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The appropriate definition of symmetry uses infinitesimal quantities, not just small quantities. Thus, in terms of your question, the Lagrangian is symmetric if $dL/d\epsilon=0$ at $\epsilon=0$. In terms of your example (rotation of a 2D harmonic oscillator), we have $$ L \to (1+\epsilon^2) L = L + \mathcal{O}(\epsilon^2) $$ Thus to first order in ...


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Classical Lagrangian field theory deals with fields $\phi: M \to N$, where $M$ is spacetime and $N$ is the target-space of the fields. We shall for convenience call $M$ and $N$ the horizontal and the vertical space, respectively. OP is in this terminology essentially asking Q: What is the meaning of horizontal transformations? A: It is a (horizontal) flow ...


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When you integrate the Lagrangian density over a certain region $\Omega$, this is in principle allowed to change and this gives you a "boundary" term in the variation. This is well discussed in, e.g., the book of Goldstein (3rd edition), where the correct proof of the Noether theorem is given.


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May I ask what text you are reading? My understanding of the stress energy tensor is as follows. The Noether condition is written as,\begin{equation} \partial _\mu \bigg[\frac{\partial \mathcal L}{\partial (\partial _\mu \phi )}\delta \phi +\mathcal L \delta x^\mu\bigg]=0 \end{equation} In the discrete case we can imagine separate infinitesimal time and ...


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Details have to be filled in, but I think the general idea went along these lines: the variation of the action with respect to the metric $g_{\mu\nu}$ is given by $$ \delta_gS \sim \int T^{\mu\nu}\delta g_{\mu\nu}\,. $$ Now specialize to particular variations of $g_{\mu\nu}$, the diffeomorphisms. For an infinitesimal diffeomorphism of the form $x^\mu\to ...


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Noether's theorem states that if a system has a continuous symmetry, there is a quantity related to this symmetry, called the Noether charge, which is conserved. It does not state anything on the fact that adding a constant term to a measurable quantity may or may not change the physical description of the system. Only some physical quantities in fact are ...



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