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61 votes

Do all Noether theorems have a common mathematical structure?

The core of the Noether theorem in all contexts where it arises is surprisingly elementary! From a very general point of view, one considers the following structure. (i) A set of "states" $x\...
Valter Moretti's user avatar
34 votes
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Why is Noether's theorem important?

We usually call equations like $$\frac{d}{dt} \frac{\partial L}{\partial \dot{q_i}} - \frac{\partial L}{\partial q_i} = 0$$ "equations of motion," because they are equations that tell us how the ...
user1379857's user avatar
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28 votes
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Elementary argument for conservation laws from symmetries *without* using the Lagrangian formalism

The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on ...
Chiral Anomaly's user avatar
27 votes
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Hamiltonian for relativistic free particle is zero

...what I would like to know is why we get a zero Hamiltonian. I suspect that this is due to the reparametrization invariance... Will this always happen? Why? Yes, it is due to reparameterization ...
Chiral Anomaly's user avatar
25 votes
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Why is Noether's theorem not guaranteed by calculus?

Since the problem here appear to be coordinates, let's just stop using coordinates, and for simplicity consider the theory of a single scalar field on space(time) $M$: Our field is a function $\phi : ...
ACuriousMind's user avatar
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22 votes

I don’t understand Noether’s theorem… there is nothing to prove?

Comments to the post (v3): Noether's theorem is just one method to determine conservation laws. If you have another, that's totally fine. Not all quasi-symmetry transformations (which in principle ...
Qmechanic's user avatar
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21 votes

Are all Lagrangians translationally invariant?

When you write $\mathcal{L}(x)=\mathcal{L}(\phi(x),\partial_\mu\phi(x))$, you're assuming translational invariance. A more general Lagrangian is written $\mathcal{L}(x)=\mathcal{L}(\phi(x),\partial_\...
Jahan Claes's user avatar
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20 votes

(A modification to) Jon Pérez Laraudogoita’s "Beautiful Supertask" — What assumptions of Noether's theorem fail?

Consider the simpler problem: a particle of mass $m_1$ moving at a speed of $v_1$ undergoes an elastic collision with a stationary ($v_2=0$) extended body of infinite mass $m_2$. What happens? (The ...
Nullius in Verba's user avatar
18 votes
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Noether's Theorem and scale invariance

First of all, let's see what Noether's Theorem says about your specific case (Klein-Gordon under global rescaling of the fields). Noether's theorem states that To every differentiable symmetry of ...
Giorgio Comitini's user avatar
18 votes

Why, really, should a practically-minded physicist care about Noether's Theorem?

Top 10 reasons This is a great question which I myself have thought long about. The way Noether's theorem is presented in textbooks, it really is just this nice shiny thing that doesn't factor much ...
user1379857's user avatar
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18 votes

What symmetries would cause conservation of acceleration?

Try and think of a Noether charge besides a momenta (linear, angular, Hamiltonian). It's rather hard to do in point particle mechanics because there really aren't any we talk about (it's easier in ...
Richard Myers's user avatar
17 votes

Why is Noether's theorem important?

The 2n constants in the system of second order differential equations (Lagrange equations) are just the arbitrary initial conditions of generalized coordinates and velocities of the system that ...
freecharly's user avatar
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16 votes
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How can Noether's Theorem be used to prove that the probability density satisfies a continuity equation?

First note that Schrödinger's equation can be understood to come from an action. The Lagrangian is $$L = \int~\mathrm d^3x \,\,\psi^†(x) \left(i \frac{\partial}{\partial t} - \frac{\nabla^2}{2m}\right)...
Luke Pritchett's user avatar
15 votes

Do all Noether theorems have a common mathematical structure?

I won't bother to reproduce the contents of the paper I want to recommend, but I will try to summarize what you'll find in it. Baez, John C. "Getting to the Bottom of Noether's Theorem." ...
rschwieb's user avatar
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14 votes

Layman's version of Noether's Theorem (or the intuition behind it)

Explaining Noether's theorem to 8th graders may be hard, as even the simplest special case of the theorem requires some calculus to state. However, if we're talking about why one example of an ...
J.G.'s user avatar
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14 votes
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What conserved quantity does Supersymmetry imply?

A supercharge is the conserved quantity that corresponds to a supersymmetry. It generates a supersymmetry & commutes with the Hamiltonian. A supercharge is a Grassmann-odd quantity. Noether's ...
Qmechanic's user avatar
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14 votes
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Equation in Noether's paper

Divergences here mean terms of the form $$\sum_{\mu=1}^n\frac{\partial F^{\mu}(x,\phi(x), \ldots)}{\partial x^{\mu}}.$$ The equation is a consequence of applying Leibniz rule (multiple times). Under ...
Qmechanic's user avatar
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13 votes

Do all Noether theorems have a common mathematical structure?

Noether's first and second theorem only apply to classical theories with an action formulation. The quantum analogs are (generalizations of) the Schwinger-Dyson equations and the Ward-Takahashi ...
Qmechanic's user avatar
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13 votes

How can the universe possess rotational symmetry yet have no center?

The laws of nature satisfy rotational symmetry about every point (or in other words, the general law of conservation of angular momentum holds everywhere in space). So you can pick any origin you'd ...
Eric Smith's user avatar
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13 votes

What makes energy "the" conserved quantity associated with temporal translation symmetry?

The OP's question is basically stating that in a system with time-translation invariant dynamics, we can define a conserved quantity by arbitrarily assigning a real number to each orbit; when the ...
Brian Bi's user avatar
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12 votes
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Conserved charges and generators

OP is wondering whether the conserved charge associated to a continuous symmetry always generates the symmetry itself. We can say, in full generality, that the answer is Yes. Let us see how this ...
AccidentalFourierTransform's user avatar
12 votes

Invariance of Action vs. Lagrangian in Noether's theorem?

No, they are not the same. To see why, even in classical mechanics, suppose we have symmetry transformation $q \rightarrow q + \epsilon K$ that leaves the Lagrangian invariant. This means that we must ...
Giuseppe Rossi's user avatar
12 votes
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Can we have conservation of momentum without conservation of energy?

Just adding a function $V(t)$ to the Hamiltonian does nothing - the equations of motion involve only the derivatives of the Hamiltonian w.r.t. $q$ and $p$, and so this changes nothing about the system,...
ACuriousMind's user avatar
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12 votes

What makes energy "the" conserved quantity associated with temporal translation symmetry?

What is special about the conserved quantity $Q(x, p) := \frac{1}{2} (x^2 + p^2)$, when also the quantity $Q_2(x, p) = \sin(x^2 + p^2)$ is conserved too, by the same temporal translational symmetry? ...
hft's user avatar
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11 votes

What conservation law corresponds to Lorentz boosts?

To supplement Marek's execllent answer, I provide an alternative derivation below and provide as many intermediate steps as possible. For an infinitesimal displacement $y^\mu=x^\mu+\xi^\mu$, a scalar ...
awct's user avatar
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11 votes
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Damped oscillator: time-reversal, time-translation and dissipation

To apply Noether's theorem, which is what you are alluding to here, one needs to look at continuous symmetries of a Lagrangian description of a system's dynamics. The damped oscillation equation you ...
Selene Routley's user avatar
11 votes
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What is the actual form of Noether current in field theory?

Eq. (5) is (up to factors of the infinitesimal parameter $\varepsilon$) the standard expression for the full Noether current. Here: $\delta x^{\mu}$ is the so-called horizontal component of the ...
Qmechanic's user avatar
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11 votes
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What is the Noether charge associated with the the color $SU(3)$ symmetry of QCD?

The $\mathrm{SU}(3)$ gauge symmetry is a local symmetry, and therefore it is not Noether's first, but Noether's second theorem that applies to it, which does not yield conserved quantities. For $\...
ACuriousMind's user avatar
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11 votes
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What symmetry is responsible for the amplitude independence of the period of a simple harmonic oscillator?

The most clean derivation is to go to the Hamiltonian formulation. Then the conserved charge is the Hamiltonian $$H~=~\frac{p^2}{2m}+\frac{kq^2}{2}\tag{A}$$ (basically the square of the amplitude $A$),...
Qmechanic's user avatar
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11 votes

Why is Noether's theorem not guaranteed by calculus?

It might be easier to see what's going on by making a few simplifications: First, we can work in $0+1$ dimensions -- in other words, we can work with ordinary particle mechanics, where the action is ...
Andrew's user avatar
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