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Noether Theorem
Description
In 1915, Emmy Noether discovered a fundamental relationship between symmetries and conservations in physics. Whenever a system is invariant under changes in a particular variable, there is a corresponding conservation of another associated variable.
Particularly notable are the symmetries in:
• Time (conservation of energy),
• Translation (conservation of momentum), and
• Rotation (conservation of angular momentum).
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Time invariant
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Invariance (=no change) with respect to time means that something does not change as time passes. In other words, if something happens in a certain way today, it will happen the same way tomorrow.
Time invariance is associated with the conservation of energy. This implies that the sum of all energies will be equal to the total energy present at the beginning:
 $ E_0 = \displaystyle\sum_i E_i$ |
An example is an object in a gravitational field that consistently behaves the same way, indicating that the gravitational field does not dissipate energy from moving objects within it.
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Position invariant
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The invariance of spatial translation means that it doesn't matter where we perform an experiment. If we can translate the experiment in a direction without changing what happens, we talk about translational invariance in that direction. When translational invariance exists, momentum is conserved. Mathematically, this can be expressed as the sum of momenta always being equal to the initial total momentum:
| $ p_0 = \displaystyle\sum_i p_i$ |
An example is the translational invariance that exists on the surface of the Earth. As long as the altitude doesn't change, an experiment conducted in the southern or northern hemisphere will yield the same result. In other words, there's translational invariance along the $x$ and $y$ axes if these are defined on the planet's surface. The situation is different with altitude, as gravitational force varies, leading to different experimental outcomes. The consequence of translational symmetry on the planet's surface is that momentum is conserved. If mass doesn't change, this implies that velocity would remain constant.
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Position invariant (vector)
Equation
The conservation of momentum, which was initially introduced for one dimension
| $ p_0 = \displaystyle\sum_i p_i$ |
can be generalized to more dimensions
| $ \vec{p}_0 = \displaystyle\sum_i \vec{p}_i $ |
Since conservation for one dimension is
| $ p_0 = \displaystyle\sum_i p_i$ |
if there is invariance in the other dimensions, it can be generalized for each dimension
$\vec{p}0=(p_x,p_y,p_z)=\left(\displaystyle\sum_i p{i,x},\displaystyle\sum_i p_{i,x},\displaystyle\sum_i p_{i,x}\right)=\displaystyle\sum_i (p_{i,x},p_{i,y},p_{i,z})=\displaystyle\sum_i \vec{p}_i$
so that
| $ \vec{p}_0 = \displaystyle\sum_i \vec{p}_i $ |
In many cases, the invariance holds in one dimension and not in the others. In such cases, the one-dimensional relationship cannot be generalized to all dimensions.
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Angle invariant
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The rotational invariance means that we can rotate an experiment and its outcome will remain unchanged. An example is a satellite rotating around the Earth. For it, the situation doesn't change if we shift it by a certain number of degrees or perform the experiment at a different point.
The rotational invariance is associated with the conservation of angular momentum. This can be mathematically expressed by the sum of all angular momenta being equal to the initial angular momentum:
| $ L_0 = \displaystyle\sum_i L_i $ |
A classic example of this conservation can be seen in the case of a dancer altering their moment of inertia by extending or contracting their arms. If the moment of inertia decreases, the angular velocity increases, and vice versa.
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Angle invariant (vector)
Equation
The conservation of angular momentum, introduced for one dimension
| $ L_0 = \displaystyle\sum_i L_i $ |
can be generalized to more dimensions
| $ \vec{L}_0 = \displaystyle\sum_i \vec{L}_i $ |
Since conservation for one dimension is
| $ L_0 = \displaystyle\sum_i L_i $ |
if there is invariance in the other dimensions, it can be generalized for each dimension
$\vec{L}0=(L_x,L_y,L_z)=\left(\displaystyle\sum_i L{i,x},\displaystyle\sum_i L_{i,x},\displaystyle\sum_i L_{i,x}\right)=\displaystyle\sum_i (L_{i,x},L_{i,y},L_{i,z})=\displaystyle\sum_i \vec{L}_i$
thus
| $ \vec{L}_0 = \displaystyle\sum_i \vec{L}_i $ |
In many cases, the invariance occurs in one of the dimensions and not in the others. In such cases, it is not possible to generalize the one-dimensional relationship to all dimensions.
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Video
Video: Conservations