61 questions linked to/from Invariance of Lagrangian in Noether's theorem
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 $\phi'(... 1answer 38 views ### What is the definition of a symmetry of an action? Symmetries of Lagrangians The definition of a symmetry of a theory is quite clear at the level of a Lagrangian. We say a Lagrangian$\mathcal{L}(\phi,\partial_\mu \phi)$is symmetric under the ... 1answer 57 views ### Why is an action built from superfields guaranteed to be supersymmetric? Given a superfield (in 0+1 spacetime + 2 superspace coordinates) $$X(t,\theta_1,\theta_2) = x(t) + \theta_i \psi_i(t) + \theta_1 \theta_2 F_{12}(t)\tag{1}$$ and given the standard supercharges ... 1answer 332 views ### Problem using Noether's theorem in time-dependent lagrangian I have some problems calculating the conserved quantity for a lagrangian of the form $$L = \frac{1}{2}m\dot{q}^2 - f(t) a q,$$ because I found the general problem too abstract, I tried at first ... 1answer 240 views ### Proving that Noether charge generates symmetries in the Lagrangian formalism I know that similar questions have been asked on this site before, but I haven't been able to find the answer to my specific question. I want to show that the Noether charge defined in Lagrangian ... 1answer 452 views ### The Lagrangian of a free particle in Landau & Lifshitz In Landau & Lifshitz's derivation of the Lagrangian of a free particle in a galilean frame of reference one finds the following argument: the equations of motion in two galilean frames must be ... 1answer 167 views ### Elementary question about global supersymmetry of a worldsheet [closed] I'm reading chapter 4 of the book by Green, Schwarz and Witten. They consider an action $$S = -\frac{1}{2\pi} \int d^2 \sigma \left( \partial_\alpha X^\mu \partial^\alpha X_\mu - i \bar \psi^\mu \rho^... 1answer 466 views ### Does an on-shell symmetry necessarily change the Lagrangian by a total derivative? This is a follow-up question to: Does a symmetry necessarily leave the action invariant? Qmechanic writes here: Here the word off-shell means that the Lagrangian eqs. of motion are not assumed to ... 1answer 93 views ### Symmetry modulo total derivative term in Noether's Theorem I came across the proof of Noether's Theorem in David Tong's notes (page 14) on QFT. He writes something like, We say that the transformation$$\delta\phi(x) = \chi (\phi) \tag{1.34}$$is a ... 1answer 41 views ### How do we define the quantity Q, in the conservation of energy? And what does it rely on? Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ... 1answer 111 views ### Lagrangian of free particle - classical case I have a question, more related to a mathematical aspect of physics, which seems I am not understanding very well. So, by applying Galilean transformation between two reference frames, which move at ... 1answer 323 views ### Conserved quantity of a relativistic free Lagrangian for a Lorentz boost Let$$L~=~-mc^2\sqrt{1- \frac{|\textbf{v}|^2}{c^2} },$$where \textbf{v} is the usual velocity of the particle in a fixed inertial frame. Then, this is the Lagrangian for a relativistic free ... 1answer 331 views ### Variation of the Action under infinitesimal arbitrary transformations and Noether's Theorem Let's consider an arbitrary infinitesimal transformation of the fields and their coordinates :$$x'^{\mu}= x^{\mu} + \delta x^{\mu} = x^{\mu} + \frac{\delta x^{\mu}}{\delta{\omega}^a}{\omega}^a\tag{1}... 1answer 190 views ### Conservation of energy for a class of Lagrangians with explicit time dependence I have read in my book that if$\frac{\partial L}{\partial t}=0$, then the quantity$ L-\frac{\partial L}{\partial \dot{q}} \dot{q} \$ is conserved, and we call it the energy of the system. But if ...
In all QFTs I know, the Lagrangian density is completely invariant under the Poincare group, $$\mathcal L \to \mathcal L.$$ On the other hand, the action would be invariant even if the Lagrangian ...