# Linked Questions

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### 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, ...
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### How are classical optics phenomena explained in QED (Snell's law)?

How is the following classical optics phenomenon explained in quantum electrodynamics? Reflection and Refraction Are they simply due to photons being absorbed and re-emitted? How do we get to Snell'...
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### 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 ...
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### 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?
1answer
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### On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
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### How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?

The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
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I'm studying the motion of light near Schwarzschild black holes, and I was wondering why the Hamiltonian of the Schwarzschild metric $$H = - \left( 1-\frac{2M}{r} \right)^{-1} \frac{p_{t}^2}{2}+\left(... 1answer 134 views ### What is the reasoning that leads one to postulate this second form for the relativistic particle action? The action for the free relativistic particle with worldline \gamma : I\subset \mathbb{R}\to M is$$S[\gamma]=-m\int d\lambda\sqrt{-\dot{\gamma}^a(\lambda)\dot{\gamma}_a(\lambda)}\tag{1} $$Now, ... 1answer 171 views ### Why Hamiltonian of gravity is zero? In paper Topological Gravity as the Early Phase of Our Universe there's statement: Hamiltonian of gravity would vanish by time reparameterization invariance. How to derive such result? 1answer 93 views ### Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero? Let's say our Lagrangian looks something like this:$$L = \int dz\, Q\cdot \dot{A},\tag{1}$$where Q and A are two generalized coordinates and \dot{Q} and \dot{A} would be the respective ... 1answer 93 views ### Hamilton-Jacobi-Einstein equation I have been looking at the Hamiltonian formalism of GR for some time and recently stumbled across the Hamilton-Jacobi-Einstein equation:$$\frac{1}{\sqrt{g}} (\frac{1}{2}g_{pq}g_{rs} - g_{pr}g_{qs}) \...

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