Emmy Noether's theorem states that there is a corresponding conservation law for every continuous symmetry of a physical system. Thus, the law of conservation of energy corresponds to the homogeneity of time, the law of conservation of momentum - the homogeneity of space, the law of conservation of angular momentum - the isotropy of space, the law of conservation of electric charge - a gauge symmetry, etc.
The theorem is usually formulated for systems with functional activities, and expresses the invariance of the Lagrangian with respect to a continuous group of transformations.
Theorem was given in the work of scientists Gottingen school of Hilbert, Klein and E. Noether. In the most common formulation, it was proved by Emmy Noether in 1918.
Noether's theorem allows obtaining significant information on the properties of solutions of differential equations, based only on their symmetry. It is also one of the methods of integration of ordinary differential equations, as it allows in some cases are the first integrals of the system of equations, and thus reduce the number of unknown functions. For example:
The momentum of the system follows from the invariance under spatial shifts. More specifically, if the shift along the X axis does not change the system of equations, the momentum is preserved along this axis.
Conservation of angular momentum follows from the invariance of the system with respect to rotations of space.
The law of conservation of energy - is a consequence of time homogeneity, allowing arbitrarily shift the origin of time.
In the case of partial differential equations must generally seek an infinite number of first integrals. Even knowing they are usually difficult to describe the general solution.
Due to its fundamental nature, Noether's theorem is used in the fields of physics such as quantum mechanics, to the introduction of the concepts of momentum, angular momentum, etc. The invariance equations for some symmetry are only the essence of these values and ensure their preservation.
Works Cited
Emmy Noether and The Fabric of Reality (2010) Retrieved from: HYPERLINK "http://www.youtube.com/watch?v=1_MpQG2xXVo" …