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 → Lagrangian mechanics ↓


Principle of minimum action


The principle of minimum action is a fundamental concept in theoretical physics, particularly in the field of Lagrangian mechanics, which is a reformulation of classical mechanics. This principle provides a powerful and elegant way of analyzing dynamical systems, yielding insights into how nature operates at a deep level. The principle holds that the path taken by a system between two states is the one for which the action is minimal, or more generally, at its peak.

Before we get into the detailed explanation, let us understand some basic concepts and terminologies related to the theory:

Action and Lagrangian

Action is a quantity that can usually be calculated from the motion of a physical system. It is calculated by integrating the Lagrange function over time, called the Lagrangian.

S = int L(q, dot{q}, t) , dt

In this expression, S denotes the action, L is the Lagrangian, which depends on the generalized coordinates q, their time derivatives dot{q} (velocities), and the time t.

Lagrangian

The Lagrangian L is often given by the difference between the kinetic energy T and potential energy V of the system:

L = T - V

This deceptively simple equation holds a universe of information about the dynamics of the system. Let's explain each component:

  • Kinetic energy (T): It is the energy that an object has because of its motion. For a particle of mass m moving with velocity v, the kinetic energy is given by: T = frac{1}{2}mv^2.
  • Potential energy (V): It is the energy stored in the system due to its position or configuration. For example, the potential energy of an object in a gravitational field is V = mgh, where g is the acceleration due to gravity, and h is the height.

Understanding the principle

The principle of least action suggests that of all the possible paths a system can take, the path actually taken is the one that minimizes action. This implies that nature plays out scenarios in a way that is most "economical" or "efficient".

Calculus of variations

To find the path that takes the action to an extreme, we use a mathematical tool called the calculus of variations. This is a technique that finds functions that optimize certain quantities.

The specific tool we use from variational calculus is called the Euler-Lagrange equation:

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

This equation provides the condition that must be fulfilled for the action to reach an extremum. By solving the Euler-Lagrange equation, we find the path taken by the system, i.e., the equations of motion in the Lagrangian framework.

An example: a simple pendulum

Let us consider a simple pendulum of mass m and attached to a string of length l, swinging under the influence of gravity.

The potential energy V of the pendulum is given by the height of the mass, h = l - l cos(theta), where θ is the angle with the vertical:

V = mgl(1 - cos(theta))

The kinetic energy T is given by the translational kinetic energy T = frac{1}{2} m (ldot{theta})^2.

Thus, the Lagrangian L for the pendulum is:

L = T - V = frac{1}{2} m (ldot{theta})^2 - mgl(1 - cos(theta))

Substituting the Lagrangian into the Euler-Lagrange equation, we obtain the equation of motion for the pendulum:

ml^2 ddot{theta} + mgl sin(theta) = 0

This is the classic equation for a simple pendulum that describes small oscillations around a stable equilibrium position.

Visualizing the action principle

Visual illustrations can help solidify understanding. Consider a particle moving from point A to point B. The theory states that of all conceivable paths, the particle follows the path for which the action S is extreme.

A B

In the above picture:

  • The red circle is the starting point A
  • The green circle is the end point B
  • The black line represents the possible path between A and B
  • The blue curve shows the actual path taken by the particle according to the principle of least action.

Further consideration

It is important to note that "minimal" action is a misnomer. The principle of minimal action is essentially an extremum principle, meaning that the action can be maximized, minimized, or limited. The important thing is that it reaches an extremum compared to neighboring paths. The reason for calling it "minimal" action probably comes from historical contexts and typical scenarios of systems tending toward minimum potential energy configurations.

Moreover, this principle is a cornerstone not only in classical mechanics but also in fields such as quantum mechanics and general relativity, reflecting a universal principle of the economy of nature. In quantum mechanics, the path followed by a particle is affected by the contribution of all possible paths, which is elegantly explained by Feynman's path integral formulation.

Mathematical legacy

The historical roots of the principle of minimum action go back to the 17th and 18th centuries, with contributions from mathematicians and philosophers such as Fermat, Maupertuis, Euler, and Lagrange. Each made unique contributions to the development of the principle, eventually formulating it into the well-established foundation of modern physics.

Applications in physics

The principle of minimum action is applied in a variety of areas of physics, providing insight and predictive power:

  • Electromagnetism: This theory helps derive Maxwell's equations, which are fundamental to understanding how electric and magnetic fields interact.
  • Quantum mechanics: The path integral formulation of quantum mechanics, developed by Richard Feynman, is an extension of the minimum action principle.
  • Relativity: In general relativity, this theory is used to derive Einstein's field equations, which describe how matter and energy curve spacetime.

Closing thoughts

The principle of minimum action in physics is a profound and beautiful principle, which reveals nature's preference for frugality in action. The range of its application across different scales and disciplines underlines its universality and its profound role in physical theories. Mastery of this principle opens the door to understanding and deriving equations that govern the behavior of innumerable physical systems, from simple pendulums to the structure of the universe.

We find that through its mathematical beauty and practical applications, the principle of least action is evidence of the enduring mysteries and orderliness within the laws of nature.


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Principle of minimum action

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