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### Variation of the Einstein-Hilbert action in D dimensions without the Gibbons-Hawking-York term

Consider the standard Einstein-Hilbert action in $D \ne 2$ dimensions spacetimes : $$S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x,$$ where $\Omega$ is ...
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### Proof of Hamilton's principle [duplicate]

Is there a anything like a proof of Hamilton's principle? Where would I find it?
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### Optimal Firing Rate to Maximize Vehicle Velocity

Suppose that you are in a vehicle in space with a power source generating a constant power of $P_0$ that you can use to fire iron pellets. The iron pellets compose 99% of the ship's mass, $m$. The ...
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### Mathematics of the Virtual Displacement

So I'm pretty certain this question has been asked to death here, but I still can't find a good explanation of a very particular aspect of the virtual displacements in physics. Background For ...
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### Prove a transformation is a variational symmetry?

My question: How to prove the family of transformations of the $(t,q)$ space, given by $(t,q) \to (t,U(\epsilon)q)$, where $U(\epsilon) \in SO(3)$, is a variational symmetry? So it depends on $L$ by ...
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### Optimal Airplane trajectory

The last time I took a plane the following problem crossed my mind. Setting: take the Earth and neglect its rotation around the Sun. It then only rotates on itself with angular velocity $\Omega$. ...
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### Noether's Theorem For Functionals of Several Variables

My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say ...
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### Is this constraint non-holonomic?

I am working on a variational problem involving elastic stability of a beam. The deformation of the beam is given by six functions of the material coordinate along the beams longitudinal axis. The ...
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### Is it possible to prove that planets should be approximately spherical using the calculus of variations?

Is it possible to use the Lagrangian formalism involving physical terms to answer the question of why all planets are approximately spherical? Lets assume that a planet is 'born' when lots of ...
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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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### Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt}$

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
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### With respect to what quantities do I vary Lagrangians in field theory?

I have recently been wondering, with respect to which quantities (covariant or contravariant) one should vary QFT Lagrangians and whether there is some rule regarding this. Let me give an example ...
Let's have the action $$S = \int (\partial_{\mu}h^{\mu \sigma}\partial^{\nu}h_{\nu \sigma} - \Lambda h^{\mu \nu}T_{\mu \nu}) d^{4}x.$$ For definiteness, $$h_{\mu \nu} = h_{\nu \mu} , \quad T_{\mu \... 3answers 3k views ### Why treat complex scalar field and its complex conjugate as two different fields? I am new to QFT, so I may have some of the terminology incorrect. Many QFT books provide an example of deriving equations of motion for various free theories. One example is for a complex scalar ... 3answers 1k views ### Does a four-divergence extra term in a Lagrangian density matter to the field equations? Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: {\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^... 1answer 73 views ### Double variation of Schwinger action principle The Schwinger action principle is given by$$\delta_{1}\big\langle b\big|a\big\rangle= i\int_{t_{a}}^{t_{b}}\text{d}t\,\sum_{c,d}\big\langle b\big|c\big\rangle\big\langle c\big|\delta_{1}L(t)\big|d\...
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 ...