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### Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{...
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### What to do when the fields are arranged in a matrix?

I am dealing with a Lagrangian in which the fields are arranged in an $N\times N$ matrix and i have to find the minima of the potential. Usually i would write the Lagrangian in components and then ...
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### Partial Differentiation without chain rule in Euler Lagrange Equations [duplicate]

The Euler-Lagrange equations for a bob attached to a spring are $$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x}$$ But $v$ is a function of $x$. Is it ...
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### What's the Propagator in the Free Particle Case? (Path Integrals with Source Term)

If I take the Lagrangian to be, $$L(t)=\frac{1}{2}m \dot q(t)^2$$ The Euclidean Path Integral is supposed to be, $$K=\int D[q(t)] \ e^{-\int L(\dot q) d \tau}$$ If I add a source term $J(\tau)$ we ...
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### General form for functional derivatives

Working on the hamiltonian formalism applied to canonical field theory, how do I deduce the general form for the functional derivatives $\frac{\delta}{\delta \pi}$ and $\frac{\delta}{\delta \phi}$ (...
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### How to relate second variation to quadratic terms?

I am working on a physics problem, but my issue is math-related. My professor skips some steps based on 'intuition' that I lack: In a conservative system, to find out the nature of equilibrium points,...
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### Derive Schwinger-Dyson equations in Srednicki

On page 135 in Srednicki he defines the functional integral $$Z(J) = \int\mathcal{D}\phi\,\exp\Big[\mathrm{i}\big(S+\int\mathrm{d^4}y \,J_a\phi_a\big)\Big], \tag{A}$$ where $S$ and $J_a$ are the ...
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### Einstein action and the second derivatives

I have naive question about Einstein action for field-free case: $$S = -\frac{1}{16 \pi G}\int \sqrt{-g} d^{4}x g^{\mu \nu}R_{\mu \nu}.$$ It contains the second derivatives of metric. When we want ...
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### Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book Quantum Field Theory in a Nutshell, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of ...
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### Free energy variations

In a paper, I found this: $\mathbf{h}=\mathbf{h}(\mathbf{r})$ is called molecular field and is defined as the variation field of the Frank free energy functional $F_{d}$ with respect to the ...
Weinberg says, page 299, The quantum theory of fields, Vol 1, that The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...