# Questions tagged [variational-principle]

Any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).

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### About variational methods, renormalization and $a$, $c$-theorems

Variational approximation Variational methods are an important technique, frequently applied for the approximation of complicated probability distributions, with the applications in statistical ...
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### Variation of the Einstein-Hilbert action in $D$ dimensions without the Gibbons-Hawking-York (GHY) 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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### When to use (and when not to use) electromagnetic field conjugates in variational formulations

I found something a little bit confusing about writing variational formulas or Lagrangians for electromagnetic fields. I was looking at the book by Schwinger and Milton (chapter 4), and saw that ...
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### Question about the applications of Gauss's principle of least constraint

Recently i've learned the formulation of Gauss's principle of least constraints, which states that the motion of a system of material points is in maximal accordance with free motion, or under least ...
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### Derive Hamilton's principle of stationary action only from unitarity in quantum mechanics?

In this Question I want to give a derivation of Hamiltons Principle of Stationary action, and my question to the community would be, whether my argument is flawed. The System I want to look at is (...
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### Global and local symmetries in Noether's theorem. And also Stress-Energy tensors

Noether's theorem for fields is usually given as follows: Given a field theory with action $S=\int\mathcal{L}(\phi,\partial\phi)d^4x$, and given a one-parameter variation of the fields $\phi_\epsilon$...
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### f(R) action and GYH term

I have the following question. I have done integration by parts and I'm now applying the Gauss-Stokes theorem to get the boundary terms. I'm not interested in ignoring the terms, rather keeping them ...
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### Kleinert's Variational Perturbation Theory

I've been reading Kleinert's book and have been very intrigued by the chapter on variational perturbation theory. Namely, Kleinert derives a very good strong-coupling approximation to the ground state ...
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### Variation of quadratic term in modified Einstein-Hilbert actions

In the context of mimetic gravity at some point one try to add to an already modified Einstein-Hilbert action also a term like $$S_\chi=\int\,d^4x\,\sqrt{-g}\frac{1}{2}\gamma\chi^2,\qquad(\star)$$ ...
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### Is the "Force" of Gravity Simply Hamilton's Principle on a Curved Spacetime?

It's my understanding that General Relativity abstracts away the concept of gravity as a force, and instead describes it as a feature of spacetime by which massive objects cause curvature. Then it ...
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### Most stable shape if Newtonian gravity was proportional to $r^\alpha$

Consider lots of mass in isolated 3D space, close to each other. Consider that only the gravitational force (Newtonian) exists. Also consider that there is no rotational motion. It is evident that a ...
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### What is the variational energy of two spinless bosons with given interaction potential?

There was a question on my exam quantum mechanics that I wasn't able to solve and I am curious how it is done, I cannot find any reference in the section of pertubation theory that describes systems ...
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### Infinitesimal geodesic motion directly from the metric?

How can I see---directly from the Schwarzschild metric---that initially stationary (w.r.t. Schwarzschild coordinates) inertial test clocks will begin to fall toward e.g. the Earth (i.e. far outside ...
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### Boundary condition from extremization

I have an AdS Schwarzschild blackhole spacetime where the metric is given by, $$ds^2 = \frac{1}{z^2} \left( -f(z) dt^2 + \frac{dz^2}{f(z)} + dx^2 \right) \tag{1}\label{1}$$ There is a plane embedded ...
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### Question about the Weiss variational of gravitational action and related equations of motion

I was reading The Weiss Variation of the Gravitational Action by Feng and Matzner, where the authors take the variations of the gravitational action with respect to the bulk metric $g$, the induced ...
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### Proof that the maximal slicing condition maximizes the volume

I am struggling with proving that the maximum slicing condition $$K = 0$$ not only extremizes, but maximizes the volume $\mathcal{V}$ enclosed within a closed two-dimensional surface $S$. The ...
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### Relation between self-adjointness and variational principle and Rayleigh's principle

In mathematical physics, why is it that when an eigen-equation is described by a self-adjoint operator we say that it can be written (formulated) as a variational action (or principle)? Does the ...
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### Brachistochrone Problem for Spacecraft

I understand how the Brachistochrone Problem works, and do understand how the friction is added to it, however I am unsure of how to use the Brachistochrone Problem for use in spacecraft. We know ...
In page 14 to get the equation of motion, it takes the variation of the action $$S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu$$ ...