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 ↓


Canonical conversion


In the field of classical mechanics, Hamiltonian mechanics stands out as a powerful reformulation of Newtonian mechanics. Unlike Lagrangian mechanics, which revolves around the concept of generalized coordinates and velocities, Hamiltonian mechanics introduces a new approach by transforming these into a set of new variables known as generalized coordinates and momenta. These transformations are at the core of canonical transformations, which provide powerful tools to simplify and solve complex mechanical systems. This exploration of canonical transformations delves into their significance, mathematical formulation, and their applications in classical mechanics.

Understanding canonical transformations

At its core, Hamiltonian mechanics is governed by Hamilton's equations, which are expressed as:

        ,
        dot{q}_i = frac{partial H}{partial p_i}, quad dot{p}_i = -frac{partial H}{partial q_i}
        ,

where (q_i) are the generalized coordinates, (p_i) are the generalized momenta, and (H) is the Hamiltonian of the system.

Canonical transformations involve transforming the coordinates and momenta ((q_i, p_i)) into a new set of coordinates and momenta ((Q_i, P_i)), so that the form of Hamilton's equations is preserved. This is important because solving the equations of motion after simplification remains equivalent to the original problem.

Formal definition

A transformation from ((q_i, p_i)) to ((Q_i, P_i)) is called canonical if it preserves the form of Hamilton's equations. This implies that the new variables also satisfy the same equations:

        ,
        dot{Q}_i = frac{partial K}{partial P_i}, quad dot{P}_i = -frac{partial K}{partial Q_i}
        ,

where (K(Q, P, t)) is the new Hamiltonian expressed in terms of the new variables ((Q_i, P_i)).

Generating function

Generating functions play an important role in the construction of canonical transformations, since they can be used to determine a functional relationship between the old and new coordinates and momenta.

There are mainly four types of generating functions:

  1. F1(q, Q, t) - a function of the old coordinates and the new coordinates.
  2. F2(q, P, t) - a function of the old coordinates and the new momenta.
  3. F3(p, Q, t) - a function of the old momenta and the new coordinates.
  4. F4(p, P, t) - a function of the old momentum and the new momentum.

For each type of generating function, the corresponding relations connect the old and new variables. Let's consider one in more detail:

Example of a generating function F2(q, P, t)

This generating function leads to the transformation equations:

        ,
        p_i = frac{partial F2}{partial q_i}, quad Q_i = frac{partial F2}{partial P_i}
        ,

From these, we can find explicit expressions for (p_i) and (Q_i) in terms of (q_i) and (P_i).

Visual representation of changes

Consider a simple harmonic oscillator system represented in phase space, where the canonical transformation simplifies the problem:

Phase space q, p q, p

In the figure, changes represented as shifts in phase space occur without changing the form of the equations.

Properties of the canonical transformation

One of the most interesting properties of canonical transformations is the symplectic nature of these mappings. They preserve the symplectic structure which is expressed as:

        ,
        sum_{i} dQ_i wedge dP_i = sum_{i} dq_i wedge dp_i
        ,

This conservation ensures that the volume in phase space remains unchanged under canonical transformations. Thus, Liouville's principle of constant density is naturally satisfied, leading to important implications in statistical mechanics and thermodynamics.

Applications of canonical transformations

Canonical transformations serve a variety of purposes:

  • They simplify the solution of dynamical problems by transforming the Hamiltonian into a simpler form, sometimes by diagonalizing it completely.
  • In quantum mechanics, they are used to demonstrate the equivalence of different quantum problems and to reveal symmetries and invariants.
  • In celestial mechanics, canonical transformations help solve the equations governing planetary motion through perturbation theories such as the Poincare method.

Example: Harmonic oscillator

The simple harmonic oscillator is a cornerstone example demonstrating canonical transformations. Consider its Hamiltonian:

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

Using canonical transformations, this Hamiltonian can be simplified by transferring the problem to the complex plane or by using action-angle variables, thereby reducing the complexity in time to trivial integrals.

Conclusion

Canonical transformations are an essential tool in Hamiltonian mechanics, providing a systematic method for simplifying and solving complex mechanical systems. Through the use of the generating function and the preservation of symplectic structure, they facilitate the solution of problems that would be extremely difficult to solve analytically under normal circumstances.

The utility of canonical transformations extends beyond classical mechanics, providing insights and techniques that apply to quantum mechanics, statistical mechanics, and beyond. As a result, understanding these transformations is a crucial step for any physicist who wishes to apply Hamiltonian mechanics in both theoretical explorations and practical applications.


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Canonical conversion

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