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The point is that eq. (1.35) should hold off-shell to have a symmetry, while eq. (1.37) may only hold on-shell. [The term on-shell (in this context) means that the Euler-Lagrange equations are satisfied. See also this Phys.SE post.] In other words: On-shell, the action will only change with at most a boundary term for any infinitesimal variation, whether ...

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The basic idea is the following. For the shake of simplicity, I henceforth assume that every function does not depend explicitly on time (with a little effort, everything could be generalized dealing with a suitable fiber bundle over the axis of time whose fibers are spaces of phases at time $t$). On a symplectic 2n dimensional manifold (a space of ...

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"Derivation" of Baryon Number Conservation - Consider the QCD Lagrangian (density) $$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$ where the symbols have their usual meaning. This is invariant under $U(1)$, which is nothing but a multiplication of $\psi$ by a global phase factor $e^{i \theta}$. This is ...

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Consider an element $g$ of the symmetry group. Say $g$ is represented by a unitary operator on the Hilbertspace $$T_g = \exp(tX)$$ with generator $X$ and some parameter $t$. It acts on an operator $\phi(y)$ by conjugation $$(g\cdot\phi)(y) = T_g^{-1}\phi(y) T_g = e^{-tX}\phi(y) e^{tX} = \big[ 1 + t[X,\cdot]+\mathcal{O}(t^2)\big]\phi(y)$$ On the other ...

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There is at least one philosopher before Plato and he is Anaximander. There are many passages in his works that relate to the concept of symmetry: The basic elements of nature (water, air, fire, earth) which the first Greek philosophers believed that constituted the universe represent in fact the primordial forces of previous thought. Their collision ...

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The phrase "the function is spherically simmetrical" means that, if $G$ is an orthogonal transformation (that sends spheres into themselves), then $$f(G\mathbf r, G\mathbf p,t)=f(\mathbf r , \mathbf p, t).$$ If you know $\mathbf r^2$, $\mathbf p^2$, $\mathbf r \cdot \mathbf p$ you can calculate $f$ by taking an orthogonal transformation which maps $\mathbf ... 1 An internal symmetry only involves transformations on the fields of a theory, and must act the same independent of the point in spacetime. For example, consider a Lagrangian, $$\mathcal{L} = \partial_\mu \psi^\star \partial^\mu \psi - V(|\psi|^2)$$ for some potential$V$, and complex field$\psi$. The theory has an internal symmetry, namely one which ... 1 Your question seems to contain two parts. First, you're asking how to set up the equations of motion for this coupled system. Second, you are asking how to use symmetry considerations to find the normal modes and frequencies. Let's first answer the bit about symmetry first Normal modes - symmetry Your observation about the reflection symmetry is spot on. ... 1 In this answer we will consider a Lie algebra$L$(rather than a Lie group). Then: If$M$is a manifold, let there be a Lie algebra homomorphism$\rho:L\to \Gamma(TM)$into the Lie algebra of vector fields on$M$. The map$\rho$is called an anchor. If the manifold$(M,\{\cdot,\cdot\}_{PB})\$ is a Poisson manifold, it is natural to require that the vector ...

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