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### Invariance of Lagrangian under Poincaré group transformations implies covariant Lagrange equations? [duplicate]

I'm taking a class on classical fields and I came across a statement that I can't think about an argument to show that its true. It says that Invariance of a Lagrangian under transformations on ...
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### What's the interpretation of Feynman's picture proof of Noether's Theorem?

On pp 103 - 105 of The Character of Physical Law, Feynman draws this diagram to demonstrate that invariance under spatial translation leads to conservation of momentum: To paraphrase Feynman's ...
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### 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 ...
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### Is there a mathematical reason for the Lagrangian to be Lorentz invariant?

The Hamiltonian is the energy, which is just one component of a four-vector and therefore not Lorentz invariant. The Lagrangian is the Legendre transform of the Hamiltonian and I was wondering if ...
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### Why is a theory Lorentz invariant if the Lagrangian is Lorentz invariant?

For if I started by trying to make the Hamiltonian Lorentz invariant, I would have failed. Indeed, the Hamiltonian is part of a covariant tensor. But how do I know that the Lagrangian is not a part of ...
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### Explicitly show covariance of Euler Lagrange equations

I know that the Euler Lagrange equation (here only in 1D) $$\left(\frac{d}{dt}\frac{\partial}{\partial\dot{x}}-\frac{\partial}{\partial x}\right)L\left(x,\dot{x},t\right)=0$$ is invariant under (...
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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 $t, q(t)$ ...
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### On-shell and off-shell transformations in Noether's theorem

For any transformation of the fields, $$\varphi\to\varphi'=\varphi+\delta\varphi$$ the change in the Lagrangian can be written as $$\delta\mathcal L = \text{EoM} + \partial_\mu j^\mu\tag{1}$$where "...
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### 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 ...
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### Isotropic of Inertial frame?

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and ...
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### Lorentz Invariance of the Euler-Lagrange equation for fields

Given an Lorentz invariant Lagrangian density $L$ of a Lorentz invariant scalar field $\phi$, How does one show that the following term in the Euler-Lagrange equation is invariant under Lorentz ...
I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...
I think that the following statement is true, but can't seem to prove it. Suppose we have a scalar field whose Lagrangian density$^1$ $\mathcal L(\phi, ~\partial_\mu\phi, ~X^\mu)$ is Lorentz invariant....