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Finding Equation of motion from Lagrangian

With the Lagrangian $$L = \frac{ml^2}{2}\left(\dot \theta^2 + \sin^2(\theta) \dot \phi^2\right) +\frac{I}{2} \dot\phi^2 + mgl\cos(\theta)$$ the Euler-Lagrange equations are : \begin{align} ml^2\ddot \...
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Obtaining the KG equation from Action

$\square \phi = V'(\phi)$ does not imply $\ddot{\phi} = V'(\phi)$ as you have implied. The covariant derivative term $\square \phi$ contains the metric tensor, and consequent connection terms, so that ...
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Variation for the Canonical Scalar Field in $f(\phi)R$

For 1 do not repeat the same index more than two times. As Eletie says, you may rename the indices in one of the terms, then your expression will be meaningful. 4,5 are the same. The whole term is a ...
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Variation for the Canonical Scalar Field in $f(\phi)R$

In your variation of (1) you shouldn't use the same indices, so write $g_{\alpha \beta} \delta g^{\alpha \beta}$ instead. Then for (4) and (5) you'll have to integrate by parts in order to get $\delta ...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action

In this answer I will first point out a general property of the stationary action concept, and with that property laid out I will proceed to your question specifically The criterium of stationary ...
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Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} \dot{u}(x,t) - ca(u')^2$$ with respect to $u$, ($\delta L / \delta u$). Following my lecture ...
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For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

Generally speaking, you must consider a displacement with $\delta x_i$ to be $t$-dependent, when applying the principle of least action. The section where $\delta\dot{x}_i = 0$ is about symmetries of ...
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Variational operator confusion

One can in principle vary infinitesimally the $S_0[X,e]$ action (2) simultaneously wrt. both the $e$ and $X$ variables. The coefficient function in front of $\delta X$ will then give the Euler-...
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Confusion with the variational operator $\delta$ and finding variations

Yes, that happened. I guess you meant $$ \delta f = \sum_i \frac{\partial f}{\partial x_i} \delta x_i $$ on your third equation. Also you've implicity fixed inital $t_0$ and final $t_1$, so that your ...
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Confusion with the variational operator $\delta$ and finding variations

If the Lagrangian only depends on time through $X$ or $\dot{X}$, then we say that the Lagrangian has implicit but not explicit time dependence. So in your example, we would write \begin{equation} L(X, ...
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Four-vector differentiation (E-M Euler-Lagrange eq.)

As mentioned in the OP's link Schwartz (in his QFT book) doesn't keep track of the index placement on tensor objects that might obscure the structure a little. However, ignoring the first derivative $\...
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Geodesic equations with varying mass and the variational principle

Hints: The variable mass $$m~\propto~\phi\tag{A}$$ gives rise to an effective metric $$\bar{g}_{\mu\nu}~=~\phi^2g_{\mu\nu}.\tag{B}$$ The relationship between the corresponding proper times is $$-\...
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