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 ↓


Principle of minimum action


The principle of minimum action is a fundamental concept in physics, particularly in the study of classical mechanics via the Lagrangian and Hamiltonian formulas. It provides a powerful and elegant method for deriving the equations of motion for a system, and has profound implications and applications in various fields of physics, from classical mechanics to quantum mechanics and beyond.

Introduction to the principle of least action

In simple terms, the principle of minimum action can be stated as follows: Of all the possible paths that a system can take between two time points, the path that the system actually follows is the one for which a particular quantity, called the "action", is constant (usually minimum).

The action, denoted as S, is calculated as the integral over time of a function called the Lagrangian, L(q, dot{q}, t), which depends on the generalized coordinates q, their time derivatives dot{q}, and possibly time t. Mathematically, the action is given as:

        S = int_{t_1}^{t_2} L(q, dot{q}, t) , dt

Understanding the Lagrangian

The Lagrangian L is a function that summarizes the dynamics of the system. For many mechanical systems, the Lagrangian is defined as the difference between the kinetic energy T and the potential energy V of the system:

        L = T – V

For example, consider a simple pendulum. The kinetic energy T depends on the velocity of the pendulum mass, and the potential energy V depends on the height of the mass in the gravitational field.

Euler–Lagrange equations

The core of the calculus of variations applied to the principle of minimum action is the Euler-Lagrange equation. To find the path that makes the action stationary, we obtain the Euler-Lagrange equation from the principle of stationary action:

        frac{d}{dt}left(frac{partial L}{partial dot{q}}right) - frac{partial L}{partial q} = 0

Each generalized coordinate q_i in the system yields such an equation. Solving these equations gives the equations of motion for the system.

An example: the simple harmonic oscillator

Let us consider a simple harmonic oscillator, such as a mass attached to a spring. This system has a mass m, an angular frequency omega, and is described by a single generalized coordinate x (representing the displacement of the mass from its equilibrium position).

The kinetic energy T is represented by T = frac{1}{2}mdot{x}^2, and the potential energy V is represented by V = frac{1}{2}kx^2, where k is the spring constant. The Lagrangian for this system is:

        L = frac{1}{2}mdot{x}^2 - frac{1}{2}kx^2

Applying the Euler-Lagrange equation, we get:

        frac{d}{dt}left(frac{partial L}{partial dot{x}}right) - frac{partial L}{partial x} = 0

Substituting the Lagrangian into it, we get the equation of motion for the harmonic oscillator:

        mddot{x} + kx = 0

Visual explanation through an example

The above visualization shows a simple harmonic oscillator in equilibrium, where the potential and kinetic energies switch back and forth, which is explained through the principle of least action.

Connecting to Hamiltonian mechanics

The Hamiltonian formulation of mechanics is very closely related to the Lagrangian formulation. It uses a different function, the Hamiltonian H, which can often be thought of as representing the total energy (kinetic and potential) of the system. The Hamiltonian is obtained from the Lagrangian through a process called the Legendre transformation.

The Hamiltonian is defined as follows:

        H = dot{q}p - L

where p = frac{partial L}{partial dot{q}} is the normalized momentum.

Example changes

The momentum p for our simple harmonic oscillator is:

        p = frac{partial L}{partial dot{x}} = mdot{x}

The Hamiltonian becomes:

        H = dot{x}(mdot{x}) - left(frac{1}{2}mdot{x}^2 - frac{1}{2}kx^2right) = frac{1}{2}mdot{x}^2 + frac{1}{2}kx^2

Illustrative example: the brachistochrone problem

To make the principle of least action more clear, let's consider the brachistochrone problem - finding the shape of the curve along which a bead will slide, under the influence of gravity, without friction, from one point to another, in the shortest time.

The solution of this problem involves calculus of variations and results in a cycloid path, again applying the principles of calculus of variations based on the method of minimum actions.

Conclusion

The principle of minimum action summarizes the behavior of a system in terms of a single scalar quantity - the action. Its beauty lies in its generality, which applies to a wide range of problems in classical mechanics, and forms a bridge to quantum mechanics and field theories in high energy physics. Through Lagrangian and Hamiltonian formulations, it provides a streamlined approach to analyze the dynamics of complex systems, which serves as a cornerstone of modern theoretical physics.


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Principle of minimum action

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