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


Hamiltonian mechanics is a reformulation of classical mechanics that emerged from the work of William Rowan Hamilton. It provides a structured framework for describing the evolution of a dynamical system. Unlike Newtonian mechanics, which uses vectors to describe force and momentum through differential equations, Hamiltonian mechanics focuses on energy and phase space. This approach has profound implications not only in classical mechanics but also in quantum mechanics where it holds an important ground.

Basics of Hamiltonian mechanics

Before delving into the intricacies of Hamiltonian mechanics, let's revisit some of the fundamental concepts of classical mechanics. In Newtonian mechanics, the motion of particles is governed by Newton's laws. The state of a system is determined by its position and velocity at any time. In Hamiltonian mechanics, we describe the state of the system using generalized coordinates and canonical momentum.

Phase space

Phase space is a conceptual space in which every possible state of a system is represented. Each state corresponds to a unique point in the phase space. In the case of N particles moving in three dimensions, the phase space is 6N-dimensional, containing all positions and velocities.

Consider a simple one-dimensional system with position ( q ) and momentum ( p ). The phase space is a plane where each point ( (q, p) ) represents a specific state of the system. The trajectory of the system motion can be viewed as a line or a curve in this phase space.

Position (q, p) P Why

Hamiltonian function

The core of Hamiltonian mechanics is the Hamiltonian function, usually denoted as ( H ). It is a scalar function representing the total energy of the system, expressed in terms of the generalized coordinates ( q_i ) and momentum ( p_i ). For many systems, the Hamiltonian includes kinetic and potential energy, although it may have different forms for different systems.

Consider the Hamiltonian for a system where the kinetic energy ( T ) and potential energy ( V ) depend on momentum and position, respectively:

H(q, p) = T(p) + V(q)

The equations of motion in such a system are given by Hamilton's equations:

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

Example: simple harmonic oscillator

Let us consider a simple harmonic oscillator. The Hamiltonian ( H ) is given as follows:

H(q, p) = frac{p^2}{2m} + frac{1}{2} kq^2

where ( q ) is the displacement from equilibrium, ( p ) is the momentum, ( m ) is the mass, and ( k ) is the spring constant. Using Hamilton's equations, the differential equations for this system are:

(dot{q} = frac{partial H}{partial p} = frac{p}{m}) (dot{p} = -frac{partial H}{partial q} = -kq)

Conservation and symmetry

One of the powerful aspects of Hamiltonian mechanics is its clear insights about symmetries and conserved quantities. Via Noether's theorem, from each symmetry in the Hamiltonian, a conservation law can be derived. For example, if the Hamiltonian does not depend explicitly on time, then the total energy is conserved.

This connection between symmetry and conservation provides a deeper understanding of physical systems. In many cases, problems can be simplified by identifying symmetries and applying the corresponding conservation laws.

Applications in various fields

Classical systems

In classical mechanics, Hamiltonian mechanics is used to analyze complex systems that may be difficult to solve using Newton's approach. This framework provides a straightforward way to integrate modern analytical techniques such as perturbation methods to study non-linear dynamics.

Quantum mechanics

Hamiltonian mechanics serves as a bridge to quantum mechanics. In quantum mechanics, the Hamiltonian operator governs the time evolution of a system and plays a central role in Schrödinger's equation:

ihbar frac{partial}{partial t} psi = hat{H} psi

where ( hat{H} ) is the Hamiltonian operator, and ( psi ) is the wave function of the physical system.

Relativistic mechanics

In relativistic mechanics, Hamiltonian formulas help deal with dynamical systems where relativistic speeds and effects are involved. Here, the Hamiltonian provides a convenient formalism for unifying forces and energy distributions consistent with the theory of relativity.

Advanced concepts

Canonical conversion

Canonical transformations refer to transformations of coordinates in phase space that preserve the form of Hamiltonian equations. These transformations are important tools in simplifying Hamiltonian problems by transforming them into more tractable forms.

For example, transforming from Cartesian coordinates ( (x, y, z, p_x, p_y, p_z) ) to spherical coordinates ( (r, theta, phi, p_r, p_theta, p_phi) ) can greatly simplify the Hamiltonian of a spherically symmetric potential.

Poisson bracket

Poisson brackets provide another layer of abstraction in Hamiltonian mechanics, providing a means of evaluating the time evolution of observables. Given two functions ( F ) and ( G ) in phase space:

({F, G} = sum_i left( frac{partial F}{partial q_i} frac{partial G}{partial p_i} - frac{partial F}{partial p_i} frac{partial G}{partial q_i} right))

The Poisson bracket is zero if and only if the two observables are in a decoupling state, meaning that they are conserved quantities. This bracket is particularly useful in studying dynamics and stability within a system.

Conclusion

Hamiltonian mechanics provides a robust and powerful framework for analyzing complex dynamical systems. Its focus on energy, symmetry, and phase space provides profound insights, unifying various branches of physics. As you move into advanced mechanics and related fields, understanding the Hamiltonian formulation becomes an essential part of your toolkit, enhancing both classical and quantum analytical capabilities.


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Hamiltonian mechanics

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