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 → Hamiltonian mechanics ↓


Poisson bracket


In the field of classical mechanics, the study of systems is often divided into different formulations such as Lagrangian and Hamiltonian mechanics. The Hamiltonian formulation is particularly beautiful because of its mathematical structure and its connection to other areas of physics. One of the key mathematical tools in Hamiltonian mechanics is the concept of the Poisson bracket. In this article, we will go deep into the world of Poisson brackets, exploring their definition, properties, and applications in classical mechanics.

Definition of Poisson bracket

Let us begin by considering a dynamical system characterized by its Hamiltonian function H(q,p,t), where q denotes the generalized coordinate, p denotes the generalized momentum, and t is time. The Poisson bracket is a binary operation defined for two differentiable functions f and g on the phase space of the system.

The Poisson bracket of two functions f and g is defined as:

{f, g} = ∑ ( ∂f/∂q_i * ∂g/∂p_i - ∂f/∂p_i * ∂g/∂q_i )

where the sum is over all normalized coordinates and momenta (q_i, p_i). The Poisson bracket is a measure of the infinitesimal change in one observable due to the flow generated by another observable.

Properties of the Poisson bracket

The Poisson bracket has several important properties that make it a powerful tool in Hamiltonian mechanics. These properties include:

1. Linearity

Poisson brackets are linear in both arguments. If we have functions a, b, f, g and constants α, β, then:

{αf + βg, a} = α{f, a} + β{g, a}

2. Antisymmetry

The Poisson bracket is antisymmetric, which means:

{f, g} = -{g, f}

3. Leibniz's rule

The Leibniz rule for the Poisson bracket states that this works as a derivation in the same way as differentiation:

{f, gh} = {f, g}h + g{f, h}

4. Jacobi identity

The Jacobi identity is a fundamental property expressing the consistency of Poisson brackets:

{{f, g}, h} + {{g, h}, f} + {{h, f}, g} = 0

The role of the Poisson bracket in Hamiltonian mechanics

In Hamiltonian mechanics, the dynamics of a system is governed by Hamilton's equations, which can be elegantly expressed using the Poisson bracket. For a function f(q, p, t) that evolves with the system, its time evolution is given by:

df/dt = {f, H} + ∂f/∂t

This makes it clear that the Poisson bracket with the Hamiltonian H determines how the observable f changes with time.

Visual example: Poisson bracket and phase space

Consider a simple visual example where we have a two-dimensional phase space with axes showing q and p. The Poisson bracket can be seen as a measure of the "twisting" or "rotation" of the phase space due to changes in q and p. Here is a simple diagram:

P Why f = constant g = const

In this diagram, the blue and red circles represent the level curves of the functions f and g, respectively. The Poisson bracket {f, g} measures the extent to which these level curves twist around each other.

Examples and applications

To understand Poisson bracket further, let's look at examples and practical applications.

Example 1: Simple harmonic oscillator

Consider a simple harmonic oscillator with the Hamiltonian:

H = (p^2 / 2m) + (1/2) mω^2 q^2

Calculate the Poisson bracket {q, p} for this system:

{q, p} = ( ∂q/∂q * ∂p/∂p - ∂q/∂p * ∂p/∂q ) = 1

Example 2: Angular momentum

Consider the components of the angular momentum L_i. The Poisson brackets of these components satisfy:

{L_x, L_y} = L_z
{L_y, L_z} = L_x
{L_z, L_x} = L_y

This shows that the angular momentum components form a closed algebra under the Poisson brackets.

Importance in classical and quantum mechanics

In classical mechanics, Poisson brackets are important for understanding the symplectic structure of phase space. They provide information about conserved quantities, symmetries, and integrability of a system. In addition, Poisson brackets serve as a bridge to quantum mechanics. In quantum mechanics, the Poisson bracket is replaced by the commutator, and this transition has many classical properties.

Conclusion

In summary, Poisson brackets are a central concept in Hamiltonian mechanics, which offer a concise and beautiful description of dynamical systems. They encompass key symmetries and conservation laws, beautifully linking classical mechanics with quantum mechanics. Understanding Poisson brackets helps physicists understand the complex behavior of systems, providing a gateway to deeper insights into theoretical physics.


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Poisson bracket

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