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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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How to take derivative with respect to Lagrangian of complex field?

Basics: The Lagrangian in field theory was written as $$\frac{\partial \mathfrak{L}}{\partial \varphi}=\partial_\mu(\frac{\partial\mathfrak{L}}{\partial(\partial_\mu\varphi)})$$. Question 1: Is $\...
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Deriving the equations of motion of a real vector field [duplicate]

I'm trying to calculate the euler-lagrange equations of the following Lagrangian density. $$\mathcal{L} = -\frac{1}{2}(\partial_\alpha A_\beta)(\partial^\alpha A^\beta)+\frac{1}{2}(\partial_\alpha A^\...
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How to derive the Einstein-Skyrme equations?

I would like to derive the Einstein-Skyrme equations. The action can be read as \begin{equation} S[g,U] = \int d^{4}x\sqrt{-g}\biggl[R + \frac{K}{4}Tr\bigg(A^{\mu}A_{\mu} + \frac{\lambda}{8}F_{\mu\...
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1answer
93 views

A Potential Euler-Lagrange Equation Alternate Derivation?

Can the Euler Lagrange Equation be derived with this overall strategy? Step 1 – Define a geodesic in flat space to be $\frac{d}{dy} \frac{ds}{dx} = \frac{d}{dx} \frac{ds}{dy}$, where $ds$ represents ...
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Field equation of motion for this Lagrangian

In my first QFT exam I was supposed to derive the equations of motion for all fields for this Lagrangian: $$\mathcal{L} = \bar{\Psi}(i\gamma^\mu\partial_\mu-M)\Psi+g\bar{\Psi}\gamma^\mu\Psi\bar{\Psi}\...
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Deriving the Maxwell Scalar field Equation from the given action principle

I am currently starting to learn general relativity. I have been trying to derive the Maxwell Scalar field equation from the given action $\mathcal{S}[\phi, A, \mathbf{g}]=\int\left(\frac{1}{2}\left(\...
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1answer
40 views

Energy-momentum Tensor for a Real Scalar Field Lagrangian

I'm currently working through Schwartz's QFT book, and I'm trying to find the energy-momentum tensor for the following Lagrangian: $$ L = -\frac{1}2\phi(\Box+m^2)\phi. $$ Am I correct in thinking ...
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133 views

Lagrangian non-relativistic limit to the non-relativistic action: lagrangian of a free particle

Let be $u=|\bar{u}|$ the speed of a free particle (at constant speed) of mass $m$ that is moving in relation to an inertial frame. Why we initially introduce the term $\epsilon$ to the free lagrangian ...
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How to go from $\delta(\dot{\Psi})$ to $\delta\Psi$ in variational calculations? [duplicate]

This must have been done somewhere before but I never saw a clear and rigorous explanation of why. This is possibly related to my recent post here about potential abuse of notation. Just to be ...
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Functional derivative and variation of action $S$ vs Lagrangian $L$ vs Lagrangian density $\mathcal{L}$ vs Lagrangian 4-form $\mathbf{L}$

I have seen many potential abuse of notation that prevents me from clearly understanding variational methods in QFT and GR that I want to get this settled once and for all. This may be a bit long but ...
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Variation of the metric w.r.t. the metric in derivation of stress tensor

Consider massless free scalar theory $$S = \int d^4x \sqrt{-g}L = \int d^4x \sqrt{-g} \;g^{cd}\nabla_c\phi \nabla_d \phi.\tag{1}$$ To compute the Hilbert stress-energy tensor we require $\sim\delta S/\...
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Proof of equivalence of variational principle and Euler-Lagrange equations on a manifold [closed]

Let M be some manifold, and TM the tangent bundle. Let $\gamma : [a,b] \to M$ be a smooth curve on M defined on an interval on $\mathbb{R}$. Let $J$ be another interval in $\mathbb{R}$ containing 0. A ...
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Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem

I'm really confused about total derivatives and partial derivatives. My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
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$\delta S=0$ only for $\frac{\partial\mathcal{L}}{\partial\phi}-\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}=0$?

Condition for the variation of action is: $$0=\delta S$$ $$=\int d^4 x [\frac{\partial \mathcal{L}}{\partial \phi}\delta\phi-\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu \phi)})\...
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1answer
98 views

Varying covariant derivatives

If we take a variation of a covariant derivative, we must take into account the connections, so we get: $$ \delta (\nabla_\beta T_{\mu \nu}) = \nabla_\beta \delta(T_{\mu \nu}) -\delta (\Gamma_{\beta ...
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Variational principle if coordinate transformation depends on fields

Assume we have a Lagrangian that is given in terms of Lagrangian density. $$ L = \int \mathcal{L} (\Phi, \partial_{\mu}\Phi, x) d^N x $$ Also assume that $\Phi : \mathbb{R}^N \to \mathbb{R}^N$ and ...
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1answer
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Why does the integral symbol disappear when applying a functional derivative?

it is known that variation is defined by following: but could anyone tell me why the integral symbol disappears after following functional derivative?
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MCS Lagrangian and Euler-Lagrange

I'm trying to solve the Euler-Lagrange equation for the MCS Lagrangian density as given by Kharzeev in this article (Eqn. 7): $$ \mathcal{L}_{\textrm{MCS}} = -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-A_\mu J^{...
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How to transform a lagrangian after a change of coordinates?

Let's consider a generic lagrangian density in classical field theory: $$L(\phi(x),\partial_{\mu} \phi(x))$$ Now suppose I want to find the lagrangian for the same system with respect to another field ...
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1answer
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Srednicki chapter 22: continuous symmetries and conserved current

In Srednicki's book he says that: The Noether current plays a special role if we can find a set of infinitesimal field transformations that leaves the lagrangian unchanged, or invariant. In this ...
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Soap film bounded by a circular wire

If a soap film is bounded by two parallel circular wires, then the equation of its surface of the minimal area is $$ r=c\cosh(z/c) $$ where $c$ is a constant and $r=\sqrt{x^2+y^2}$. This can be ...
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Path integral and least action principle

I'm reading Sakurai's book. And there is a part, where it says: let's consider the path that satisfies $$\delta S(N,1) = 0,$$ where the change in $S$ is due to a slight deformation of the path with ...
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“Chain Rule” for functional derivatives in the context of a derivation of the geodesic equation by the stationary proper-time principle

I have been working on deriving the geodesic action via finding the stationary points of the proper-time integral for a massive point particle. Consider a space-time manifold $M$ ($\dim M=4)$ equipped ...
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Is metric compatibility an on-shell statement?

In General Relativity (GR), it is standard to endow the spacetime manifold with a metric compatible connection, i.e. $\nabla_{\alpha}g_{\mu\nu}=0$. My question is: is metric compatibility of the ...
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Solving the Euler-Lagrange equation with the Axion Lagrangian

I am trying to show that for a constant axion field $\theta(\textbf{x},t)=const.$ the axion Lagrangian $\mathcal{L}_\theta=-\frac{\kappa\theta}{4\mu_0}F_{\mu\nu}\tilde{F}^{\mu\nu}$ does not lead to a ...
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Independents fields and the Lagrange Density of Schrodinger equation [duplicate]

I have a doubt about the lagrangian of the Schrodinger equation. If we consider the wave function $\psi(\textbf{x},t)$ that satisfy the Schrodinger equation as a field, one way of construct the ...
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Tensor Notation - David Tongs Notes [duplicate]

I'm trying to understand the Maxwell's Equation example from David Tongs QFT notes. He uses the Lagrangian: $$ L = -\frac{1}{2}(\partial_{\mu}A_{\nu})(\partial^{\mu}A^{\nu})+\frac{1}{2}(\partial_{\mu}...
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Why are these two variables being treated differently in the action?

I'm trying to understand the derivation provided in the section 2.4 of this paper. I have modified the notation and cut out the unimportant parts of the equations for clarity purposes, but for ...
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1answer
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Inconsistency in variation of the metric tensor in an action

While doing some exercises on the variation of the metric tensor $g_{\mu\nu}$ and of its inverse $g^{\mu\nu}$, I came across the following identity: $$\begin{align} & \delta(g_{\mu\nu}g^{\mu\nu})=...
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How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?

The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
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Doubts in an introduction to classical field theory

I started to study classical field theory using the book "Field Quantization" of Greiner and Reinhardt, and I have some doubts. First, the book write the Lagrangian $L(t)$ as a functional of a field $\...
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Variations of tensors are tensors?

Recently I posted a question about variation of metric. I thought I understood it and talked with my friend about it. After that he said he's not convinced because he can't prove variation of metric ...
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1answer
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How can dissipative/friction terms be incorporated into a Lagrangian?

I'm trying to find a suitable Lagrangian for a damped harmonic oscillator, a system that satisfies the following equation of motion: $$m \ddot{x} + \gamma \dot{x} + \frac{d\phi}{dx} = 0.$$ What I ...
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3answers
96 views

Functional derivative of metric

To do functional derivative of some actions, we need to know a functional differential of metrics $g_{\mu \nu}(x)$. One of the formulae about that is: $$g_{\mu\nu}\delta g^{\mu\nu} = - g^{\mu\nu} \...
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2answers
85 views

Action principle and Functional derivative in CM

I want to extremize this well known action. $$S[\phi]=\int \mathcal{L}(\phi(t),\dot{\phi}(t)) dt $$ The result is also well known. It turns out to be E-L equation. The Action principle states that the ...
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1answer
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Varying the Dirac action with differential forms

The Dirac action in a curved spacetime can be written in terms of the vierbein $\{ e^a \}$ and spin connection $\{ \omega^{ab} \}$ differential forms. Let the spinor field $\psi$ be interpreted as a ...
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2answers
112 views

Derivation of Hamilton-Jacobi equation

I am trying my own way of deriving the Hamilton Jacobi equation $$\frac{\partial S}{\partial t} = -H \tag{1}$$ through direct variation. I think the difficulty of doing this is that the upper limit ...
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2answers
63 views

Taylor expansion in derivation of Noether-theorem

In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by: $$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
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2answers
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Why the zero-order term in a variational transformation of coordinates should be identically the same as the old coordinates?

In the Ref.[1, page 61] the author proposes that transformations between two coordinate systems can be described by a continuous parameter $\varepsilon$ such that when $\varepsilon=0$ the original ...
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30 views

Help with an specific example of a higher derivative Lagrangian

I want to find the equation of motion that comes from the following Lagrangian density $$\mathscr{L}=\mathbf{E}\cdot\left(\nabla^{2}\mathbf{E}\right)$$ where $E_{i}=\partial_{i}\phi\;(i=x,y,z)$ . In ...
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Variation under infinitesimal reparametrization

Say that under the infinitesimal reparametrization $\sigma \rightarrow \sigma' = \sigma-\xi(\sigma)$, $x^\mu$ transforms as a scalar, i.e. $x'^\mu(\sigma')=x^\mu(\sigma)$. I would like to show the ...
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1answer
68 views

Integration by parts, Weinberg Cosmology p.526 [closed]

How do I perform this integration by parts done explicitly? $$0 = \delta I_m = \int d^4 \sqrt{-g} T^{\mu \nu} \left[- \frac{\partial \epsilon^\rho}{\partial x^\mu} g_{\nu \rho} - \frac{\partial \...
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1answer
52 views

Derivation of gradient of the expectation of local energy

Background: In Variational Monte Carlo, given a Hamiltonian $H$ and a wave function $\psi_\alpha$ dependent on some parameter(s) $\alpha$, we have defined a quantity known as the local energy, $$E_L =...
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2answers
66 views

Variation in Hamiltonian mechanics

I have a question about a property of variational calculus used in following bachelor thesis: http://users.physik.fu-berlin.de/~pelster/Bachelor/fraessdorf.pdf Here the excerpt: Why it is possible ...
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1answer
53 views

Variation in field theory with respect to one quantity

In my QFT course we are supposed to vary the action of a for a scalar field coupled to an electromagnetic field with the following Lagrangian density: $$\mathcal{L} = [D_\mu\phi(x)]^*D^\mu\phi(x)-m^2\...
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1answer
48 views

Free boson Equation motion from action

So in David tongs notes we have $$S=\frac{m}{8\pi}\int d^2x\partial_i\varphi\partial^i\varphi$$ and he finds that the equation of motion is $$[\partial_{t}^2-v^2\partial_{x}^2]\varphi=0$$ now my ...
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2answers
179 views

$\int (f(x+\delta x) - f(x)) dx = \int \left ( \frac{df(x)}{dx} \delta x \right) dx$

From Landau and Lifshitz's Mechanics Vol: 1 $$ \delta S= \int \limits_{t_1}^{t_2} L(q + \delta q, \dot q + \delta \dot q, t)dt - \int \limits_{t_1}^{t_2} L(q, \dot q, t)dt \tag{2.3b}$$ $$\Rightarrow ...
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1answer
100 views

Confusing with the equation $(2.4)$ and $(2.5)$ of Landau and Lifshitz, Mechanics, Chapter 1, The principle of Least Action

I'm a 12th Grader and I'm interested in Lagrangian Mechanics and having a bit of knowledge about the Newtonian Mechanics. So, I found a book of Landau and Lifshitz's Mechanics and started reading from ...
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3answers
116 views

Calculus of Variations commutes with Integrals

I have a question about the variational calculus. Assume a function $q(t,x)$ gives rise for another function $$f(x) := \int dt q(t,x)$$ My question is why the variation $\delta$ commutes with the ...
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14 views

Invariance with respect to time dilations of the free particle

Consider the action of a free particle in the space $$ s=\int_{t_1}^{t_2} \frac{m v^2}{2} d t.\tag{*} $$ The change of time coordinates $t'=\alpha t$, where $\alpha\in(0,1]$, preserves the form of the ...