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 ↓


Noether's theorem and conservation laws


In the field of classical mechanics, it is important to understand the fundamental principles that govern the motion of particles and systems. The two most powerful formalisms in this field are Lagrangian and Hamiltonian mechanics, named after mathematicians Joseph-Louis Lagrange and William Rowan Hamilton. Within these frameworks, Noether's theorem provides deep insights by linking symmetries and conservation laws, demonstrating a profound connection between the physics of a system and its underlying symmetries.

Understanding symmetries and their role

In physics, symmetry implies that a particular transformation or transformation in a system results in no observable difference. Classical mechanics often deals with symmetries such as spatial translation, rotation, and time translation. When the equations of a system remain unchanged under such transformations, it exhibits a symmetry.

Take the simple example of a free particle moving in space. The laws governing its motion do not depend on its specific location. If the particle were at a different location in space but with the same velocity, it would behave in the same way. This indicates symmetry under spatial translation.

Lagrangian mechanics: basis

Lagrangian mechanics provides a reconstruction of classical mechanics, introducing a Lagrangian function L that is defined as:

L = T – V

where T is the kinetic energy, and V is the potential energy of the system.

The principle of minimum action states that the path taken by the system between two states is the one for which the action integral is constant. The action S is given by:

S = ∫ L dt

The Euler–Lagrange equation derived from this principle provides the equations of motion:

d/dt (∂L/∂ẋ) – ∂L/∂x = 0

where ẋ denotes the derivative of the coordinate x with respect to time.

Noether's theorem: bridging symmetry and conservation

Emmy Noether's theorem is a cornerstone of theoretical physics, fundamentally linking symmetries and conservation laws. It states that every differentiable symmetry of the action of a physical system corresponds to a conservation law.

For example, if the Lagrangian of a system does not depend explicitly on time, then the system exhibits time translation symmetry, leading to conservation of energy.

Example: conservation of momentum

Consider a one-dimensional system whose Lagrangian depends only on the velocity v, not on the position x:

L = 1/2 m v²

Applying the Euler–Lagrange equation, the spatial transfer symmetry implies:

d/dt (mẋ) = 0

This equation reveals the conservation of linear momentum, p = mẋ.

The idea of conservation

Particle 1 Particle 2

The above diagram shows two particles on a frictionless surface. Unless an external force acts on them, their combined momentum remains constant, which is an example of the conservation principle.

Hamiltonian mechanics: an alternative approach

The transition from Lagrangian to Hamiltonian mechanics involves a change in phase space, characterized by coordinates and momenta. The Hamiltonian H represents the total energy of the system:

H = T + V

Hamilton's equations, derived from the principle of stationary action, give the time evolution of the system:

ẋ = ∂H/∂p
ṗ = -∂H/∂x

Here, p is the generalized momentum, and the equations provide a set of first-order differential equations compared to the second-order Euler–Lagrange equations.

Example: energy conservation

For a simple harmonic oscillator with the Hamiltonian:

h = (p²/2m) + (1/2)k x²

Hamilton's equations yield two coupled equations that conserve the total energy of the oscillator over time.

Visualization of phase space

H = constant

Above is a phase space diagram for the harmonic oscillator, which shows the path as a closed loop, symbolizing constant energy. The system always follows the same trajectory, which underlines energy conservation.

Noether's theorem in Hamiltonian mechanics

In Hamiltonian mechanics, Noether's theorem can be applied in a similar way. The central symmetry of the Hamiltonian leads to a conserved quantity. More generally, time translation symmetry in the Hamiltonian formalism implies energy conservation.

Example: conservation of angular momentum

For a rotationally symmetric system, angular momentum is conserved. Consider:

h = (p²/2m) + (1/2) m ω² r²

We can conclude that the angular momentum L = r × p remains constant, given the rotational symmetry.

Visualization of rotational symmetry

R P

The position r and momentum p vectors reflect conservation of angular momentum through continuous rotational symmetry.

Influence and wider implications of Noether's theorem

Noether's theorem plays an important role in a variety of fields beyond classical mechanics:

  • Quantum mechanics: The symmetries and conservation laws derived from Noether's theorem are important in understanding particle interactions.
  • General relativity: In Einstein's theory, the theorem explains energy–momentum conservation in curved spacetime symmetries.
  • Particle physics: The Standard Model relies heavily on symmetry principles provided by Noether's theorem.

The conceptual beauty and universality of Noether's theorem emphasize its importance. By systematically linking symmetries to conservation laws, it greatly enhances our understanding of physical phenomena.


Graduate → 1.2.3


U
username
0%
completed in Graduate

← Prev (1.2.2)
Euler–Lagrange equations
Next (1.2.4) →
Normalized Speed

Comments 



Noether's theorem and conservation laws

https://www.physicsflow.com/g/1.2.3?pfal=en