Last update: January 13, 2026

Economical Nature: The Secrets of Conserved Quantities

Fundamental symmetries of the universe (translations, rotations) and the conserved quantities they imply (energy, momentum, angular momentum)

The Principle of Universal Laziness

Since ancient times, philosophers and scientists have sought a unifying principle explaining the order of the world. In the 18th century, such a principle emerged.

The Principle of Least Action

The principle of least action, formulated by thinkers such as Pierre-Louis Moreau de Maupertuis (1698-1759) and Leonhard Euler (1707-1783), postulates that nature always acts in the most "economical" way possible. Whether it is a light ray refracting as it passes from air to water or the orbital motion of a planet around its star, the observed trajectories are never arbitrary. They correspond to those that minimize, or more precisely make "critical" (neither increasing nor decreasing), a mathematical quantity called action. This logic of parsimony, where nature favors certain evolutions among all possible ones, appears as a fundamental principle structuring the entire Universe.

Conservation Laws: Nature's Economy

At all scales, nature seems to obey a logic of economy. Certain physical quantities are neither created nor destroyed, but simply transformed or redistributed. This regularity is manifested through conservation laws, pillars of modern physics. Whether it is energy, momentum, or electric charge, each quantity can be compared to a volume of water circulating in a network of rivers and lakes. Just as water moves and changes form between rivers, lakes, and underground aquifers without disappearing, these "physical volumes" circulate and transform, their total remaining strictly constant in an isolated system.

One Symmetry, One Law: Noether's Revelation

Since Isaac Newton (1643-1727), physicists had observed and empirically verified fundamental conservation laws. The energy of an isolated system is neither created nor lost, its total momentum remains constant, and its angular momentum is conserved. These principles worked perfectly, but one question persisted: why does the Universe obey such rules?

The answer, as profound as it is elegant, was provided in 1915 by the German mathematician Amalie Emmy Noether (1882-1935). Her revolutionary theorem established a fundamental link between the geometry of spacetime and the laws of physics.

The core of her discovery can be summarized in a powerful principle:

Every continuous symmetry of the laws of Nature implies a conserved physical quantity.

Imagine a strategy game whose fundamental rules remain strictly identical, no matter when you play (symmetry in time), where you are (symmetry in space), or the orientation of the board (rotational symmetry).

Noether's theorem reveals that this perfect stability of the rules mathematically guarantees the conservation of a specific "value" throughout the game. Thus, just as in a classic game the total number of pieces remains constant, the continuous symmetries of the Universe ensure the conservation of fundamental quantities such as energy, momentum, and angular momentum.

The theorem also works the other way around, serving as a discovery tool:

Every conserved quantity we observe betrays the existence of an underlying continuous symmetry.

Thus, Noether showed that conservation laws are not accidents or arbitrary postulates. They are the inevitable and mathematical consequences of the fundamental symmetries that structure our Universe. The economy of nature finds here its deepest explanation.

The Dictionary of Symmetries

Noether's theorem provides us with a precise dictionary to translate symmetries into conservation laws. Here are the most fundamental correspondences: