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The condition 3 is the main assumption that goes into Noether's (first) theorem. It states that the action functional is invariant under a (global, continuous, off-shell) symmetry transformation (of the fields and spacetime). Referring to condition 3 as scale invariance, as Goldstein does, is non-standard terminology, and probably a bit confusing, since ...

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There is no inconsistency. $L_z$ is conserved, but $h^s$ is not; hence it is not a paradox that $h^s$ has no action on the trajectory at a certain time and nontrivial action at others. On the other hand, if the trajectory ever does cross the $z$ axis, then $L_z=m(\dot y x-\dot x y)$ will also vanish. Since $L_z$ is conserved, you can conclude that ...

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If we have some coordinates $q_i$ and some momenta $p_i$, then a generator of a transformation is defined as a function $g(q_i, p_i)$. By definition, this generates the transformation $$q_i \to q_i + \epsilon \frac{\partial g}{\partial p_i}$$ $$p_i \to p_i + \epsilon \frac{\partial g}{\partial q_i}$$ So if we want the generator of translations, we want ...

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You may use the following notation for hypersurfaces in four dimensions : $d\sigma_\mu = \epsilon_{\mu\alpha\beta\gamma}dx^\alpha dx^\beta dx^\gamma$ For instance $d\sigma_0= d^3x$ The expression of the momentum-energy is then : $P_\nu = \int d\sigma^\mu \Theta_{\mu\nu}$ The same kind of expression could be used with the charge : \$Q = \int ...

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