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 ↓


Poisson bracket and Hamilton–Jacobi theory


Lagrangian and Hamiltonian mechanics are two important formalisms in classical mechanics that help us understand and describe the motion and dynamics of systems. Among the multitude of concepts within these formalisms, the Poisson bracket and Hamilton-Jacobi theory are of particular importance, providing powerful insights into the symmetries and integrability of a dynamical system.

Poisson bracket

The Poisson bracket is an important structure in Hamiltonian mechanics. It is an operation that assigns a number to every pair of functions defined on the phase space of a dynamical system. For functions f and g on a 2n-dimensional phase space with coordinates (q₁, q₂, ..., qₙ, p₁, p₂, ..., pₙ), the Poisson bracket is defined as:

{f, g} = Σ ( (∂f/∂qᵢ)(∂g/∂pᵢ) - (∂f/∂pᵢ)(∂g/∂qᵢ) )

Here, the sum runs over all i from 1 to n.

The Poisson bracket satisfies several important properties:

  • Bilinearity: {af + bg, h} = a{f, h} + b{g, h} for any functions f, g, h and scalars a, b.
  • Antisymmetry: {f, g} = -{g, f}.
  • Jacobi identity: {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0.
  • Leibniz law: {f, gh} = {f, g}h + g{f, h}.

A fundamental role of the Poisson bracket is to express the time evolution of a function on phase space. The time evolution of an observable f(q, p, t) is given by:

df/dt = {f, H} + ∂f/∂t

where H is the Hamiltonian of the system. This is known as Hamilton's equation of motion in terms of generalized functions, not just positions and momenta.

Illustrative example: simple harmonic oscillator

For a simple harmonic oscillator with the Hamiltonian:

h = (p²/2m) + (1/2)mω²q²

We calculate the Poisson bracket as follows:

Consider position and momentum, q and p. The relevant Poisson brackets are:

{q, p} = 1
{q, q} = {p, p} = 0

Thus, the time derivatives are:

dq/dt = {q, H} = p/m
dP/dt = {p, h} = -mω²q
the mass is oscillating on the spring

The simple harmonic oscillator provides a clear example of how accurately the Poisson brackets allow us to derive the equations of motion from the Hamiltonian.

Hamilton–Jacobi theory

Hamilton–Jacobi theory offers a third, alternative approach to the dynamics of a system, focusing on action rather than path, position, or momentum.

The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation. The main idea is to find a function, Hamilton's principal function S(q, α, t), that satisfies:

H(q₁, q₂, ..., qₙ, ∂S/∂q₁, ∂S/∂q₂, ..., ∂S/∂qₙ, t) + ∂S/∂t = 0

where α represents constants. This equation allows us to obtain the dynamics by considering the level curves of S

In particular, if S is completely separable, then it is of the form:

S(q, α, t) = W(q, α) − eₜt

Then, W is called the characteristic function of the Hamiltonian. By solving for W, we can determine the complete integrals of the motion.

Visualization of Hamilton–Jacobi theory

Consider a separable system where the Hamiltonian is:

H = T(p) + V(q)

The Hamilton–Jacobi equation transforms as follows:

T(∂W/∂q) + V(q) = Eₜ
Characteristic curve

Solving this equation allows us to trace the evolution of a system along these characteristic curves in phase space, which is a graphical depiction of the integral of the motion. These level curves are valuable for visualizing the evolution of different parts of a system in time.

Applications and implications

Both the Poisson bracket and Hamilton–Jacobi theory have profound implications and applications in physics:

  • Quantum mechanics: Poisson brackets form the basis of the commutation relations in quantum mechanics. The transition from classical to quantum mechanics often involves replacing the Poisson bracket with a commutator.
  • Integrable systems: In the Hamilton-Jacobi formalism, if one can find a complete integral of the motion, then the system is said to be integrable. These systems are characterized by the ability to resolve motions exactly.
  • Canonical transformations: Both theories provide the basis for the idea of canonical transformations, which are important for simplifying complex mechanical systems. These transformations preserve the form of Hamilton's equations, allowing for much simpler analysis.

These tools provide a powerful set of principles essential for understanding not only classical mechanics, but also modern theoretical physics. From both theoretical and computational perspectives, mastering the Poisson bracket and Hamilton-Jacobi theory gives physicists fundamental insights into the nature of dynamical systems.


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Poisson bracket and Hamilton–Jacobi theory

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