Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

Graduate → Classical mechanics → Lagrangian and Hamiltonian mechanics ↓


Hamiltonian chaos and integrability


The field of classical mechanics provides us with powerful tools to describe the motion of particles and systems in the universe. Of these tools, Lagrangian and Hamiltonian mechanics provide two complementary approaches to understanding dynamics. Within this vast landscape, the concepts of chaos and unification play a key role in predicting the motion of a system and determining its structure. In this talk, we will explore Hamiltonian chaos and unification, highlighting these phenomena through the lens of Lagrangian and Hamiltonian mechanics.

Classical mechanics: Lagrangian and Hamiltonian formulation

Classical mechanics was revolutionized by the formalism of Joseph Louis Lagrange, whose work changed the way we understand motion. Instead of focusing on forces, as in Newtonian mechanics, the Lagrangian formulation focuses on the principle of stationary action. The basic quantity here is the Lagrangian L, defined as the difference between the kinetic energy T and potential energy V of a system:

L = T – V

Given the generalized coordinates q_i and their time derivatives (velocities) dot{q}_i, the dynamics of a system is determined by the Euler-Lagrange equations:

(frac{d}{dt} left( frac{partial L}{partial dot{q}_i} right) - frac{partial L}{partial q_i} = 0)

In contrast, the Hamiltonian approach, developed by William Rowan Hamilton, transforms the equations of motion into a form involving generalized momenta p_i. The Hamiltonian H is expressed as:

H = sum_i p_i dot{q}_i - L

The equations describing this system are Hamilton's equations:

(dot{q}_i = frac{partial H}{partial p_i})
    (dot{p}_i = -frac{partial H}{partial q_i})

Integrability in Hamiltonian systems

Integrability in the context of Hamiltonian systems refers to the ability to solve the equations of motion analytically. This is possible when the system has a sufficient number of conserved quantities (or constants of motion) and symmetries.

Definition of Integrability

A Hamiltonian system with n degrees of freedom is absolutely integrable if there exist n independent conserved quantities which are involutions, i.e. their Poisson brackets vanish:

({F_i, F_j} = 0 text{ for all } i, j)

Example: Simple Pendulum

A classic example of this is the simple pendulum. Its Hamiltonian can be written as:

H = frac{p_theta^2}{2mell^2} + mgh(1 - costheta)

where p_theta is the momentum conjugate for angle theta, m is the mass, ell is the length, and h is the height of the pendulum. This system is integrable because it can be solved exactly, with energy being the integral of the momentum.

prime base Mass

Hamiltonian chaos

Chaos theory deals with systems that exhibit sensitive dependence on initial conditions. Hamiltonian chaos describes such behavior within the realm of Hamiltonian systems, which are typically deterministic and energy-conserving.

Characteristics of anarchy

Characteristics of chaotic systems include:

  • Sensitivity to initial conditions: small differences in initial conditions can produce very different trajectories.
  • Non-linear dynamics: systems often have non-linear equations that amplify small perturbations.
  • Fractal structures: Phase space often exhibits fractal boundaries, indicating complexity.

Example: Double Pendulum

A well-known example of Hamiltonian chaos is seen in the double pendulum. Its motion depends sensitively on initial conditions, and small changes can result in very different paths. The Hamiltonian for a simple double pendulum is non-trivial and is given as:

H = frac{1}{2}m_1 ell_1^2 dot{theta_1}^2 + frac{1}{2}m_2 (ell_1^2 dot{theta_1}^2 + ell_2^2 dot{theta_2}^2 + 2 ell_1 ell_2 dot{theta_1} dot{theta_2} cos(theta_1 - theta_2)) + m_1 g ell_1 costheta_1 + m_2 g (ell_1 costheta_1 + ell_2 costheta_2)
prime base month 1 month 2

Transition from integration to anarchy

Although integrable systems are solvable and predictable, small disturbances or parameter changes often lead to chaotic behavior. The transition from integrability to chaos can be explored through Kolmogorov-Arnold-Moser (KAM) theory and related phenomena.

KAM principle

KAM theory deals with the persistence of quasi-periodic trajectories in near-integrable systems. If a system is integrated, perturbations can only partially disrupt its motion, provided that certain non-resonance conditions are satisfied.

Disturbed harmonic oscillator

Consider a harmonic oscillator with a small non-linear disturbance:

H = frac{1}{2}(p^2 + q^2) + epsilon q^4

Without perturbations ((epsilon = 0)), the system is integrable and can be solved using the action-angle variable. However, when ε is non-zero, resonances and chaotic regions can emerge.

Oscillator mess Phase space path

Conclusion

Understanding Hamiltonian chaos and integrability in classical mechanics requires an appreciation of the delicate balance between order and disorder. While integrable systems offer predictability through conservation laws and symmetries, chaos introduces unpredictability and complexity, even in deterministic systems. By studying these concepts through Lagrangian and Hamiltonian formulations, physicists continue to uncover the rich dynamics that govern our universe.

Through examples such as the simple and double pendulums, as well as the perturbed harmonic oscillator, it is clear that the universe of Hamiltonian systems is vast, with chaos and integration presenting themselves as two sides of the same coin. Addressing them opens up discussions about non-linear dynamics, deterministic chaos, and complex patterns of motion within the classical paradigm.


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Hamiltonian chaos and integrability

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