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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 ...

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In what sense does variational calculus help us graph the path of a moving object represented by the stationary action formula? [on hold]

This is off the beaten path if you allow me to build a short history of the question. Let us assume we have the function in hand we know that function will produce a stationary action so I assume if ...
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Gravity train solution through symmetries

I've been having trouble feeling okay with all of the solutions I've found to the Brachistochrone problem inside earth thus far. To me the way to do it is to show: The time that it takes a path $\...
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2answers
108 views

In what sense is the stress-energy tensor the derivative with respect to the metric?

In Di Francesco et al (the big yellow book), section 2.5.2, it is suggested that the (symmetrized) stress energy tensor can be interpreted as the functional derivative of the action with respect to he ...
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1answer
81 views

Functional derivative

I am not able to derive Eq. 21 of this paper F. Zahariev, S. S. Leang, and Mark S. Gordon, "Functional derivatives of meta-generalized gradient approximation (meta-GGA) type exchange-correlation ...
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30 views

Derivation of Lorentz law in Curved Spacetime?

In the presence of external forces or rather presence of fields to which a particle "couples" in curved spacetime. Ex: $$S[\gamma;A]=\int d\lambda m\sqrt{g_{\gamma(\lambda)}(v_{\gamma,\gamma(\lambda)}...
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32 views

Palatini action: variation of spin connection: show that torsion vanishes

Consider the tetrad-Palatini action: $$S[e,\omega] = \int e \wedge e \wedge F[\omega]^\star,$$ where $\star$ denotes the Hodge dual, i.e. $F_{IJ}^\star = \frac{1}{2} \varepsilon_{IJKL} F^{KL}$. The ...
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Why is the shape of a hanging chain not a “V”?

From Wikipedia: To answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy. ...
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1answer
27 views

How to take into account the symmetry of the metric tensor when doing the Functional derivative in GR? [duplicate]

I have a Straightforward question. When the functional derivative of the Ricci scalar to get the GR field equations. As the derivative is done using the metric which is symmetric do I have to ...
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2answers
50 views

General relativity - scalar gravitational field, variation principle

I have a basic question about the variation principal when applied to a scalar gravitational field in general relativity. Consider the action $$S_M = \int d^4 x\sqrt{|g|}g^{uv}\partial_u \phi\...
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1answer
56 views

Does the integral in the action formula regarding the principle of stationary action represent an area or a length?

I am referring to the Feynman Lectures. The second volume has the "Principle of Least Action" as one of his lectures. (See after the 2nd paragraph below figure 19-6.) Although he does not explicitly ...
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1answer
44 views

Why isn't it important, after which coordinates the Variation of the action integral is done?

I often read,that if the lagrangian $L=p\dot{q}-H$ of a pair of coordinates in phase space $(q,p)$ and $P\dot{Q}- K $, for some new pair of coordinates $(Q,P)$ only differ by a total time derivative $...
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34 views

Variational calculus and KKR method for band structure calculation

I’ve been studying the KKR method from the original Kohn and Kostoker’s paper (https://journals.aps.org/pr/abstract/10.1103/PhysRev.94.1111). On the text, they use variational calculus for dealing ...
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1answer
58 views

Notation question in calculus of variations — QFT

these two integrals below are equal, but I am not understanding where the $x'$ variable comes from. \begin{align} I_0&=e^{ i\int d^4x \left\{ \frac{1}{2}\left[ \left( \partial\varphi(x) \right)^...
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1answer
161 views

Equation of motion from $D=3$ Lorentz Chern-Simons action

In three dimensions, the well known Lorentz Chern-Simons action is $$ S_{\text{CS}}=\int\text{d}^3x\varepsilon^{\mu\nu\rho}\bigg(\omega_{\mu}{}^{ab}R_{\nu\rho ab}+\frac{2}{3}\omega_{\mu a}{}^{b}\...
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36 views

Reduction of the mechanical equilibrium condition given by Gibbs

In the "The scientific papers of J.Willard Gibbs: Volume One Thermodynamics" p.276-281, the following mechanical equilibrium condition is given for a heterogeneous liquid system with gravity ...
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2answers
70 views

Derivative of the electromagnetic tensor invariant $F_{\mu\nu}F^{\mu\nu}$

The electromagnetic field tensor is $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$. I am trying to calculate the quantity $$ \frac{\partial(F_{\alpha\beta}F^{\alpha\beta})}{\partial(\partial_{\...
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1answer
271 views

How to find the Lagrangian of this system?

I am trying to find the Lagrangian $L$ of a system I am studying. The equations of motion is: $$\left\{ \begin{array}{c l} r \ddot{\phi} + 2\dot{r} \dot{\phi}+k(r) \cdot r \dot{r} \dot{\phi} = ...
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3answers
119 views

Lagrangian gauge invariance $L'=L+\frac{df(q,t)}{dt}$

So, I have to prove directly (e.g. by substitution) that if a path satisfies the Euler-Lagrange equations for the Lagrangian $L$ it does so for $$L'=L+\frac{df(q,t)}{dt}.$$ Let me tell you what I have ...
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27 views

Is there a formal proof that the equilibrium figure of fluid under its own gravity is a sphere? [duplicate]

Every textbook that presents the hydrostatic equilibrium condition of stars starts by assuming a spherical distribution of matter. This all makes sense intuitively, yet I haven't been able to find a ...
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2answers
53 views

Why can we consider the endpoint fixed in the derivation of the Euler-Lagrange equation in mechanics?

In mechanics, we obtain the equations of motion (Euler-Lagrange equations) via Hamilton's principle by considering stationary points of the action $$ S = \int_{t_i}^{t_f} L ~ dt $$ where we have $L=T-...
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1answer
83 views

Energy-momentum tensor from Matter Field Action

I'm currently in the process of familiarising myself with some basic concepts of general relativity and have stumbled upon a problem that is probanly quite simple. I'm referring to the book by Hobson, ...
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1answer
124 views

Equation of Motion for non-linear sigma model (WZW)

I am struggling with deriving the equations of motion for the non-linear sigma model that becomes the WZW model later on, in the CFT book by Francesco et. al. The relevant snippet is below. The ...
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1answer
203 views

Euler-Lagrange Equation Proving Maxwell Equation [duplicate]

When quantizing the EM Field, we get the Lagrangian density, $$L=\frac{1}{2}\left(\epsilon \vert E\vert ^2 - \frac{1}{\mu}\vert B\vert^2\right) = \frac{\epsilon}{2}\vert\nabla\phi + \dot{\textbf{A}}\...
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1answer
64 views

Decomposition of variation of metric derivatives (and why is Gibbons Hawking York term not identically zero?)

So, I've been studying the Gibbons Hawking York (GHY) boundary term, which involves variations of the derivatives of the metric, $\delta(\partial_\lambda g_{\mu\nu})$. But, since the metric and its ...
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38 views

Variation of electromagnetic part of action

I've got the same problem as Gabriel Luz Almeida had out here: Variation of Maxwell action with respect to the vierbein - Einstein-Cartan Theory I try to vary the electromagnetic part of action i.e. $...
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51 views

State of minimum uncertainty of a particle in an infinite square well

Let $\sigma_x$ and $\sigma_p$ be the standard deviation of position and momentum of a particle. The ordinary uncertainty relation tells us that in general we have $$\sigma_x\,\sigma_p\geq\hbar/2.$$ ...
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23 views

Finding well-posed variational form for contour shape optimization

In a given physical problem near resonance, it is required that two energy terms, $\iint_{A} [f(x,y)]^{2}dxdy$ and $\iint_{A}[g(x,y)]^{2}dxdy$, in the system under consideration be almost equal. In ...
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1answer
34 views

Variations with respect to objects with symmetries and indices in different positions

I was studying variational methods in theoretical physics and I got stuck with a few simple questions. I have possible answers but I cannot see clearly and rigorously if they are correct. Suppose we ...
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40 views

Light Ray Trajectory through Periodic Refractive Index

Consider a ray of light travelling between two points A and B on the $xy$ plane. Using the calculus of variations and Fermat's Principle we can derive equations which give the trajectory of a ray of ...
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1answer
93 views

Lagragian density of a massless scalar field

I have seen in some books that the simplest Lagrangian density of a massless scalar field is $$\mathscr{L}=\dfrac{1}{2}\partial^\mu\phi\partial_\mu\phi=\dfrac{1}{2}\left(\partial_\mu\phi\right)^2.$$ ...
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1answer
119 views

Deriving Supergravity Equations of Motion

The other day I decided to quickly make sure I can derive the supergravity equations of motion for the NS/NS sector using the following action: $$ S=\int_{M_{10}}d^{10}x \space \sqrt{-g} e^{-2\phi}\...
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1answer
39 views

Box form of the kinetic term and Euler-Lagrange equation

When consulting Schwartz's QFT and Standard Model, I observe that he writes the kinetic term for field species in a form somewhat a little bit different from the ones appearing in other literature. ...
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1answer
73 views

Proof of Thomson's theorem on electrostatics using variational calculus

I'm following a proof of Thomson's theorem but I'm a bit confused when they use a lagrange multiplier to include the charge conservation constraint, could someone explain to me how is this multiplier ...
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38 views

Functional derivative of a symmetrized field

I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
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1answer
72 views

Calculus of Variations: Refractive Index Problem [closed]

The problem is as follows: "Given that the refractive index $µ(r)$ of some material equals $|∇f|$ for some function $f(r)$, show that the optical path length $\int_A^B \mu(r) dl$ between points A ...
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1answer
228 views

Variation of scalar field action

I am reading Polchinski's review on AdS/CFT . I have a very simple question, and please help me out. Thanks in advanced. The question abou formula (3.19) The scalar effective bulk action is given by ...
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Proving the most efficient profile for gear teeth is an epicycloid

I'm reading 'Revolution in time' by David S. Landes. It says that the most efficient (i.e. with least friction) profile for meshing gear teeth (for the old watches which used balance spring and wheel ...
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1answer
107 views

2D Liouville Stress-Energy tensor

I am working on 2D Liouville field theory and trying to follow mostly Harold Erbin's note on 2d quantum gravity and Liouville theory. I have a really simple question: One consider the Euclidean ...
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2answers
100 views

Finding equation of motion of Lagrangian density: What does the location of the indices mean?

We are given the following Lagrangian density: $$\mathcal{L}=F_{\mu \nu} A^{\mu} \mathcal{J}^{\nu}$$ where $F_{\mu \nu}$ is the electromagnetic field tensor, $ A^{\mu}$ the 4-vector of the vector ...
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67 views

Lagrangian density for General Relativity

Walter Thirring in his book proposes lagrangian density $$\mathcal L = -\frac{1}{2} J \wedge \star J -\frac{1}{2}F \wedge \star F+\frac{1}{16 \pi G}R_{\alpha \beta}\wedge \star e^{\alpha \beta},$$ ...
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1answer
51 views

Lagrangian of a single scalar field

Hello all, I've been trying to understand how Sean Carroll is able to come to the conclusion that he does in $1.153$. I tried replacing my dummy index $\sigma$ with $\nu$ based on the substitution ...
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0answers
63 views

Derive a Lagrangian containing an integral

I am studying a packing of spheroidal particles, contained in a space $\mathcal{B}$ of volume $V$, with number density $\rho_0$, and interacting through a potential $\mathcal{V}_{ij}(\vec{r}_i,\vec{r}...
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1answer
78 views

How to model the path of a particle using geodesics?

While I was studying the Euler-Lagrange equation, Hamilton's principle of least action and geodesics, I started to wonder how to find the equations of motion of a particle restricted to a particular ...
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2answers
93 views

Functional derivatives of inverse tensor field

The short-hand notation here is $1 = x_1 , 2 = x_2 ,... $and $\int_{1}=\int{dx_1},\int_{2}=\int{dx_2}.... $ In appendix A of this paper https://arxiv.org/abs/hep-th/9908172 it is said that the basic ...
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1answer
40 views

Finding a path of beam in a gradient-index media

I'm trying to find a polynomial that describes a path of a beam in a gradient index media. It is a path of least time (Fermat's principle) meaning that it takes path that takes the least time to get ...
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37 views

Proving gravity deforms mass into spheres [duplicate]

It is my understanding that reason why planets are roughly spherical is because gravity is a center directed force that scales by the inverse square. So matter ends up being uniformly packed together ...
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1answer
76 views

Second variation of a functional

I am trying to find the second variation of the Hartree energy functional $E_{H} [\rho]$: $$ \dfrac {\delta^2 E_{H}}{\delta \rho (r)\delta \rho (r')}=\dfrac {\delta^2}{\delta \rho (r)\delta \rho (r')}\...
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25 views

Spatially uniform VEV for gauged Ginzburg-Landau functional

The Ginzburg-Landau functional reads: $$F = \int dV \left \{\alpha |\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*} \mid (\frac{\hbar}{i}\nabla - \frac{e^*}{c}A)\psi \mid^2 + \frac{h^2}{8\pi}\...
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37 views

Solving $\frac{d^2 \theta}{d x^2} - m^2\theta = 0$ using the Variational Ritz method

This is my first question. I don't have much background on physics, and just found out about the knowledge this community seems to have about the variational method and, more specifically, the Ritz ...
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2answers
116 views

What is the “surface term”?

In Peskin's quantum field theory book, There is a sentence in page 17: ... More generally, we can allow the action to change by a surface term, since the presence of such a term would not ...