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


Lagrangian and Hamiltonian mechanics present two reformulations of classical mechanics that provide great insight and many advantages over the Newtonian approach. These formalisms provide deep connections between physics and geometry and serve as precursors to advanced theories such as quantum mechanics and general relativity. In this article, we will explore these concepts in detail, using simple language to make the subject clear for students and anyone interested in physics.

Introduction to Lagrangian mechanics

Lagrangian mechanics, named after Joseph Louis Lagrange, is a powerful method for analyzing the motion of systems under the influence of forces. Unlike Newtonian mechanics, which relies on vector equations of motion, Lagrangian mechanics uses scalar quantities based on the energy of the system.

Principle of minimum action

The cornerstone of Lagrangian mechanics is the principle of minimum action. This principle states that the path taken by a physical system between two states is the one for which a certain quantity, the action, is minimal. The action is defined as the integral over time of the Lagrangian, denoted as L(t, q, dot{q}), where q denotes the generalized coordinates and dot{q} their time derivatives.

        S = ∫ L(t, q, dot{q}) dt

Lagrangian

The Lagrangian is usually defined as the difference between the kinetic energy, T, and the potential energy, V, of the system:

        L = T – V

This simple mathematical function encapsulates the dynamics of the system. Let's use the classic example of a simple pendulum to illustrate this concept:

        t = 0.5 * m * l^2 * dot{theta}^2
        v = m * g * l * (1 - cos(theta))
        L = T – V

Here, m is the mass of the bob, l is the length of the pendulum, g is the gravitational acceleration, and θ is the angle with the vertical.

Euler–Lagrange equations

The equations of motion of a system are derived using the Euler–Lagrange equation, which arises from the principle of least action:

        d/dt (∂L/∂dot{q_i}) - ∂L/∂q_i = 0

This equation, where q_i are generalized coordinates, provides a systematic way to derive the equations of motion for any system, such as multiple particles or complex mechanical systems.

Example: Atwood machine

Consider the Atwood machine, which consists of two masses connected by a string over a pulley. The masses, m1 and m2, experience gravitational forces. The Lagrangian for this system is:

        L = 0.5 * (m1 + m2) * dot{x}^2 - (m2 - m1) * g * x

Applying the Euler-Lagrange equation gives the acceleration:

        (m2 - m1) * g = (m1 + m2) * ddot{x}

Introduction to Hamiltonian mechanics

Hamiltonian mechanics, developed by Sir William Rowan Hamilton, is a further reformulation of classical mechanics that arises from Lagrangian mechanics. Hamiltonian mechanics restructures the equations and emphasizes concepts important to modern physics, such as symplectic geometry and phase space.

Hamiltonian function

The Hamiltonian, H, represents the total energy of the system and is obtained by performing the Legendre transformation on the Lagrangian:

        H(q, p, t) = ∑ p_i dot{q}_i - L

In this context, p_i are conjugate momenta defined as p_i = ∂L/∂dot{q}_i.

Hamilton's equations

The dynamics of a system is described using Hamilton's equations, a set of first-order differential equations:

        dot{q}_i = ∂H/∂p_i
        dot{p}_i = -∂H/∂q_i

These equations highlight the beauty of Hamiltonian mechanics by relating time evolution to time through differential equations and are the basis of many modern theoretical physics frameworks.

Phase space

In Hamiltonian mechanics, the state of a system is represented in a multidimensional space, known as phase space, where each point corresponds to a unique configuration of position and momentum.

Phase space pathPoint in phase space

Advantages and applications

The Lagrangian and Hamiltonian formalisms offer various advantages:

  • Complex systems: They are particularly useful for complex systems that involve constraints, such as in electromagnetism or general relativity.
  • Simplification: By focusing on energy rather than forces, these methods can simplify the derivation of the motion equations.
  • Quantization: These formalisms extend naturally to quantum mechanics, where positions and momenta are quantized.

Example: Harmonic oscillator in Hamiltonian mechanics

Consider a one-dimensional harmonic oscillator with mass m and spring constant k. The Hamiltonian is given as follows:

        h = 0.5 * m * p^2 + 0.5 * k * q^2

Applying Hamilton's equations gives the equations of motion for the system:

        dot{q} = p/m
        p = -k * q

Conclusion

Lagrangian and Hamiltonian mechanics provide elegant and powerful ways to tackle classical mechanics problems. Their focus on energy rather than forces, combined with their flexibility in handling different coordinate systems and constraints, make them invaluable tools in theoretical and applied physics. Whether you are analyzing the stability of planetary orbits or building the foundations of quantum field theories, these formalisms provide essential insights that extend far beyond traditional approaches.


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Lagrangian and Hamiltonian mechanics

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