61 questions linked to/from Invariance of Lagrangian in Noether's theorem
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### 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 $D$-...
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### Why do the Lagrangian and Hamiltonian formulations give the same conserved quantities for the same symmetries?

The connection between symmetries and conservation laws can be viewed through the lens of both Lagrangian and Hamiltonian mechanics. In the Lagrangian picture we have Noether's theorem. In the ...
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### Invariance of Action vs. Lagrangian in Noether's theorem?

I have recently started studying classical field theory. Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. But I find ...
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### What symmetry gives you charge conservation?

This is a popular question on this site but I haven't found the answer I'm looking for in other questions. It is often stated that charge conservation in electromagnetism is a consequence of local ...
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### In what sense are the equations of motion conserved by symmetries?

I am studying variational principles and I have been reading this set of notes by Townsend. In the first paragraph of Section 9, Townsend defines what it means for a transformation to be a symmetry of ...
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### Big puzzle about Noether's theorem of coordinate transformation (spacetime symmetry)

For Noether theorem with only internal symmetry, I've found there has been a very clear proof. But I still struggle with the proof of coordinate transformation. Because there are so many different ...
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### Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
Setup Consider a mapping $F$ that takes every point $x$ on the manifold $M$ to the point $x'$ on the same manifold. Under this mapping the field $\phi(x)$ evaluated at the point $x$ changes to \$\phi'(...