Noether’s (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action.
Noether’s theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.
For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether’s theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry – it is the laws of motion that are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether’s theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. (These examples are just for illustration; in the first one, Noether’s theorem added nothing new – the results were known to follow from Lagrange’s equations and from Hamilton’s equations.)
Noether’s theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether’s theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.
There are numerous different versions of Noether’s theorem, with varying degrees of generality. The original version only applied to ordinary differential equations (particles) and not partial differential equations (fields). The original versions also assume that the Lagrangian only depends upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative. There is also a quantum version of this theorem, known as the Ward–Takahashi identity. Generalizations of Noether’s theorem to superspaces also exist.