# What is Noether's theorem: Definition and 96 Discussions

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.
Noether's theorem is used in theoretical physics and the calculus of variations. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.

View More On Wikipedia.org
1. ### I Noether's Theorem in the Presence of a Charged Operator

I am trying to understand the following idea that I found from some notes: Generally, a system with U(1) symmetry will have a conserved current: ##\partial_{\mu}j^{\mu} = 0##. The notes then state that in the presence of a local operator ##\mathcal{O}(x)## with charge ##q\in \mathbb{Z}## under...
2. ### I Taylor expansion about lagrangian in noether

I was studying a derivation of noether's theorem mathematically and something struck my eyes. Suppose you have ##L(q, \dot q, t)## and you transform it and get ##L' = L(\sigma(q, a), \frac{d}{dt}\sigma(q,a), t)##. ##\sigma## is a transformation function for ##q## Let's represent ##L'## by...
3. ### I Noether's second theorem: two questions

A technical subject, well above my level it seems (I'm still learning about quantum physics and special relativity), but one about which I absolutely must get some clear ideas as soon as possible. From what I 'understand', Noether's second theorem applies to infinite-dimensional symmetry...

9. ### I Vacuum energy and Energy conservation

Also, I have heard from physicists that vacuum energy fluctuation (creation and destruction of virtual particles) violates energy conservation. The reason, they justify, is based on uncertainty principle (energy-time form of uncertainty principle), energy can exist and disappear for a very short...
10. ### A Noether's theorem for finite Hamiltonian systems

The Noether's theorem for finite Hamiltonian systems says that: My question is: If I know a symmetry how can I write the first integral?
11. ### Lagrangian for the electromagnetic field coupled to a scalar field

It is the first time that I am faced with a complex field, I would not want to be wrong about how to solve this type of problem. Usually to solve the equations of motion I apply the Euler Lagrange equations. $$\partial_\mu\frac{\partial L}{\partial \phi/_\mu}-\frac{\partial L}{\partial \phi}=0$$...

14. ### Noether's theorem with non-finite transformations

Hi! I am given the lagrangian: ## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ## (Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case). I am...
15. ### I Conflict of domain and endpoints in Noether's theorem

In the derivation of energy conservation, there is the transformation ##q(t)\rightarrow q'(t)=q(t+\epsilon)##, whose end points are kind of fuzzy. The original path ##q(t)## is only defined from ##t_1## to ##t_2##. If this transformation rule is imposed, ##q'(t_2-\epsilon)=q(t_2)## to...
16. ### Trick for conserved currents in classical field theory

First I found the equations of motion for both fields: $$\partial_\mu \partial^\mu \psi = -\frac{\partial V(\psi^* \psi)}{\psi^*}$$ The eq. of motion with the other field is simply found by ##\psi \rightarrow \psi^*## and ##\psi^* \rightarrow \psi## due to the symmetry between the two fields...
17. ### A Physical meaning of Noether's theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
18. ### I Noether's theorem for discrete symmetry

I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ). The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have...
19. ### Understand Logic of Wald & Zoupas' Expression on Conserved Quantities

Wald and Zoupas discussed the general definition of conserved quantities" in a diffeomorphism invariant theory in this work. In Section IV, they gave one expression (33) in the linked article. I cannot really understand the logic of this expression. Would you please help me with this?

41. ### In an infinite quantum well, why Δn=0?

I've been reading up a bit on semiconductor quantum wells, and came across a selection rule for an infinite quantum well that says that "Δn = n' - n = 0", where n' is the quantum well index of an excited electron state in the conduction band, and n is the index of the valence band state where...
42. ### Newtonian formulation/proof of Noether's theorem

Hi. I've only ever seen Noether's theorem formulated ond proven in the framework of Lagrangian mechanics. Is it possible to do the same in Newtonian mechanics, essentially only using F=dp/dt ? The "symmetries" in the usual formulation of the theorem are symmetries of the action with respect to...
43. ### Help Me Understand This Author's Point: Noether's Theorem

I don't understand how the author get to these point. Please help me as i have been spending so much time trying to figure this out but to no avail. Thanks for your help Source: http://phys.columbia.edu/~nicolis/NewFiles/Noether_theorem.pdf
44. ### Noether's currents under dilatations (scaling transformations)

Hello, 1. Homework Statement Suppose we have the following Lagrangian density, in ## 3 + 1## dimensions: $$L = \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - g \phi^4$$ Under the dilatation (scaling transformation): ##x \rightarrow \lambda x^{\mu}, \phi (x) \rightarrow \lambda^{-1}...
45. ### Noether's theorem -- Time inversion

Noether's theorem said that because of homogeneity in time the law of conservation of energy exists. I am bit of confused and I am not sure is also time inversion some consequence of this. For example in the case of free fall we have symmetry ## t \rightarrow -t##. I am sometimes confused of...
46. ### Relation of Noether's theorem and group theory

I'm doing a small research project on group theory and its applications. The topic I wanted to investigate was Noether's theorem. I've only seen the easy proofs regarding translational symmetry, time symmetry and rotational symmetry (I'll post a link to illustrate what I mean by "the easy...
47. ### What is the relationship between dynamical symmetry and Noether's theorem?

Hi, I am learning classical mechanics right now, Particularly Noether's theorem. What I understood was that those kinds of transformations under which the the Hamiltonian framework remains unchanged, were the key to finding constants of motion. But here are my Questions: 1. What is...
48. ### Noether's Theorem and the real world

Hi. Noether's theorem comes from the symmetries of the world. In the real world the distribution of galaxies and materials are inhomogeneous. Noether's theorem does not stand for the real world, so conxervations of energy, momentum, angular momentum do not stand exactly. Is it OK...
49. ### Is Weak Isospin Conservation Provable Using Noether's Theorem?

Is it possible to prove that weak isospin associated with SU(2) is conserved using Noether's theorem?
50. ### Noether's theorem and constructing conserved quantities

Homework Statement A particle of mass m and charge e moving in a constant magnetic field B which points in the z-direction has Lagrangian ##L = (1/2) m( \dot{x}^2 + \dot{y}^2 + \dot{z}^2 ) + (eB/2c)(x\dot{y} − y\dot{x}). ## Show that the system is invariant under spatial displacement (in any...