PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Classical mechanics → Chaos and nonlinear dynamics ↓


Hamiltonian Chaos


Hamiltonian chaos is a fascinating concept in classical mechanics and dynamical systems, exploring how chaotic dynamics can emerge from energy-conserving deterministic systems. In the field of classical mechanics, a system is described by a Hamiltonian, a function that contains the total energy of the system, usually expressed as the sum of the kinetic and potential energies. Understanding how chaos arises in Hamiltonian systems is crucial to advancing fields such as celestial mechanics, quantum physics, and even modern electronics.

Basis of Hamiltonian systems

A Hamiltonian system is mathematically described by Hamilton's equations. These are a set of first-order differential equations that describe the evolution of the energy configuration of a system over time. The Hamiltonian H(q, p) is a function of the generalized coordinate q and momentum p :

    (frac{dq}{dt} = frac{partial H}{partial p}) (frac{dp}{dt} = -frac{partial H}{partial q})
    (frac{dq}{dt} = frac{partial H}{partial p}) (frac{dp}{dt} = -frac{partial H}{partial q})

In Hamiltonian mechanics, the state of a system is represented as a point in phase space, a multidimensional space constructed along axes of position and momentum. The evolution of a system traces out a path or trajectory in this phase space.

Phase space and energy conservation

An essential feature of Hamiltonian systems is the conservation of their phase space volume, a property known as Liouville's theorem. This means that although a trajectory can trace a complicated path through phase space, it does so without losing information about its initial position, similar to a die moving in a conserved volume of liquid.

Energy Surface

This diagram illustrates the concept of trajectories confined to energy surfaces in phase space. Such constraints are due to the energy conservation principles inherent in Hamiltonian systems, which restrict the evolution of the system to these surfaces.

Routine and disorganized behavior

In a traditional Hamiltonian system, the solutions exhibit regular, predictable behavior, characterized by periodic or quasi-periodic orbits. However, when perturbations are introduced, often even minor ones, they can alter the trajectories substantially leading to chaotic behavior.

Visualization of regular versus chaotic dynamics

Let's consider a simple example - integrable and well-ordered harmonic oscillator, where chaos does not arise:

    H = frac{1}{2m}p^2 + frac{1}{2}kq^2
    H = frac{1}{2m}p^2 + frac{1}{2}kq^2

Here, the energy surface is an ellipse, and the trajectory is circular, repeating over time. In contrast, chaotic behavior in Hamiltonian systems can be characterized by examining slight modifications, such as elliptical orbits in a perturbed nonlinear pendulum:

Nonlinear trajectories

Discovery of the KAM principle

Kolmogorov-Arnold-Moser (KAM) theory provides a basis for understanding how some degree of regular behavior persists even when such a system is perturbed. KAM theory states that for small perturbations, most quasi-periodic orbits of an integrable system survive, while exhibiting some degree of chaotic behavior. Nevertheless, as the perturbation strength increases, the system may slide further into chaos.

Poincaré sections and chaos

A useful tool for exploring chaotic behavior in Hamiltonian systems is the Poincare section. This method involves taking cross-sections through a phase space plot to assess how the trajectories evolve. A notable hallmark of chaos is the scattering of points filling substantial portions of the Poincare section.

Poincaré sections

Dual nature: determinism and chaos

The concept of Hamiltonian chaos is a profound reflection of the duality of the universe, where deterministic laws of motion can unexpectedly give rise to chaos and unpredictability. Despite these systems being deterministic in principle, meaning that the future state can be accurately predicted given initial conditions, chaos is an example of how sensitive dependence on initial conditions can lead to unpredictable outcomes on long time scales.

Double pendulum: a classic example

A classical example of chaotic dynamics is the double pendulum, a system of two pendulums connected end-to-end. The mathematical complexity of this system makes it highly sensitive to initial conditions and is a perfect example of chaos in action:

    theta_1(t), theta_2(t) = f(theta_{1,0}, theta_{2,0}, omega_{1,0}, omega_{2,0}, t)
    theta_1(t), theta_2(t) = f(theta_{1,0}, theta_{2,0}, omega_{1,0}, omega_{2,0}, t)

The complexity of the motion in the double pendulum shows how initial deviations significantly alter future evolution, which is challenging to predict directly.

Symbolic dynamics and chaos

Symbolic dynamics provides a different approach by encoding the complex trajectories of chaotic systems into sequences of symbols that encapsulate the essence of the dynamic behavior. This technique helps to represent and track the manifestation of complexity within chaotic systems.

Implications and applications of Hamiltonian chaos

The understanding of Hamiltonian chaos goes beyond classical mechanics and affects a variety of fields. For example, in celestial mechanics, it helps to understand the complex gravitational interactions that determine planet and asteroid paths. In quantum systems, the semiclassical analysis of chaotic Hamiltonian systems aids in the exploration of phenomena such as quantum tunneling and statistical mechanics.

Chaotic dynamics in celestial mechanics

Long-term predictions about planetary motions naturally require taking into account Hamiltonian chaos due to the complexity of the n-body problem, where multiple gravitational interactions create complex and sometimes unpredictable orbital paths.

Ultimately, Hamiltonian chaos provides a bridge connecting deterministic classical systems to the probabilistic interpretations seen in modern physics, and expands our understanding of how simple rules govern complex realities, in both theoretical and practical contexts.


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Hamiltonian Chaos

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