Linked Questions
68 questions linked to/from Invariance of Lagrangian in Noether's theorem
60
votes
5answers
7k views
Is the converse of Noether's first theorem true: Every conservation law has a symmetry?
Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law.
Is the converse true: Any conservation law of a physical ...
49
votes
6answers
13k views
What symmetry causes the Runge-Lenz vector to be conserved?
Noether's theorem relates symmetries to conserved quantities. For a central potential $V \propto \frac{1}{r}$, the Laplace-Runge-Lenz vector is conserved. What is the symmetry associated with the ...
35
votes
5answers
9k views
Noether charge of local symmetries
If our Lagrangian is invariant under a local symmetry, then, by simply restricting our local symmetry to the case in which the transformation is constant over space-time, we obtain a global symmetry, ...
24
votes
4answers
6k views
Galilean invariance of Lagrangian for non-relativistic free point particle?
In QFT, the Lagrangian density is explicitly constructed to be Lorentz-invariant from the beginning. However the Lagrangian
$$L = \frac{1}{2} mv^2$$
for a non-relativistic free point particle is ...
34
votes
4answers
3k views
When can a global symmetry be gauged?
Take a classical field theory described by a local Lagrangian depending on a set of fields and their derivatives. Suppose that the action possesses some global symmetry. What conditions have to be ...
21
votes
5answers
4k views
Noether Theorem and Energy conservation in classical mechanics
I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
36
votes
1answer
4k views
Do an action and its Euler-Lagrange equations have the same symmetries?
Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations.
Can ...
18
votes
2answers
3k views
Is there a kind of Noether's theorem for the Hamiltonian formalism?
The original Noether's theorem assumes a Lagrangian formulation. Is there a kind of Noether's theorem for the Hamiltonian formalism?
22
votes
2answers
1k views
What symmetry is associated with conservation of Lipkin's zilch?
The 'zilch' of an electromagnetic field is the tensor
$$
Z^{\mu}_{\ \ \ \nu\rho}=^*\!\!F^{\mu\lambda}F_{\lambda\nu,\rho}-F^{\mu\lambda}\,{}^*\!F_{\lambda\nu,\rho}
\tag1
$$
given in terms of the ...
13
votes
4answers
2k views
Does the action and Lagrangian have identical symmetries and conserved quantities?
From the book Introduction to Classical Mechanics With Problems and Solutions by David Morin, page 236 states:
Noether's Theorem: For each symmetry of the Lagrangian, there is a conserved quantity.
...
6
votes
4answers
1k views
Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
24
votes
1answer
5k views
Noether's Theorem and scale invariance
Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e.
$$\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$$
...
10
votes
4answers
666 views
What constitutes a symmetry for Noether's Theorem?
I have some confusion over what exactly constitutes a symmetry when trying to apply Noether's theorem. I have heard both that a symmetry in the action gives a conserved quantity, and that a symmetry ...
10
votes
3answers
1k views
What is the actual form of Noether current in field theory?
Let us consider $N$ independent scalar fields which satisfy the Euler-Lagrange equations of motion and are denoted by $\phi^{(i)}(x) \ ( i = 1,...,N)$, and are extended in a region $\Omega$ in a $D$-...
9
votes
2answers
2k views
Why the Galileo transformation are written like this in Quantum Mechanics?
In Quantum Mechanics it is said that the Galileo transformation $$\mathbf{r}\mapsto \mathbf{r}-\mathbf{v}t\quad \text{and}\quad \mathbf{p}\mapsto \mathbf{p}-m\mathbf{v}\tag{1}$$
is given by the ...