotators on Mylar balloons.https://www.zbmath.org/1456.530102021-04-16T16:22:00+00:00"Kovalchuk, Vasyl"https://www.zbmath.org/authors/?q=ai:kovalchuk.vasyl"Mladenov, Ivaïlo"https://www.zbmath.org/authors/?q=ai:mladenov.ivailo-mSummary: This paper starts with the derivation of the most general equations of motion for the infinitesimal rotators moving on arbitrary two-dimensional surfaces of revolution. Both geodesic and geodetic (i.e., without any external potential) equations of motion on surfaces with nontrivial curvatures that are embedded into the three-dimensional Euclidean space are discussed. The Mylar balloon as a concrete example for the application of the scheme was chosen. A new parameterization of this surface is presented, and the corresponding equations of motion for geodesics and geodetics are expressed in an analytical form through the elliptic functions and elliptic integrals. The so-obtained results are also compared with those for the two-dimensional sphere embedded into the three-dimensional Euclidean space for which it can be shown that the geodesics and geodetics are plane curves realized as the great and small circles on the sphere, respectively.The globalization problem of the Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework.https://www.zbmath.org/1456.530732021-04-16T16:22:00+00:00"Esen, Oğul"https://www.zbmath.org/authors/?q=ai:esen.ogul"de León, Manuel"https://www.zbmath.org/authors/?q=ai:de-leon.manuel"Sardón, Cristina"https://www.zbmath.org/authors/?q=ai:sardon.cristina"Zając, Marcin"https://www.zbmath.org/authors/?q=ai:zajac.marcinSummary: In this paper, we aim at addressing the globalization problem of Hamilton-DeDonder-Weyl equations on a local \(k\)-symplectic framework and we introduce the notion of \textit{locally conformal k-symplectic (l.c.k-s.) manifolds}. This formalism describes the dynamical properties of physical systems that locally behave like multi-Hamiltonian systems. Here, we describe the local Hamiltonian properties of such systems, but we also provide a global outlook by introducing the global Lee one-form approach. In particular, the dynamics will be depicted with the aid of the Hamilton-Jacobi equation, which is specifically proposed in a l.c.k-s manifold.Energy dissipation in Hamiltonian chains of rotators.https://www.zbmath.org/1456.370652021-04-16T16:22:00+00:00"Cuneo, Noé"https://www.zbmath.org/authors/?q=ai:cuneo.noe"Eckmann, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:eckmann.jean-pierre"Wayne, C. Eugene"https://www.zbmath.org/authors/?q=ai:wayne.c-eugeneHerman's approach to quasi-periodic perturbations in the reversible KAM context 2.https://www.zbmath.org/1456.700382021-04-16T16:22:00+00:00"Sevryuk, Mikhail B."https://www.zbmath.org/authors/?q=ai:sevryuk.mikhail-borisovichSummary: We revisit non-autonomous systems depending quasi-periodically in time within the reversible context 2 of KAM theory and obtain Whitney smooth families of invariant tori in such systems via Herman's method. The reversible KAM context 2 refers to the situation where the dimension of the fixed point manifold of the reversing involution is less than half the codimension of the invariant torus in question.New bi-Hamiltonian systems on the plane.https://www.zbmath.org/1456.370592021-04-16T16:22:00+00:00"Tsiganov, A. V."https://www.zbmath.org/authors/?q=ai:tsiganov.andrey-vladimirovichSummary: We discuss several new bi-Hamiltonian integrable systems on the plane with integrals of motion of third, fourth, and sixth orders in momenta. The corresponding variables of separation, separated relations, compatible Poisson brackets, and recursion operators are also presented in the framework of the Jacobi method.{
\copyright American Institute of Physics}Explicit estimates on the measure of primary KAM tori.https://www.zbmath.org/1456.370632021-04-16T16:22:00+00:00"Biasco, L."https://www.zbmath.org/authors/?q=ai:biasco.luca"Chierchia, L."https://www.zbmath.org/authors/?q=ai:chierchia.luigiSummary: From KAM theory it follows that the measure of phase points which do not lie on Diophantine, Lagrangian, ``primary'' tori in a nearly integrable, real-analytic Hamiltonian system is \(O(\sqrt{\varepsilon })\), if \(\varepsilon\) is the size of the perturbation. In this paper, we discuss how the constant in front of \(\sqrt{\varepsilon }\) depends on the unperturbed system and in particular on the phase-space domain.A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann's translated curve theorem to higher dimensions.https://www.zbmath.org/1456.370662021-04-16T16:22:00+00:00"Massetti, Jessica Elisa"https://www.zbmath.org/authors/?q=ai:massetti.jessica-elisaSummary: We prove a discrete time analogue of \textit{J. Moser}'s normal form [Math. Ann. 169, 136--176 (1967; Zbl 0149.29903)] of real analytic perturbations of vector fields possessing an invariant, reducible, Diophantine torus; in the case of diffeomorphisms too, the persistence of such an invariant torus is a phenomenon of finite codimension. Under convenient nondegeneracy assumptions on the diffeomorphisms under study (a torsion property for example), this codimension can be reduced. As a by-product we obtain generalizations of Rüssmann's translated curve theorem in any dimension, by a technique of elimination of parameters.A new approach to the parameterization method for Lagrangian tori of Hamiltonian systems.https://www.zbmath.org/1456.370672021-04-16T16:22:00+00:00"Villanueva, Jordi"https://www.zbmath.org/authors/?q=ai:villanueva.jordiSummary: We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov's non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of \({\mathcal {O}}(\varepsilon ^{1/2})\), where \(\varepsilon \) is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation.https://www.zbmath.org/1456.830162021-04-16T16:22:00+00:00"Tanzi, Roberto"https://www.zbmath.org/authors/?q=ai:tanzi.roberto"Giulini, Domenico"https://www.zbmath.org/authors/?q=ai:giulini.domenico-j-wSummary: We investigate the asymptotic symmetry group of the free \(\mathrm{SU} (N)\)-Yang-Mills theory using the Hamiltonian formalism. We closely follow the strategy of Henneaux and Troessaert who successfully applied the Hamiltonian formalism to the case of gravity and electrodynamics, thereby deriving the respective asymptotic symmetry groups of these theories from clear-cut first principles. These principles include the minimal assumptions that are necessary to ensure the existence of Hamiltonian structures (phase space, symplectic form, differentiable Hamiltonian) and, in case of Poincaré invariant theories, a canonical action of the Poincaré group. In the first part of the paper we show how these requirements can be met in the non-abelian \(\mathrm{SU} (N)\)-Yang-Mills case by imposing suitable fall-off and parity conditions on the fields. We observe that these conditions admit neither non-trivial asymptotic symmetries nor non-zero global charges. In the second part of the paper we discuss possible gradual relaxations of these conditions by following the same strategy that Henneaux and Troessaert had employed to remedy a similar situation in the electromagnetic case. Contrary to our expectation and the findings of Henneaux and Troessaert for the abelian case, there seems to be no relaxation that meets the requirements of a Hamiltonian formalism \textit{and} allows for non-trivial asymptotic symmetries and charges. Non-trivial asymptotic symmetries and charges are only possible if either the Poincaré group fails to act canonically or if the formal expression for the symplectic form diverges, i.e. the form does not exist. This seems to hint at a kind of colour-confinement built into the classical Hamiltonian formulation of non-abelian gauge theories.On the measure of KAM tori in two degrees of freedom.https://www.zbmath.org/1456.370642021-04-16T16:22:00+00:00"Biasco, Luca"https://www.zbmath.org/authors/?q=ai:biasco.luca"Chierchia, Luigi"https://www.zbmath.org/authors/?q=ai:chierchia.luigiThis paper is devoted to the following
conjecture of \textit{V. I. Arnold} et al. [Mathematical aspects of classical and celestial mechanics. Transl. from the Russian by E. Khukhro. 3rd revised ed. Berlin: Springer (2006; Zbl 1105.70002)]: ``It is natural to expect that in a generic (analytic) system with two degrees of freedom and with frequencies that do not vanish simultaneously the total measure of the `non-torus' set corresponding to all the resonances is exponentially small.''
The authors of this paper investigate real analytic nearly-integrable systems with two degrees of freedom, and look specifically at Hamiltonians of the form (in action-angle coordinates):
\[H_\epsilon (y,x) = \frac{y_1^2 + y_2^2}{2} + \epsilon f(x_1,x_2),\]
where \(y = (y_1, y_2) \in \mathbb{R}^2\) and \(x = (x_1, x_2) \in \mathbb{T}^2\), \(f: \mathbb{T}^2 \rightarrow \mathbb{R}\) is real-analytic and \(\epsilon \ge 0\) is a small parameter.
The phase space \(\mathbb{R}^2 \times \mathbb{T}^2\) carries the standard symplectic form and the flow \(\Phi^t_{H_\epsilon}\) induced by \(H_\epsilon\) is a solution of the Hamiltonian equations \(\dot{y} = - \partial_x H_\epsilon = -\epsilon f_x\), and \(\dot{x} = \partial_y H_\epsilon = y + \epsilon f_y\).
The system is integrable when \(\epsilon = 0\). The integrals of motion are the action variables \(y_1\) and \(y_2\), and the trajectories are \(y(t) = y_0\) and \(x(t) = x_0 + \omega t\), where \(\omega\) is the frequency corresponding to \(y_0\). The 2-tori \(\{y_0\} \times \mathbb{T}^2\) are invariant under the Hamiltonian flow. According to KAM theory most integrable tori \(\{y_0\} \times \mathbb{T}^2\) persist for small \(\epsilon\).
The authors focus on the non-torus set of analytic systems with two degrees of freedom. They consider the Banach space \(\mathbb{B}^2_s\) of real analytic functions on the torus \(\mathbb{T}^2_s\) consisting of \(x = (x_1, x_2) \in \mathbb{C}^2\) for which the absolute value of the imaginary parts of \(x_1\) and \(x_2\) are less than \(s\). The space \(\mathbb{B}^2_s\) consists of real analytic functions on \(\mathbb{T}^2_s\) having zero-average and finite \(l^\infty\) norm.
The main result is as follows. For \(s > 0\) there is a set \(\mathcal{P}_s \in \mathbb{B}^2_s\) containing an open and dense set so that the following conditions hold: for fixed \(r\) and \(R\) with \(0 < r < R\), take \(D = \{y \in \mathbb{R}^2 : r \le |y| \le R\}\) and consider the Hamiltonian system with phase space \(D \times \mathbb{T}^2\) and Hamiltonian \(H_\epsilon\) (as described above) with potential \(f \in \mathcal{P}_s\). Then there exist \(\epsilon_0\) and \(a > 0\) small enough so that if \(0 < \epsilon < \epsilon_0\), the Liouville measure of the complementary set of the \(\Phi^t_{H_\epsilon}\)-invariant tori in the phase region \(D\) is smaller than \(R^2 e^{-k/ \epsilon^a}\) where \(k > 0\) is a constant.
A sketch of the proof is provided, but a detailed proof is deferred to a later paper.
Reviewer: William J. Satzer Jr. (St. Paul)Topological classification of Hamiltonian systems on two-dimensional noncompact manifolds.https://www.zbmath.org/1456.370622021-04-16T16:22:00+00:00"Nikolaenko, Stanislav S."https://www.zbmath.org/authors/?q=ai:nikolaenko.stanislav-sThis paper presents a topological classification of Hamiltonian systems on two-dimensional noncompact symplectic manifolds. The author continues along the line of research established in [\textit{D. A. Fedoseev} and \textit{A. T. Fomenko}, Fundam. Prikl. Mat., 21, No. 6, 217---243 (2016)].
The author investigates topological invariants of foliations of finite type defined by smooth functions on two-dimensional noncompact orientable manifolds. The goal is to describe a complete topological classification of noncompact bifurcations for foliations like this.
Fomenko's approach uses the compactness of leaves in the Liouville foliation (the partition of the phase manifold into connected components of common level surfaces of the first integrals, known as Liouville leaves). Liouville foliations with noncompact leaves appear in many systems in mechanics, so there is motivation to extend the theory to systems with noncompact leaves.
This paper completely solves the problem of trajectory classification for Hamiltonian systems with noncompact foliations in the case of systems with one degree of freedom of finite type. Such Hamiltonian systems are said to be of finite type if the foliation defined by the Hamiltonian \(H\) is of finite type, and this means that the number of bifurcation values of \(H\) is finite and that the atoms corresponding to the bifurcation values are atoms of finite type. The author does not assume that the system is nondegenerate or that the Hamiltonian flow is complete.
In systems with one degree of freedom almost all Liouville leaves are one-dimensional; they are the level curves of the Hamiltonian, and each leaf consists of one or more trajectories. Liouville and trajectory classification are essentially identical in this case except for the orientation of trajectories. The symplectic structure is not relevant, except for enforcing orientability.
Special difficulties arise with noncompact manifolds. Much of the paper is devoted to handling those issues. In the end the author provides a natural one-to-one correspondence between the set of noncompact bifurcations of foliations and a set of oriented colored graphs that have a special form.
Reviewer: William J. Satzer Jr. (St. Paul)A study of a generalized first extended (3+1)-dimensional Jimbo-Miwa equation.https://www.zbmath.org/1456.350872021-04-16T16:22:00+00:00"Khalique, Chaudry Masood"https://www.zbmath.org/authors/?q=ai:khalique.chaudry-masood"Moleleki, Letlhogonolo Daddy"https://www.zbmath.org/authors/?q=ai:moleleki.letlhogonolo-daddySummary: This paper aims to study a generalized first extended (3+1)-dimensional Jimbo-Miwa equation. Symmetry reductions on this equation are performed several times and it is reduced to a nonlinear fourth-order ordinary differential equation. The general solution of this ordinary differential equation is found in terms of the incomplete elliptic integral function. Also exact solutions are constructed using the \(({G'}/{G})\)-expansion method. Thereafter the conservation laws of the underlying equation are computed by invoking the conservation theorem due to Ibragimov. The conservation laws obtained contain an energy conservation law and three momentum conservation laws.The sign exchange bifurcation in a family of linear Hamiltonian systems.https://www.zbmath.org/1456.370552021-04-16T16:22:00+00:00"Cushman, Richard"https://www.zbmath.org/authors/?q=ai:cushman.richard-h"Robbins, Johnathan M."https://www.zbmath.org/authors/?q=ai:robbins.johnathan-m"Sadovskii, Dimitrii"https://www.zbmath.org/authors/?q=ai:sadovskii.dimitriiThis paper considers a one-parameter family of Hamiltonian systems that describe two classical particles with angular momenta under the influence of a magnetic field. The phase space for the system is \(\mathbb{R}^3 \times \mathbb{R}^3\) with coordinates \((S, L) = (S_x, S_y, S_z, L_x, L_y, L_z)\).
For \(\gamma \in [0, 1]\) the Hamiltonian is defined by \[H_{\gamma} (S, L) = \frac {1 - \gamma}{s} S_z + \frac {\gamma}{sl} S \cdot L,\] where \(s^2 = S \cdot S\) and \(l^2 = L \cdot L\). The motion of the system arises from the Hamiltonian vector field \(X_{H_\gamma}\) with integral curves that satisfy \(\frac {dS}{dt} = \frac {1 - \gamma}{s} e_3 \times S - \frac {\gamma}{sl} S \times L\) and \(\frac{dL}{dt} = \frac{\gamma}{sl} S \times L\) where \(\{e_1, e_2, e_3\}\) is the standard basis of \(\mathbb{R}^3\).
The functions \(C_1 = S_x^2 + S_y^2 +S_z^2\) and \(C_2 = L_x^2 + L_y^2 + L_z^2\) are integrals of the vector field \(X_{H_\gamma}\), so the smooth manifold \(S^2_s + S^2_l\) defined by \(C_1 = s^2\) and \(C_2 = l^2\) is invariant under the flow of \(X_{H_\gamma}\). The authors then consider the vector field \(X_{J_z}\) where \(J_z = S_z + L_z\). \(X_{J_z}\) has integral curves that satisfy \(\frac{dS}{dt} = e_3 \times S\) and \(\frac{dL}{dt} = e_3 \times L\). The flow of \(X_{J_z}\) defines an \(S^1\)-action on \(\mathbb{R}^3 + \mathbb{R}^3\) that is a symmetry of the vector field \(X_{J_z}\). Because the flows of \(X_{J_x}\) and \(X_{H_\gamma}\) commute, the \(S^1\)-action is also a symmetry of \(X_{H_\gamma}\) .
The only equilibrium points of \(X_{H_\gamma}\) have the form \(p = (\eta_s se_3, \eta_l le_3)\) where \(\eta_s^2 = \eta_l^2 = 1\); \(p\) lies on the invariant manifold \(S^2_s + S^2_l\). After linearizing \(X_{H_\gamma (p)}\) to get \(DX_{H\gamma}(p)\), the authors assume that \(s = \epsilon^2 l\) where \( 0 < \epsilon \ll 1\) and consider the \(S^1\)-invariant curve \(\mathcal{X}\) that maps \(\gamma\) in \([0,1]\) to \(DX_{H\gamma}(p)\). Their goal is to show that as \(\gamma\) increases, the curve at \(p = (se_3, -le_3)\) sees \(S^1\)-equivariant bifurcations at values of \(\gamma\) where \(DX_{H_\gamma}\) has multiple eigenvalues (all near \(\frac{1}{2}\) because of scaling). In particular, they show that at \(\gamma = \frac{1}{2 - \epsilon^2}\) the curve has a switch twist bifurcation where the sense of the rotational part of \(DX_{H_\gamma}(p) \) changes. They also show that at \(p = (-se_3, -le_3)\), the curve \(\mathcal{X}\) has a sign exchange bifurcation at \(\gamma = \frac{1}{2 + 2\epsilon + \epsilon^2}\).
For the entire collection see [Zbl 1280.37002].
Reviewer: William J. Satzer Jr. (St. Paul)Homoclinic solutions for a class of nonlinear fourth order \(p\)-Laplacian differential equations.https://www.zbmath.org/1456.340442021-04-16T16:22:00+00:00"Dimitrov, Nikolay D."https://www.zbmath.org/authors/?q=ai:dimitrov.nikolay-d"Tersian, Stepan A."https://www.zbmath.org/authors/?q=ai:tersian.stepan-agopIn this paper, the authors deal with a class of fourth-order differential equation involving \(p\)-Laplacian
\[
|u''(x)|^{p-2}u''(x))''+\omega (|u'(x)|^{p-2}u'(x))'+\lambda V(x) |u(x)|^{p-2} u(x)=f(x,u(x)
\]
where \(\omega\) is a constant, \(\lambda\) is a parameter and \(f\in C (\mathbb{R},\mathbb{R})\). with the aid of critical point theory and variational methods, they prove that under suitable growth conditions, the above equation possesses at least one nontrivial homoclinic solution, i.e., a nontrivial solution satisfying \(u(x)\longrightarrow 0\) as \(x\longrightarrow \mp\infty\).
Reviewer: Mohsen Timoumi (Monastir)The Foucault pendulum (with a twist).https://www.zbmath.org/1456.370442021-04-16T16:22:00+00:00"Moeckel, Richard"https://www.zbmath.org/authors/?q=ai:moeckel.richardQuasiperiodic Hamiltonian motions, scale invariance, harmonic oscillators.https://www.zbmath.org/1456.700322021-04-16T16:22:00+00:00"Gallavotti, Giovanni"https://www.zbmath.org/authors/?q=ai:gallavotti.giovanni-mSummary: The work of Kolmogorov, Arnold, and Moser appeared just before the renormalization group approach to statistical mechanics was proposed by \textit{ K. G. Wilson} [Phys. Rev. B (3) 4, No. 9, 3174--3183 (1971; Zbl 1236.82017)]: it can be classified as a multiscale approach which also appeared in works on the convergence of Fourier's series [\textit{ L. Carleson}, Acta Math. 116, 135--157 (1966), 135--157 (1966; Zbl 0144.06402)] and C. Fefferman, Ann. Math. (2) 98, 551--571 (1973; Zbl 0268.42009); erratum ibid. 146, 239 (1997)] or construction of Euclidean quantum fields [E. Nelson, J. Math. Phys. 5, 1190--1197 (1964)] or the scaling analysis of the short scale behavior of Navier-Stokes fluids [\textit{ L. Caffarelli, R. Kohn}, and \textit{ L. Nirenberg}, Commun. Pure Appl. Math. 35, 771--831 (1982; Zbl 0509.35067)] to name a -- few, which resulted in numerous problems. In this review, the KAM theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a ``trivial'' fixed point.{\par\copyright 2019 American Institute of Physics}Classification of a modified de Sitter metric by variational symmetries and conservation laws.https://www.zbmath.org/1456.580112021-04-16T16:22:00+00:00"Beesham, A."https://www.zbmath.org/authors/?q=ai:beesham.aroonkumar"Gadjagboui, B. B. I."https://www.zbmath.org/authors/?q=ai:gadjagboui.b-b-i"Kara, A. H."https://www.zbmath.org/authors/?q=ai:kara.abdul-hamidIntegrable homogeneous dissipative dynamical systems of an arbitrary odd order.https://www.zbmath.org/1456.370582021-04-16T16:22:00+00:00"Shamolin, M. V."https://www.zbmath.org/authors/?q=ai:shamolin.m-vSummary: We establish the integrability of homogeneous (in some variables) dynamical systems with dissipation in the case of an arbitrary odd order and thereby generalize the results earlier obtained by the author in particular cases of such systems.