# What is the Noether's theorem and how it break the law of conservation?

I was in http://worldbuilding.stackexchange.com when I read a question about how to explain magic breaking physical laws, and in one answer they talk about the Noether's theorem and how to break the law of conservation. I couldn't understand nothing (I am not sure if I can't understand the law or it's a translation barrier) so I decided to search a little more and I found a lot of information in Physics.SE but their are too complicate and I can't understand anything. Could someone explain me what is the Noether's theorem in a simple way? I mean a basic explanation, no hard-science.

• – Qmechanic Jun 13 '17 at 19:38

## 4 Answers

Could someone explain me what is the Noether's theorem in a simple way? I mean a basic explanation, no hard-science.

(1) Start with something called the Lagrangian of a system and the rules for deriving the equations of motion from the Lagrangian.

(2) A symmetry of the Lagrangian is a transformation (of coordinates etc.) that leaves the Lagrangian unchanged and thus, the equations of motion unchanged.

(3) Noether showed that there is a conservation law associated with a (differentiable) symmetry of the Lagrangian.

Here are some examples.

If the Lagrangian is unchanged by a translation in space (loosely, moving the origin of the coordinate system) we say that the Lagrangian has a spatial translation symmetry and the associated conservation law is conservation of (linear) momentum.

If the Lagrangian is unchanged by a rotation, there is a rotational symmetry and the associated conservation law is conservation of angular momentum.

If the Lagrangian is unchanged by a translation in time (change the zero on the clock), there is a temporal translation symmetry and the associated conservation law is conservation of energy.

So, for example, to 'break' conservation of (linear) momentum would require that physical law somehow depend on location so that the Lagrangian is changed by a spatial translation.

On a more advanced note, the Standard Model (the best model we have of the elementary particles and their interactions) is based in part on the idea that the fundamental interactions are most elegantly introduced by promoting a global symmetry of a Lagrangian to a local symmetry which requires the introduction of new terms in the Lagrangian called gauge fields.

For example, the electromagnetic field is a gauge field that is required to give the Lagrangian a local U(1) symmetry

This is well beyond the scope of your question but I put this here just to emphasize how important symmetries of the Lagrangian and Noether's theorem are to fundamental physics.

• When you said "If the Lagrangian is unchanged..." do you mean with unchanged all the "normal" physics laws? – Ender Look Jun 13 '17 at 20:51
• @EnderLook, I'm not sure I understand what you're asking; I don't understand "with unchanged all the "normal" physics laws?" – Hal Hollis Jun 13 '17 at 21:02
• @EnderLook, for example, think of Newton's theory of gravity (and assume it is the correct description of gravity) and the gravitational constant $G$. Since $G$ is constant in time, energy is conserved in Newtonian gravity. However, if $G$ changed with time, energy would not be conserved. For example, if $G$ increased with time, then one could hoist a large mass up to the top of a tower, wait some time for $G$ to increase, and then get 'free' energy by letting the mass do work as it falls. The amount of work to raise the mass would be less than the amount of work done lowering the mass. – Hal Hollis Jun 13 '17 at 21:12

Noether's theorem relates conservation laws to symmetries of a system.

What is a conservation law?

A conservation law is a statement that a measurable property of a system does not change with time. For example, if we denote the total energy of a system to be $H$, then energy conservation is the statement that,

$$\frac{dH}{dt} = 0$$

which if you're not familiar with calculus is essentially saying the change in energy over some period of time is zero, as expected if it were conserved.

What is a symmetry?

Mathematically, we often describe systems in physics using a Lagrangian, and one can make precise the meaning of a symmetry.

In layman's terms, it essentially means that the physics of a system remain the same if we make a certain change. For example, time translation invariance means that if we shift time $t\to t+c$, then the physical laws remain the same.

Noether's Theorem

Noether's theorem relates to every conservation law, a physical symmetry of a system. In fact, in general, it shows us how to compute the conserved quantity given a symmetry. For example, in the case of time translation, one can show the conserved quantity is $H$, the total energy.

Symmetry Breaking

As it now relates to your world-building concern, if you want to break a conservation law, this can only be possible if the system no longer possesses the symmetry, or a more subtle case.

We have explicit symmetry breaking which means the system flat out does not possess the symmetry. The other case is spontaneous symmetry breaking which is when the equations that describe a system have the symmetry, but a state of the system does not.

In the more complicated case of spontaneous breaking, Goldstone's theorem shows there are rather deep implications; the Higgs boson in the Standard Model relates to this, for example.

• Thanks for your answer but could you please explain a little about that 3 items that have the Worldbuilding.SE question? • Time invariance, • Translational invariance, •Rotational invariance. The first I belive that you have already explained, the second I belive that I get it, but I didn't understand the third. – Ender Look Jun 13 '17 at 19:57
• @EnderLook If a system has rotational invariance, it means the physics stays the same if we rotate it. The conservation law this gives rise to is that of conservation of angular momentum. Basically, momentum conservation is because we can make translations in position and the physics stays the same, so the angular momentum is because we can make "translations" in "angles" and the physics stays the same. – JamalS Jun 13 '17 at 19:59

The short version is this: every conserved quantity in physics (like energy and momentum) corresponds to some way in which the physical laws that govern reality are invariant (don't change) when you transform the situation somehow. For example, if I hit a puck on an air hockey table why does it move in a straight line at constant speed (ignoring friction)? The answer is because the table is flat - it's invariant under translations. If the table were curved then the puck wouldn't move in a straight line, even without gravity.

See Wikipedia's article on "Noether's Theorem"

Noether's (first)1 theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.

For example, I would think, if one has a ring of charge and rotates that ring in its plane, the charge is still conserved:

Charge conservation can also be understood as a consequence of symmetry through Noether's theorem, a central result in theoretical physics that asserts that each conservation law is associated with a symmetry of the underlying physics...

.

• Hi Stephen, the down vote is probably because the symmetry associated with conservation of electric charge is the global gauge invariance of the electromagnetic Lagrangian – Hal Hollis Jun 13 '17 at 20:00
• Hello Hal Hollins - Could you explain yourself further with references? I feel hesitant to ask a question why Noether's theorem does not apply to charge as it seems like a conserved quantity, even under a single rotation in the plane of the ring to me from the definition. – Stephen Elliott Jun 19 '17 at 5:50
• Stephen, Noether's theorem does apply to conservation of charge and I didn't imply otherwise. The problem with your answer is that the symmetry associated with conservation of charge, by Noether's theorem, has nothing at all to do with the rotation of a ring of charge. Instead, it has to do with an abstract symmetry called global U(1) gauge invariance. See, for example, this – Hal Hollis Jun 20 '17 at 17:20
• The edit has hardly improved things - if anything, it's only made a misleading answer more misleading. The conservation of electric charge has nothing to do with physical rotations (it's a rather more involved symmetry; see physics.stackexchange.com/q/230712 and physics.stackexchange.com/q/48305, or indeed the Wikipedia article already linked to, for details). At present this is being misattributed in a way that's directly misleading. – Emilio Pisanty Jun 21 '17 at 9:08
• ... none of which, however, touches the core problem of this answer, which is that it is simply not relevant to the question as posed. – Emilio Pisanty Jun 21 '17 at 9:10

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