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 :
\begin{equation}
S_{EH} = \frac{1}{2 \kappa} \int_{\Omega} R \; \sqrt{- g} \; d^D x,
\end{equation}
where $\Omega$ is ...
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Variational principle with $\delta I \neq 0$
In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
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BTZ Black Hole Central Charge and Conformal Weight
I have been trying to reproduce a calculation (equation 4.12) in this paper http://arxiv.org/pdf/1107.2678v1.pdf by Carlip reviewing the derivation of the effective central charge of the BTZ Black ...
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Variations of Boundary Actions and Bulk Physics
In physics, we are often taught that the action principle generates only bulk equations of motion on-shell and that boundary terms can be neglected provided the fields in question fall off ...
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Auto-parallel Transport or Principle of Extremum Action?
In an affinely connected spacetime with a metric compatible connection, the equation of the curve in which the tangent vector at each point is the result of the parallel transport of every tangent ...
user87745
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General relativity from helicity 2 massless field theory by using Deser's arguments
Recently I have discovered the method of constructing of GR from massless field with helicity 2 theory. It is considered here, in an article "Self-Interaction and Gauge Invariance" written by Deser S. ...
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Boundary conditions for Lagrangian formulation of General Relavitiy
I am reading section 4.1.3. of Poisson's book "A relativist's toolkit" and I am a bit perplexed by condition (4.13), namely that the variational principle for General Relativity has to be ...
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Is there a Lagrangian $L$ (equivalently an action functional $S$) which yields the Navier-Stokes equation?
The Navier-Stokes equation or the Euler equation are usually derived as the conservation laws.
However, I wonder if there exists a Lagrangian $L$ or equivalently, an action functional $S[\...
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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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Energy in dynamical variational principle
In quantum mechanics we use variational principle in order to find approximate expression for the ground state. Lets assume our probe wavefunction $|\Psi\rangle$ can be expanded in orthonormal basis
$...
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Variations of S-matrix functional and Feynman diagrams in Weinberg QFT
Weinberg on p. 287 of his QFT vol. 1 introduces the extended interaction operator:
$$
\tag 1 \hat{V}(t) \to \hat{V}(t) + \sum_{a}\int d^{3}\mathbf x \hat{o}_{a}(\mathbf x ,t)\varepsilon_{a}(x).
$$
...
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Variational approach to the Laplace and Poisson equation in Jackson's book
In Section 1.12, Chapter 1 in Jackson's Classical Electrodynamics, he considered an "energy-like" functional,
$$
I[\psi]=\frac{1}{2}\int_V\nabla\psi\cdot\nabla\psi\ d^3x - \int_V g\psi\ d^3 ...
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What does it mean for a force to 'produce' virtual displacement?
Book: Variational Principles of Mechanics by Lanczos, 1st edition, 1949.
Statement (page 80):
"Two systems of forces which produce the same virtual displacements are dynamically equivalent."...
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What's the path of least action for fermions off-shell?
The Lagrangian of fermions is first order both in space-derivatives and time-derivatives. In the variation of the action one usually fixes both the initial point and end point. I have the following ...
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Is it possible to obtain the Bianchi identities (i.e. equations of motion for the free gravitational fields) from a variational principle?
Einstein's field equations (EFEs) describe the pointwise relation between the geometry of the spacetime and possible sources described by an energy-momentum tensor $T^{\mu\nu}$. As well known, such ...
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Variation of Gibbons-Hawking-York term. General boundary condition and total derivatives
It is actually a comment and question to the answer of Robert McNees in the following post:
Explicit Variation of Gibbons-Hawking-York Boundary Term
In deriving the variation of the extrinsic ...
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equation of motion for the scalar field via variational principle in general relativity
I would like to find the equation of motion for the scalar field $\phi$ by varying the following action in General Relativity.
Special Relativity:
$$
S = -\tfrac{1}{2}\int d^4\xi\, \eta^{ab} \...
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On Cohomological Gauge theory Calculation
I am in trouble with calculation details of Witten's Two dimensional Gauge Theories Revisited. My questions is about (3.21) and (3.27).
From section 3, we have
$$\delta A_i=i\epsilon \psi_i\\
\delta \...
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Coulomb force from a variational principle
See the attached discussion from Zangwill's Modern Electrodynamics, and in particular footnote 9. The point of this question is to understand how to recover Coulomb’s force law from an assumed form ...
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Can the full set of II. Bianchi identities be derived from the symmetries of the action?
In pseudo-Riemannian geometry we can derive the II. Bianchi identities by considering, e.g. the expression of the Riemann tensor in Riemann normal coordinates. They read
$$R_{\mu\nu\kappa\lambda;\...
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Euler-Lagrange equation in a differential form notation
Treating the Lagrangian density as a $d$-form in $d$-dimensional spacetime, how can one write the Euler-Lagrangian equation basis independently in the form notation? If possible, can you also apply ...
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What happens when the same action extremal value can be obtained in more than one path in the configuration space?
I'm trying to understand the logic underneath the concept of action and lagrangian. I know this kind of questions have been asked many times, but I was unable to find an answer to this one.
I've ever ...
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Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?
I want to arrive to Hamilton-Jacobi equation using the Riemannian geometry.
So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
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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 ...
user145190
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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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2
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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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Boundary term for Chern-Simons action
As discussed in David Tong's lecture series on the edge modes in the quantum Hall effect (http://www.damtp.cam.ac.uk/user/tong/qhe.html) (page 203), varying the 2+1D Chern-Simons action yields:
$$\...
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Convert a quasi-symmetry of the action into a strict symmetry
A quasi-symmetry of an action $S$ is a transformation of the fields that leaves the action invariant up to a boundary term (see e.g. the answer to this question). In contrast, let us call a ...
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Writing a non-minimally coupled Einstein-Maxwell action
Usually you study a GR system with an electromagnetic field using the standard action
\begin{equation}
S=\int{(R-\frac{1}{4}F^2)\sqrt{-g} d^4 x}
\end{equation}
(where $F_{\mu\nu}=A_{\mu,\nu}-A_{\nu,\...
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answer
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How does derivation of Lagrange equation from d’Alembert principle differ from the derivation of it from principle of least action?
Using d’Alembert principle, one doesn't require any assumption like the one made in other case where particle has to follow the path of least action.
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The gothicized metric and the Palatini formalism
In the Palatini formalism of GR, we had two results treating the metric $g_{\mu\nu}$ and the connection $\Gamma^\alpha_{\mu\nu}$ separately as dynamical variables, which are
The vaccum field ...
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votes
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Adjoint operators and the method of variation for the Orr-Sommerfield problem
My question relates to the transient Orr-Sommerfield Squire problem, described on page 20 of the thesis by Eaves.
Here I briefly describe the context of my question.
We consider an infinitesimal ...
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How to translate field equation into Lagrangian density?
I'm learning about particle physics, and I was troubled by how one could write the Lagrangian density out of the field equation.
Klein-Gordon equation $$(\partial_\mu\partial^\mu +m^2)\phi(t,\vec x)=...
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Reference request: Scattering from action
Consider a separable solution to a Hamilton-Jacobi equation of an $n$-dimensional autonomous system of the form
$$W(\alpha_1,...,\alpha_n,x^1,...,x^n) = \sum_j \int_0^{x^j} w^j(\alpha_1,...,\alpha_n,x'...
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votes
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Finite differences as a variational method
I think I should be able to derive finite differences for the Schroedinger equation by starting from a variational method. More specifically: in finite differences we approximate
$$\{\psi(x),x\in\...
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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 ...
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A simple question about equation of motion in polchinski's String theory?
In page 14 to get the equation of motion, it takes the variation of the action
$$
S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu
$$
...
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Lagrangian Mechanics to solve arbitrary maximization problem
I've been thinking for some time about how to better find the optimal weights for neural networks and it struck me that when solving Lagrangian mechanics problem you are optimizing the action function....
