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Page "Laws of science" ¶ 9
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Noether's and theorem
See Noether's theorem.
* Noether's theorem
According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.
That is, energy is conserved because the laws of physics do not distinguish between different instants of time ( see Noether's theorem ).
That is, conservation of momentum is a consequence of the fact that the laws of physics do not depend on position ; this is a special case of Noether's theorem.
Specifically, Noether's theorem connects some conservation laws to certain symmetries.
An interesting point is that energy is also a symmetry with respect to time, and momentum is a symmetry with respect to space, and these are the reasons why energy and momentum are conserved-see Noether's theorem.
Noether's theorem implies that there is a conserved current associated with translations through space and time.
Noether's ( first ) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations.
As an 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 dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.
As another example, if a physical process exhibits the same outcomes regardless of place or time ( having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Friday ), 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.
Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool.
Using Noether's theorem, the types of Lagrangians that conserve X through a continuous symmetry may be determined, and their fitness judged by further criteria.
There are numerous versions of Noether's theorem, with varying degrees of generality.
Generalizations of Noether's theorem to superspaces are also available.
All fine technical points aside, Noether's theorem can be stated informally
In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants:
This is the seed idea generalized in Noether's theorem.
The essence of Noether's theorem is generalizing the ignorable coordinates outlined.
Then Noether's theorem states that the following quantity is conserved,
Since field theory problems are more common in modern physics than mechanics problems, this field theory version is the most commonly used version of Noether's theorem.
For such systems, Noether's theorem states that there are N conserved current densities
In that case, Noether's theorem corresponds to the conservation law for the stress – energy tensor T < sub > μ </ sub >< sup > ν </ sup >

Noether's and Any
Any conserved quantum number is a symmetry of the Hamiltonian of the system ( see Noether's theorem ).

Noether's and quantity
Furthermore, whenever a system of physical laws admits a one-parameter group of differentiable symmetries, then there is a conserved quantity, by Noether's theorem.
Conserved currents play an extremely important role in theoretical physics, because Noether's theorem connects the existence of a conserved current to the existence of a symmetry of some quantity in the system under study.

Noether's and which
One particularly important physical result concerning conservation laws is Noether's Theorem, which states that there is a one-to-one correspondence between conservation laws and differentiable symmetries of physical systems.
Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the action, which is defined as the integral of the Lagrangian density over the given region of spacetime.
For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation.
The same principle, and the Lagrangian formalism, are tied closely to Noether's theorem, which connects physical conserved quantities to continuous symmetries of a physical system.
If either L or H are independent of a generalized coordinate q, meaning the L and H so not change when q is changed, which in turn means the dynamics of the particle are still the same even when q changes, the corresponding momenta conjugate to those coordinates will be conserved ( this is part of Noether's theorem, and the invariance of motion with respect to the coordinate q is a symmetry ).
This is a consequence of Noether's theorem, which can be proven mathematically.
The Open University course Statistics in Society ( MDST 242 ), took the above ideas and merged them with Gottfried Noether's work, which introduced statistical inference via coin-tossing and the median test.
A very general result from classical analytical mechanics is Noether's theorem, which fuels much of modern theoretical physics.
In what is referred to in physics as Noether's theorem, the Poincaré group of transformations ( what is now called a gauge group ) for general relativity defines symmetries under a group of transformations which depend on a variational principle, or action principle.
* 1918 – Emmy Noether: Noether's theorem – conditions under which the conservation laws are valid
In the paper on Galois theory she studied the problem of parameterizing the equations with given Galois group, which she reduced to " Noether's problem ".
If let's say gravity is an emergent theory of a fundamentally flat theory over a flat Minkowski spacetime, then by Noether's theorem, we have a conserved stress-energy tensor which is Poincaré covariant.
Noether's theorem gives a current which is conserved and Poincaré covariant, but not gauge invariant.

Noether's and continuous
This is a consequence of the fact that the lattice symmetry is discrete as opposed to continuous, and thus its associated conservation law cannot be derived using Noether's theorem.

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