Last update: November 6, 2022

The Magic of Noether's Theorem: From the Principle of Least Action to Conservation Laws

Image description: Amalie Emmy Noether (1882-1935), the German mathematician and her famous equation E=K+U, one of the deepest theorems in physics. Noether's theorem, demonstrated in 1915 and published in 1918, states that the laws of physics are symmetric, and this symmetry leads to conservation laws.
Image source: astronoo.com

The Principle of Least Action

The principle of least action states that nature always takes the path that minimizes a certain quantity called action. Initially observed by Pierre de Fermat (1607-1665) for light in the 17th century, this principle was formalized in the 18th century by Joseph-Louis Lagrange (1736-1813), laying the foundation for Noether's theorem.

The principle of least action determines the motion of a system from the difference between kinetic and potential energy (K-U), allowing the prediction of the dynamics of any physical system, from pendulums to galaxies.

Statement of Noether's Theorem

"To every infinitesimal transformation that leaves the action integral invariant corresponds a conserved quantity."

What lies behind this statement?

If a physical law remains the same when a small transformation is applied to a system (moving it, rotating it, or advancing it in time), then there is a particular physical quantity (energy, momentum, charge, etc.) that will remain constant and never vary.

Thus, Noether establishes that for every symmetry there corresponds a conservation law.

What is a symmetry?

Symmetries describe the invariances in the laws of nature. They are not just aesthetic; they form the foundation on which many fundamental principles of physics rest.

In physics, a symmetry is a property of a system whose behavioral laws remain unchanged when certain transformations are applied, such as translation, rotation, or time shift.

Symmetries corresponding to a conservation law

Spatial translation invariance: conservation of momentum.
Rotational invariance: conservation of angular momentum.
Time invariance: conservation of energy.
Electromagnetic gauge symmetries: conservation of electric charge.
Color gauge symmetries: conservation of color charge (quantum chromodynamics).
Weak gauge invariance: conservation of lepton and baryon number.
CPT symmetry (charge, parity, time): fundamental invariance of quantum field theories.
Global phase transformation invariance: conservation of particle number.
Isospin symmetries: conservation of isospin in strong interactions.
Lorentz invariance: conservation of the four-momentum vector.

These symmetries testify to a fundamental property of the Universe: its physical laws are universal and immutable, regardless of spatial position, temporal moment, chosen orientation, or gauge transformations applied to quantum fields.

Concretely, this means that the same principles governed matter billions of years ago and will continue to apply in the distant future. An experiment conducted at the equator will produce identical results to one conducted at the North Pole. Similarly, the orientation of a system in space does not influence its dynamics, which explains why planetary orbits have remained stable since the formation of the solar system.

Application of Noether's Theorem to a Pendulum

Let's observe a simple pendulum swinging gently. Its total mechanical energy E is composed of two distinct but closely related forms: the kinetic energy K associated with motion and the potential energy U related to position, combined in the equation E = K + U.

When the pendulum is in motion, its kinetic energy is expressed as K = ½mv². The faster it moves (high v), the greater this energy. It reaches its peak at the lowest point of the arc, where the speed is maximum.

Conversely, the gravitational potential energy U = mgh depends on the height h of the pendulum. At the extremes of its oscillation, as it rises above its equilibrium position, this stored energy is maximum while the speed is zero.

Thanks to Noether's theorem and the invariance of physical laws over time, we know that the total energy E remains strictly constant. The pendulum thus operates a perpetual conversion between its two forms of energy: descending, it transforms its potential energy into kinetic energy; ascending, it reconverts this kinetic energy into potential energy. This energetic ballet concretely illustrates how a symmetry (here temporal) generates a conservation law (that of energy).

Key Takeaways

Conservation laws represent the structuring principles of contemporary physics. Calling them into question would lead to the collapse of our conception of the Universe, render our predictive models obsolete, and invalidate the theoretical bases on which our entire technological arsenal rests.