Pós-graduação
  1. Mecânica clássica  1.1. Cinemática Avançada  1.1.1. Generalized coordinate systems  1.1.2. Equações paramétricas de movimento  1.1.3. Referenciais não inerciais e forças fictícias  1.1.4. Formulação covariante do momento  1.1.5. Variável ação-ângulo  1.2. Mecânica Lagrangiana e Hamiltoniana  1.2.1. Princípio da ação mínima  1.2.2. Equações de Euler-Lagrange  1.2.3. Teorema de Noether e leis de conservação  1.2.4. Velocidade Normalizada  1.2.5. Conversão canônica  1.2.6. Bracket de Poisson e teoria de Hamilton-Jacobi  1.2.7. Caos Hamiltoniano e Integrabilidade  1.3. Mobilidade de corpo rígido  1.3.1. Euler's equations of motion  1.3.2. Momentos principais de inércia  1.3.3. Movimento giroscópico e precessão  1.3.4. Tensor de inércia  1.3.5. Estabilidade da velocidade rotacional  1.4. Dinâmica não linear e caos  1.4.1. Análise de espaço de fase e estabilidade  1.4.2. Seções de Poincaré e teoria de bifurcação  1.4.3. Atractores Estranhos e Fractais  1.4.4. Expoente de Lyapunov  1.4.5. KAM theorem and quasi-periodic motion
  2. Eletromagnetismo  2.1. Eletrodinâmica Avançada  2.1.1. Função de Green e teoria do potencial  2.1.2. Expansão multipolar  2.1.3. Método de carga de imagem  2.1.4. Condições de contorno e teorema de unicidade  2.2. Propagação de ondas eletromagnéticas  2.2.1. Ondas planas em dielétricos e condutores  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarização e cálculo de Jones  2.3. Eletrodinâmica relativística  2.3.1. Formulação covariante da eletrodinâmica  2.3.2. Potenciais de Liénard–Wiechert  2.3.3. Radiação de síncrotron e bremsstrahlung  2.3.4. Tensor de campo eletromagnético e transformações de Lorentz  2.4. Física de plasma  2.4.1. Magnetohidrodinâmica  2.4.2. Blindagem de Debye e oscilações do plasma  2.4.3. Ondas de Alfvén e confinamento em tokamak  2.4.4. Instabilidades de plasma e turbulência
  3. Mecânica estatística e termodinâmica  3.1. Termodinâmica Avançada  3.1.1. Transformadas de Legendre e potenciais termodinâmicos  3.1.2. Teorema da Flutuação–Dissipação  3.1.3. Relações interpessoais de Onsager  3.1.4. Critical events and phase transitions  3.2. Mecânica estatística quântica  3.2.1. Teoria de Matriz Densidade e Conjuntos  3.2.2. Condensados de Bose–Einstein  3.2.3. Transição de fase quântica  3.2.4. Estatísticas de Fermi–Dirac e Bose–Einstein  3.2.5. Funções de partição e ensembles grand canônicos
  4. Mecânica quântica  4.1. Mecânica de ondas avançada  4.1.1. Aproximação WKB  4.1.2. Formulação de integrais de caminho  4.1.3. Oscilador harmônico quântico e estados coerentes  4.2. Momento angular e spin  4.2.1. Spherical Harmonics  4.2.2. Coeficiente de Clebsch–Gordan  4.2.3. Teorema de Wigner–Eckart  4.2.4. Acoplamento spin-órbita  4.3. Teoria do espalhamento quântico  4.3.1. Born approximation  4.3.2. Análise de onda parcial  4.3.3. Teorema Óptico  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Segunda Quantização  4.4.2. Diagramas de Feynman e propagadores  4.4.3. Teoria da Renormalização  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Quebra espontânea de simetria e o mecanismo de Higgs
  5. Relatividade geral e cosmologia  5.1. Cálculo tensorial e geometria diferencial  5.1.1. Equações de campo de Einstein  5.1.2. Métricas de Schwarzschild e Kerr  5.1.3. Geodésicas e símbolos de Christoffel  5.2. Cosmologia e o Universo  5.2.1. A métrica FLRW e a inflação cósmica  5.2.2. Energia escura e formação de estruturas  5.2.3. Radiação cósmica de fundo em micro-ondas
  6. Física da matéria condensada  6.1. Estrutura de bandas e teoria do transporte  6.1.1. Teorema de Bloch e o modelo de Kronig–Penney  6.1.2. Topologia da superfície de Fermi  6.1.3. Efeito Hall Quântico  6.2. Supercondutividade  6.2.1. Teoria BCS e pares de Cooper  6.2.2. Efeito Meissner e quantização de fluxo  6.2.3. The Josephson Effect and SQUIDs  6.3. Fases topológicas da matéria  6.3.1. Isolantes Topológicos  6.3.2. Fermions de Majorana em fases topológicas da matéria
  7. Física Nuclear e de Partículas  7.1. Cromodinâmica Quântica (QCD)  7.1.1. Plasma de quarks e glúons  7.1.2. Liberdade Assintótica  7.1.3. Confinamento e hadronização  7.2. Além do Modelo Padrão  7.2.1. Teorias de Grande Unificação  7.2.2. Supersimetria e dimensões extras  7.2.3. Oscilações de neutrinos

Pós-graduação → Mecânica clássica → Dinâmica não linear e caos ↓


KAM theorem and quasi-periodic motion


The Kolmogorov-Arnold-Moser (KAM) theorem is a profound result in the field of nonlinear dynamics and chaos, dealing with the behavior of dynamical systems when they are subjected to small perturbations. It focuses in particular on the persistence of quasi-periodic motions in Hamiltonian systems. To understand the KAM theorem, it is important to explore the fundamental concepts of dynamical systems, Hamiltonian mechanics, and quasi-periodic motion.

Understanding dynamical systems

A dynamical system is a system that evolves over time according to a set of fixed rules. These systems can be described using differential equations in continuous time or differential equations in discrete time. In physics, many systems can be modeled using dynamical systems. For example, the motion of planets in the solar system, the swinging of a pendulum, and the flow of water in a river can all be described using dynamical models.

Hamiltonian systems

In classical mechanics, a Hamiltonian system is a type of dynamical system characterized by a Hamiltonian function, which typically represents the total energy (kinetic and potential) of the system. Hamiltonian mechanics provides a powerful framework for analyzing the evolution of a system over time.

The equations of motion for a Hamiltonian system are given by the Hamiltonian equation:

    ,
    frac{dq_i}{dt} = frac{partial H}{partial p_i}
    ,
    ,
    frac{dp_i}{dt} = -frac{partial H}{partial q_i}
    ,

where (q_i, p_i) are the normalized coordinates and momentum, respectively, and H is the Hamiltonian function.

Quasi-periodic motion

Quasi-periodic motion is a type of motion that occurs in certain dynamical systems where any given point in phase space returns to its initial position after a long time interval but never exactly the same one. This occurs when the motion is composed of many incoherent (non-rationally related) frequencies. Such motions are common in systems such as celestial bodies where the orbits can be viewed as quasi-periodic.

The above visual example shows a torus, which is often used to depict the phase space of a system exhibiting quasi-periodic motion.

Birth of the KAM theorem

In 1954, Andrey Kolmogorov proposed a groundbreaking theory that was refined by later mathematicians, notably Vladimir Arnold and Jürgen Moser. The KAM theorem addresses the stability of motion in Hamiltonian systems subject to small perturbations. It asserts that under certain conditions, if the perturbation of the Hamiltonian system is small enough, many of the originally present invariant tori (representing quasi-periodic motions) will persist.

Irreplaceable zucchini

An invariant torus is a toroidal surface in phase space on which the motion of a system can be confined. When a system exhibits quasi-periodic motion, it is confined to such a torus, with each point on the torus representing a specific state of the system.

Kolmogorov's insight

Kolmogorov's insight was to show that if you start with a non-degenerate integrable Hamiltonian system (a system that can be solved exactly), and make a small perturbation, there remain at most quasi-periodic motions on the invariant tori. These remaining tori are slightly deformed but retain their quasi-periodic character.

Role of Arnold and Moser

Arnold and Moser extended and refined Kolmogorov's ideas, providing rigorous mathematical proofs and extending the results to a wider class of systems. Their work showed that these deformed invariant tori remain stable and keep the dynamics of the system regular amid the potential chaos surrounding the perturbations.

Conditions and implications of the KAM theorem

For the KAM theorem to apply, several conditions must be met:

  • The system must start out close to integrable, which means that it can be described by a Hamiltonian which is a sum of terms involving only coordinates or only momenta.
  • The disturbance must be sufficiently small.
  • The system must satisfy the non-degeneracy condition, i.e. the frequency map is non-degenerate.

When these conditions are met, the theorem guarantees the continuity of many quasi-periodic orbits. This result has implications for many physical systems such as:

  • Planetary motions where small gravitational perturbations do not destroy the stability of the orbits.
  • Electrical circuits that maintain constant oscillation even with slight fluctuations.
  • Mechanical systems such as pendulums that remain stable even under small disturbances.

Limitations and non-applicability

While the KAM theorem is powerful, it does not mean that all quasi-periodic motions survive under perturbations. As perturbations increase, at some point, the invariant tori may break down, leading to chaotic behavior. These limits highlight the delicate balance between order and chaos within dynamical systems.

Example of quasi-periodic motion in the solar system

One of the most famous real-world examples of quasi-periodic motion, explained by the KAM theorem, is the motion of celestial bodies in our solar system. The planets revolve around the Sun in orbits that are not perfectly periodic due to the gravitational effects of the other planets. Their orbits are quasi-periodic instead.

Planet A Planet B

In the visual example above, we see elliptical orbits that exhibit quasi-periodic motion. Despite gravitational disturbances from other planets, the orbits maintain quasi-periodic stability, completing each cycle identically, but with slight variations over time.

Mathematical formulation

The mathematical formulation of the KAM theorem involves complex function analysis, but a simplified approach involves considering a Hamiltonian system of the following type:

    ,
    H(theta, I) = H_0(I) + varepsilon H_1(theta, I, varepsilon)
    ,

Here, (H_0(I)) is the integrable Hamiltonian part, and (varepsilon H_1(theta, I, varepsilon)) represents the perturbation, with (varepsilon) being a small parameter. The action variables (I) are constant for the integrable system, and the angle (theta) changes linearly with time.

When (varepsilon = 0), the system is integrable. But for small (varepsilon neq 0), under favorable conditions (such as non-degeneracy conditions on (H_0)), the KAM theorem implies the persistence of most invariant tori, while retaining the quasi-periodic nature.

Non-degenerate state

A key aspect is the non-decaying condition, which ensures that the frequencies on the tori do not resonate with each other, causing no instability in the system. Mathematically:

    ,
    frac{partial omega(I)}{partial I} neq 0
    ,

where (omega(I)) is the frequency function derived from (H_0(I)).

Conclusion

The KAM theorem is a cornerstone in understanding the transition between order and chaos in dynamical systems. It explains why many Hamiltonian systems exhibit robust quasi-periodic behavior, even under small perturbations. This stability, found in physical and abstract systems alike, underscores the profound, yet complex balance inherent in nonlinear dynamics.

Understanding the KAM theorem and its implications helps us understand the complexity and beauty of systems like our solar system, where order coexists with elements of unpredictability, all governed by beautiful laws of motion and dynamics.


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KAM theorem and quasi-periodic motion

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