Noether’s theorem

Five Fast Facts About Mathematician Emmy Noether | Department of Energy

Noether’s theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This theorem only applies to symmetries that are continuous and smooth in physical space. 

It was first published in the year 1918, although it was proven a few years earlier by a mathematician named Emmy Noether. Noether was a German mathematician, born in 1882, who contributed significantly to abstract algebra in the 20th century. She also established various ring field and algebraic theories. Early on, Noether intended to follow the footsteps of her mathematician father, Max Noether. When Noether reached college age, however, German universities would not admit women, so she had to audit classes. She eventually received an undergraduate degree after performing well on final examinations.

Further explanation of this theorem; If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. In other words, if a coordinate system transformation meets a specific requirement, such as being continuous, then there must be a preserved quantity. Symmetry, in this context, refers to an operation that can be done to an object or system that leaves it unchanged.  A conservation law, meanwhile, refers to a physical quantity that remains fixed and hence does not fluctuate over time.

Noether’s theorem revealed an unknown link between two fundamental ideas: symmetries and conserved quantities that had previously been addressed independently. The theorem establishes a mathematical formula for determining the symmetry that underpins a given conservation law, as well as the conservation law that corresponds to a particular symmetry. Noether’s theorem is applicable to any action, Lagrangian, Hamiltonian, or any other. Lagrangian mechanics is a formulation of classical mechanics that is based on the principle of stationary action and in which energies are used to describe motion. Hamiltonian mechanics is based on the Lagrangian formulation and is used to describe the sum of kinetic and potential energies. The Lagrangian symmetry is associated with the conservation of momentum, and the Hamiltonian symmetry is associated with the conservation of energy. Time translation symmetry gives conservation of energy. Space translation symmetry gives conservation of momentum, and rotational symmetry gives conservation of angular momentum. Noether’s theorem is based upon mathematical proof, however, it’s not a theory in itself but can be applied to physics to further develop theories.

Long before Noether’s theorem, physicists were well aware of the conservation of momentum, angular momentum, and energy. However, it was unknown that these were all linked to a certain symmetry. This new understanding, which sprang from Noether’s work, has become a guiding principle that pervades physics study and shapes our understanding of the cosmos as a whole. Physicists today are still unpacking its implications. Ruth Gregory, a theoretical physicist was once quoted saying, ‘It’s hard to overestimate the importance of Noether’s work in modern physics’. As well, it has been called one of the most important mathematical theorems ever proved in guiding the development of modern physics.

Writer: Golda Abs

Editors: Lamar Albukhari/Omar Alturki

References:

1-Noether, math.ucr.edu/home/baez/noether.html. 

2-Nadis, Steve. “How Mathematician Emmy Noether’s THEOREM Changed Physics.” Discover Magazine, Discover Magazine, 26 Apr. 2020, http://www.discovermagazine.com/the-sciences/how-mathematician-emmy-noethers-theorem-changed-physics.