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Questions tagged [variational-calculus]

Variational calculus is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find extrema of functionals: mappings from a set of functions to the real numbers. The archetype application in physics is Lagrangian mechanics, seeking extrema of action functionals.

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Varying $\Box h_{\lambda\kappa} \Box h^{\lambda\kappa}$ with respect to $h_{\mu \nu}$ [closed]

I'm trying to gain a working understanding of the basic calculus of variations used in field theories, and I'm a little stuck trying to understand a step I've seen in a derivation. I'm sure my ...
cosmologia's user avatar
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Deriving the Noether's theorem

I am familiar with how Noether's theorem is derived in some sources/books, the answer in 534699 is particularly clear. However, I'm reading A First Book of Quantum Field Theory by Pal, and although ...
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Changing coordinate system [migrated]

Someone please explain how did we get second term in equation 2.15.
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Minimum or Stationary Value of a Mixed Boundary Problem

Take the volume integral of the dissipated DC current in a finite volume $\mathcal V$ of conductivity $\sigma$ and stationary potential distribution $\phi$ while assuming charge conservation $\nabla \...
hyportnex's user avatar
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Find curve minimizing energy loss due to friction [closed]

I am looking for an ansatz of the following problem: Given a mass $m$ moving in a constant gravitational field along curves $C$ connecting two fixed points I want to find the curve $C_0$ that ...
TomS's user avatar
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Energy-momentum tensor and equation of motion in Einstein-Dilaton theory [closed]

I am following this paper (see eq. 19-22) and trying to derive the equation of corresponding to Einstein-Dilaton gravity (ignoring the Maxwell part for now) \begin{align} S_{\text{E-D}} = \int d^4 ...
Faber Bosch's user avatar
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Weyl variation of a generic action

In this paper https://arxiv.org/abs/hep-th/9906127 (see eq. 15) The following identity appears $$ \delta_{W} \int d^d x \sqrt{-\gamma} \tilde{\mathcal{L}}^{(n)} = \int d^d x \sqrt{-\gamma} \sigma\left(...
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How to do Variational Principle in QFT? ($SU(2)$-Yang-Mills)

I am currently familiarizing myself with QFT and have a question about this article. My understanding is that the Lagrangian density in (2) couples my gauge fields to the Higgs field. And with ...
Hendriksdf5's user avatar
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Are the quantity $\frac{\delta \sqrt{-g}}{\delta g_{\mu \nu}}$, $\frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}} $ are computable? [duplicate]

From $\delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g} g_{\alpha \beta}\delta g^{\alpha \beta} = \frac{1}{2}\sqrt{-g} g^{\alpha \beta} \delta g_{\alpha \beta}$, can we compute \begin{align} &\frac{\...
phy_math's user avatar
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Derivative of line element in general relativity is zero?

The Lagrangian for a point particle in general relativity is $$ L= -m \sqrt{-g_{\mu\nu}\dot{x}^\mu \dot{x}^\nu} $$ where $x^\mu(\lambda)$ is the world line of a particle with mass $m$. The world line ...
jojo123456's user avatar
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Varying the Chern-Simons action

Summary/TL;DR I want a detailed calculation of the derivation of classical equations of motion from the Chern-Simons action using differential forms, using variational derivatives. I mentioned "...
Sanjana's user avatar
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Equation of motion of free field Lagrangian

I tried to derive the equation of motion obtained by varying Lagrangian (2) in https://arxiv.org/abs/0804.4291 wrt the metric. It is supposed to give the second equation in (5) of the paper but my ...
vyali's user avatar
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Understanding certain boundary conditions of functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$

A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!) The General Question: Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
PhysicsIsHard's user avatar
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Variation of the Lagrangian expressed as a time derivative of a function

In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as $$\...
tugboat2's user avatar
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Chain rule with functional derivative?

I posted the same question on math exchange but no answer yet, so I post it also here: "I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier ...
Filippo's user avatar
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Using functional derivatives and Euler-Lagrange to obtain wave equation in 3d elastic media

I'm trying to solve exercise (1.5) from Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur wherein we consider a 3D elastic material whose potential energy is given by $$ V = \frac{\...
Keshav Balwant Deoskar's user avatar
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Some details about variational calculation on variational bi-complex

I am reading 1801.07064, where the covariant phase space formalism is elaborated. From what I have learnt from classical mechanics, the variation of Lagrangian for field theory $L[\Phi,\partial\Phi]$ ...
LaplaceSpell's user avatar
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Is there a practical distinction between functions of state and functionals in thermodynamics?

In thermodynamics, and more precisely when talking about continuous systems, some sources [1, 2] introduce functionals of state: $$F[s(x), \dots]:=\int_VdV(x)f(s(x),\dots,x)$$ In order to derive ...
GvPStack's user avatar
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1 answer
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Difference in definition of conserved current in Quantum Field Theory

In David Tong Lecture Notes (page 14), it is written that Proof of Noether's Theorem: We'll prove the theorem by working infinitesimally. We may always do this if we have a continuous symmetry. We ...
darkphysics's user avatar
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Variation of the Einstein-Hilbert action to derive the metricity condition

Consider the Einstein-Hilbert action: $$S=\int d^{4} x \sqrt{-g} g^{\mu \nu} R_{\mu \nu}$$ If we vary it with respect to the connection, assuming no prior relation between the metric and the ...
MSHD's user avatar
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How did the boundary term vanish in deriving equation of motion from Lagrangian? [closed]

I was deriving the equation of motion from Lagrangian, by using the principle of least action. Usually, at this point in derivation, $$\int dt \frac{\partial L}{\partial \dot{q}} \frac{\partial}{\...
NamikazeMinato's user avatar
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(Opinionish) Feynman Lectures Exercise: Use principle of virtual work to show the following about $n$ forces in static equilibrium

I would normally not want someone to provide a complete and well-stated solution to an exercise. In this case, that is exactly what I am seeking. The following exercise appears in Chapter 2 of the ...
Steven Thomas Hatton's user avatar
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Variation of action of non-critical string under Weyl transformation (worldsheet cosmological constant term)

In David Tong's lecture notes on string theory, section 5.3.2 An Aside: Non-Critial Strings, page 121, he describes the non-critical string with the following action: $$S_{\text{non-critical}} = \frac{...
Jens Wagemaker's user avatar
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Variation of nonminimal derivative coupling term

all. Can I request you assistance about the following problem? How do I vary this action with respect to metric $\delta g_{ab}$ $$ \int d^4x \sqrt{-g} \Big[\kappa R+ G_{ab}\nabla^a \phi \nabla^b \phi \...
trickymindful's user avatar
3 votes
1 answer
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Computing variation of triple wedge term in Chern-Simons action

For simplicity, assume $G$ is simple, compact, connected, and simply-connected. The Chern-Simons action for a non-abelian structure group $G$ on a trivial bundle with closed base manifold is given by $...
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2 votes
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Free scalar field deriving Ehrenfest using the path integral

In his lecture notes on String theory, David Tong derives Ehrenfest theorem using the path integral: $$S = \frac{1}{4\pi \alpha'}\int d^2\sigma\ \partial_\alpha X\ \partial^\alpha X\tag{4.19}$$ $$ 0 =...
Jens Wagemaker's user avatar
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Variation of the kinetic term wrt the metric in scalar field theory

Varying $\partial_\lambda\phi\,\partial^\lambda\phi$ wrt the metric tensor $g_{\mu\nu}$ in two different ways gives me different results. Obviously I'm doing something wrong. Where am I going wrong? ...
vyali's user avatar
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Palatini variation of Ricci tensor

I was looking at Problem 2 of chapter 4 of Sean Carroll's General Relativity book, where you were supposed to demonstrate starting from the Einstein-Hilbert action, and assuming that the connection is ...
Andreas Christophilopoulos's user avatar
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Ghost-free quadratic gravity

This is an question about how to write an equivalent of "energy-squared" in terms of a gravitational metric. i.e. a spin-2 term that approximates to the spin-0 term $\int (\nabla^\mu \phi \...
bob's user avatar
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How would I apply Euler-Lagrange to this action?

I am trying to apply the Euler-Lagrange equations to the following action which is a type of $f(R)$ model: $$S=\int\left(\sqrt{-g}\sqrt{ R^2 - \Lambda R + \Lambda^2} + T\right)dx^4$$ Where $R$ is the ...
bob's user avatar
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1 answer
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Question about functional derivative computation in Quantum Field Theory for the Gifted Amateur

I'm confused about the evaluation of the functional derivative of Equation 1.12, $$J[f] = \int [f(y)]^p \phi(y) dy$$ on page 13 of Quantum Field Theory for the Gifted Amateur in Chapter 1. Here are ...
aadithyaa's user avatar
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In Lagrangian mechanics, do we need to filter out impossible solutions after solving?

The principle behind Lagrangian mechanics is that the true path is one that makes the action stationary. Of course, there are many absurd paths that are not physically realizable as paths. For ...
Cort Ammon's user avatar
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-1 votes
1 answer
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Four-divergence term in Lagrangian

It is known that adding a four-divergence term, $\partial_\mu A^\mu$ does not affect the equations of motion. I am trying to reason this based on the Euler-Lagrange equation. But I want to show this ...
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Equations of motion for Lagrangian of scalar QED [closed]

I really would appreciate your help with this exercise. I have the Lagrangian for scalar electrodynamics given by: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)+(D_\mu\varphi(x))^*(D^\mu\varphi(...
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Variation of the derivative of field in 2d Sigma model

Consider 2d Sigma model, the Lagrangian is: $$ \mathcal{L} = \frac{1}{2}\eta_{\mu\nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\sqrt{-h}h^{\alpha\beta} $$ where $\eta=(-,+,+,\dots)$ is $D$-...
LaplaceSpell's user avatar
3 votes
3 answers
263 views

How to Relate the Functional Derivative to Infinitesimal Change in Noether's Theorem

When the Euler-Lagrange equation or the expression for Noether current are derived the term infinitesimal change is often used. For example, we write $\phi\rightarrow \phi + \delta\phi$ and say that $\...
ICOR's user avatar
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0 answers
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Howe-Tucker to Nambu-Goto Action

Aim to find from the Howe-Tucker action: $$S_{\text{HT}}=-\frac{1}{2}\int d^d\sigma\sqrt{-\gamma}(\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}\eta_{\mu\nu}-m^2(d-2))$$ (which is a Polyakov-like ...
cable's user avatar
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0 answers
69 views

Deriving Klein-Gordon equation from Euler-Lagrangian equation: Taking partial derivative inside [duplicate]

Lagrangian for Klein-Gordon equation is given by $$L=\frac{1}{2}\partial_\mu \phi \partial^\mu\phi - m^2\phi^2/2.$$ To derive Klein-Gordon Equation I have to Compute derivative in Euler-Lagrange ...
Vivek's user avatar
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Units for the Calculus of Variations [duplicate]

Just a quick question regarding the units for a quantity. I just started reading a QFT textbook, and it starts out with a little bit of Calculus of Variations. Specifically, there is a result that ...
Hobson Carion's user avatar
4 votes
2 answers
78 views

Functional variation about metric

Recently I’m learning about lagrangian fomulation in GR. And I’m doing some calculations for some toy models. I precisely understand the variance of christoffel, Riemann tensors about metric, but when ...
Positron3873's user avatar
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0 answers
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Weinberg gravitation variational principle in free falling bodies [duplicate]

In weinberg's gravitation and cosmology in page 77 appears this I can't see why the equation and the symmetry of Christoffel symbols and equation 3.3.5 makes that equation 3.3.10 appears I ask my ...
Alberto Alejandro Blanco Rojas's user avatar
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1 answer
38 views

Identification of the variation on the boundary and why $\delta S_{\partial V}=0$

I recently asked this question about variational principles and how it all works. The essential answer I got was to go read a book on the calculus of variations, which I did, and this helped me make ...
Alex Byard's user avatar
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0 answers
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How to understand variational principles and the math underlying them? [duplicate]

I work in finance, and studied math in college. I'm trying to use QFT statistics to model some aspects of the market. (I've already made some progress by deriving the Black-Karasinski Hamiltonian for ...
Alex Byard's user avatar
2 votes
1 answer
86 views

Evaluating functional derivatives

I am new to evaluating functional derivatives and I am having difficulty evaluating the following derivative: $$I = \frac{\delta}{\delta x(t)}\frac{\delta}{\delta x(t')}\int_{u_i}^{u_f}\frac{du}{2}\...
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1 vote
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Derive the ideal shape for a dome and a cylindrical arch

A catenary is the ideal shape for an arch whose job is to support its own weight. First, would this be true for a surface of revolution about a vertical axis through the lowest point, i.e. would a ...
user121330's user avatar
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1 vote
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The explaination of Einstein-Hilbert action

I've recently been studying about the General relativity and Einstein field equation. When I reading of the derivation of the field equation, I encounterd a method called Einstein-Hilbert action. This ...
PermQi's user avatar
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3 answers
230 views

Derivation of Maxwell's equations using Lagrangian formalism [duplicate]

Some time ago, I read in Landau's Theoretical Physics Course you could derive Maxwell's equations using the Lagrangian formalism, and I find this to be exciting. Unfortunately, I don't have access to ...
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1 answer
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Derivation of Dirac equation in curved spacetime by varying the action

I want to derive the Dirac massless equation in curved spacetime from the action. I have the symmetric form of the Dirac action: $$S = \frac{1}{2} \int \bigg[i\bar{\psi} \gamma^\mu D_\mu \psi - i D_\...
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2 votes
0 answers
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Equation of motion in conformal gravity theory?

In conformal gravity theory, the action is given by $$L=\int \sqrt{-g}C^{abcd} C_{abcd} d^4x=\int \sqrt{-g}(R^{ab}R_{ab}- \frac{1}{3}R^2)d^4 x.$$ However, the variation of the first term $\int \sqrt{-...
user392063's user avatar
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1 answer
77 views

Variation of action under coordinate transformations

I am currently studying General Relativity from M.P. Hobson's "General Relativity: An Introduction for Physicists" and I had difficulty in understanding some concepts in variational field ...
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