[Talk] Noether's theorem

[uniserv1002 (uniserv)]

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, 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
, 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.


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