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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Obtaining the KG equation from Action

After solving the field equation for $$S = \int \sqrt{-g}dx^4[f(\phi)R + h(\phi)g^{\mu \nu}\nabla_{\mu}\phi\nabla_{\nu}\phi - V(\phi)]$$ I have obtained $$2h\square \phi + \frac{\partial h}{\partial \...
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Variation for the Canonical Scalar Field in $f(\phi)R$

I am trying to find the Field equation for $$S = \int \sqrt{-g}dx^4[f(\phi)R + h(\phi)g^{\mu \nu}\nabla_{\mu}\phi\nabla_{\nu}\phi - V(\phi)$$ but I could not take the variation of $$\delta(\sqrt{-g}h(\...
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Equation of motion in quadratic gravity

I am going through the paper https://arxiv.org/abs/1502.01028 which considers the quadratic gravity with the action \begin{align} S = \int d^4x \sqrt{-g} (R - \alpha C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\...
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Variational derivative of a Lagrangian

I would like to know how exactly to calculate the variational derivative of: $$L = \oint dx \: \frac{m}{a} \dot{u}(x,t) - ca(u')^2$$ with respect to $u$, ($\delta L / \delta u$). Following my lecture ...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action

To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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Derivation of Hamiltonian $H=T+V$ from Lagrangian $L=T-V$

I understand that the Hamiltonian is the Legendre transform of the Lagrangian: $$ \begin{split}H(q,p,t) &= \frac{\partial L}{\partial \dot{q}}\dot{q} - L(q,\dot{q},t) \\ \implies H&=p\dot{q} -...
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For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$ you can see my explanation leading up to it below. I would greatly ...
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Euler-Lagrange equations in GR [closed]

This is the action: $$S = \int d\tau \left[ -e^{2a} \left(\frac{dt}{d\tau}\right)^2 + e^{-2a} \left(\frac{dr}{d\tau}\right)^2 + r^{2} \left(\frac{d\theta}{d\tau}\right)^2 + r^2 \sin(\theta)^2 \left(\...
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Variational operator confusion

Let $L=L(X, \dot X)$ such that the first variation of $L$ is given by $$\delta L=\frac{\partial L}{\partial X}\delta X+\frac{\partial L}{\partial \dot X}\delta \dot X.\tag{1}$$ This is pretty standard ...
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Confusion with the variational operator $\delta$ and finding variations

I have recently started studying String Theory and this notion of variations has come up. Suppose that we have a Lagrangian $L$ such that the action of this Lagrangian is just $$S=\int dt L.$$ The ...
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Geodesic equations with varying mass and the variational principle

Consider the action, $$ S = \int d\lambda\ \phi(x) \left( -g_{\mu\nu}\frac{d x^\mu}{d \lambda}\frac{dx^\nu}{d\lambda} \right)^{1/2}. $$ Using the variation principle we obtain, $$ \delta S = \int d\...
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Notation and Terminology Questions from Schwartz' QFT Book

I am finding some of the notation confusing in Chapter 3 on Classical Field Theory in Schwartz' QFT book a bit confusing. First off, on page 34 he defines a translation of a field to first order as $$...
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Time dependent Schrodinger equation through variation principle - questions about derivation

I'm reading a text which discusses time dependent variation principle (Geometry of the Time-Dependent Variational Principle in Quantum Mechanics by Kramer and Saraceno), and there is some part of a ...
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Four-vector differentiation (E-M Euler-Lagrange eq.)

$$\partial_{\mu} \frac{\partial(\partial_{\alpha}A_{\alpha})^2}{\partial(\partial_{\mu}A_{\nu})} = \partial_{\mu}\left[2(\partial_{\alpha}A_{\alpha})\frac{\partial(\partial_{\beta}A_{\gamma})}{\...
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Inverse of a metric under variation

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as: $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as: $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the ...
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Stress tensor for a real massive vector field in General Relativity

Let's consider the classical Lagrangian density for a real vector field $A_\mu$, $$ \mathcal{L}_v=\sqrt{-g}\left(-\frac{1}{2}A_{\mu;\alpha}A^{\mu; \alpha}-\frac{1}{2} R_{\mu \nu} A^\mu A^\nu+\frac{1}{...
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Variation of the metric determinant

I know this question has been answered here for example but I want to make emphasis in a new aspect. Consider the variation of the metric determinant $\delta g$ with respects to variations of the ...
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If $F^2 = g_{pq} \dot{x^p}\dot{x^q}$ , where $g_{pq}$ is a metric tensor, then find $\frac{\partial F}{\partial{\dot{x^k}}}$

I am trying to find the geodesics in a Riemannian space, using Tensor analysis. I am also using the Principle of Variation. I want to minimize the geodesics integral whose integrand is $F$. Then, ...
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Bosonic closed string effective action

Neil Lambert in his lecture notes https://nms.kcl.ac.uk/neil.lambert/SBQG.pdf in section 3.9 states that imposing conformal invariance at one-loop imposes the following equations on the spacetime ...
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Euler-Bernoulli equation for a periodically supported static beam

The Euler-Bernoulli equation for a homogeneous beam is $$ EI w^{(4)}(x) = q(x),$$ where $w$ is beam height and $q$ is load density. Inspired by the deflection in a multi-support cantilever bridge ...
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Computing functional derivative of exchange-correlation functional

Sakurai and Napolitano's chapter on density functional theory has claims that it is "straightforward" to find $\delta U_{\text{xc}}/\delta n$ for $$U_{\text{xc}}[n]=\int d^3 x n(\mathbf{x})\...
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Calculating the functional derivative of $\partial_\mu\phi$ with respect to $\phi$

Given $F_\mu=\partial_\mu\phi$, I need to find the functional derivative $\frac{\delta F}{\delta \phi}$. I am not familiar with the treatment of functional derivatives outside the context of finding ...
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Proof that a topological term in the Maxwell Lagrangian density does not contribute to the EOM

Given the Lagrange density $\mathscr{L} =c \cdot \epsilon^{\mu\nu\alpha\beta} F_{\mu\nu}F_{\alpha\beta} -A_\mu j^\mu$ where $A_\mu$ is a vector potential, $j^\mu$ a 4 dimensional current and $F^{\mu\...
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D'Alembert Principle and Euler-Lagrange. Virtual displacement

I have a little trouble with d'Alembert Principle and with virtual displacement. Imagine that with the d'Alembert Principle: $$ \sum_i \boldsymbol{\mathrm{F_i}} \; \cdot \delta \boldsymbol{\mathrm{...
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How to calculate the variation of the metric on a compact manifold?

For example, given a torus with a modular parameter $\tau$ and an action \begin{equation} I=\frac{g}{2}\int_\mathcal{M} d^2 z \sqrt{-g}\ g_{ij}(z) \partial^i\phi \partial^j\phi \end{equation} ...
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Second functional derivative and its units

Say I have a functional $I[\phi,g]$ with $\phi(p)$ and $g(p)$ functions from $\mathbb{R} \to \mathbb{R}$. Also say that this functional obeys the property: $$\frac{\delta I}{\delta g(p)} = -(g(p))^{-1}...
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Equations of motion forced by tensors with different Young structure

I can't understand if varying with respect to a field with less irreducible components can force a field with more components to be zero. In my specific situation I work in 4 dimensions and have the ...
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A Lagrangian of sufficient generality

Consider the Lagrangian of the form \begin{equation} \mathcal{L} = \frac{1}{2}g_{\beta\gamma}\dot{q}^\beta\dot{q}^\gamma+A_\beta\dot{q}^\beta-V \end{equation} where $g_{\beta\gamma} = g_{\gamma\beta}$ ...
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The meaning of symmetry in field theories (probably a notational problem)

I'm quite confused about the meaning of "symmetry" in the context of field theories. After reading many posts like 1-, 2-, 3-, 4- and 5-, I'm even overwhelmed. My first approach would be the ...
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Understanding a difference between a functional derivative and discrete case

I can take the following functional derivative $$ C(p)=\frac{\delta}{\delta \phi(p')} \frac{\delta}{\delta \phi(-p')} \int_{-\infty}^{\infty} dp \phi(p)\phi(-p) = 2\delta(0). $$ where I am left with ...
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Assumption of affine parametrization in the geodesic equation derivation

The derivation of geodesic equation is straight from Padmanabhan's book on General relativity. Consider the action $$A = \int d\tau=\int\sqrt{-g_{ab}dx^adx^b}.\tag{4.39}$$ We impose the condition $\...
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Variation of induced metric in Nambu-Goto action

I'm working with the Nambu-Goto action $$ S=-\mu\int d^2\zeta\sqrt{\gamma} $$ with $\gamma$ the determinant of the pull-back metric $$ \gamma_{ab}= \begin{pmatrix} \dot{X}^2 & \dot{X}\cdot X' \\ \...
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Functional derivative for the action $S$

From Lancaster and Blundell's Quantum Field Theory for the Gifted Amateur, p. 15: Example 1.3 The Lagrangian $L$ can be written as a function of both position and velocity. Quite generally, one can ...
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Functional derivative for $J[f]=\int [f(y)]^p \phi(y)dy$

In QFT for gifted amateur pg. 13, the functional derivative for the functional $$J[f]=\int [f(y)]^p \phi(y)dy$$ is given by $$\frac{\delta J[f]}{\delta f(x)}= \lim_{\epsilon\rightarrow0} \frac{1}{\...
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Demonstration of Noether's Theorem [closed]

So, as many, many people before me, I'm trying to get some insight on Noether's Theorem and its demonstration. As I'm in the process of self-teaching here, there are several things I'm "missing&...
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Variation with respect to metric in the Einstein-Cartan formalism

Consider the variation of the EH action with respect to the metric $$\delta S_{EH} = \int d^4x ~(\delta\sqrt{-g} R + \sqrt{-g}\delta g^{\mu\nu} R_{\mu\nu})$$ Now I make use of $$ \delta\sqrt{-g} = \...
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How to determine $n(x)$ when the functional depends exclusively on $n(x)$ and $x$? (Fermat's principle)

Recently I was taught an introduction to calculus of variations in reference to a course on analytical mechanics, where one problem involved Fermat's principle, stating that the path taken by a light ...
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If the Lagrangian included acceleration [duplicate]

If the Lagrangian included acceleration, then how many conditions would be needed to unambiguously define the trajectory, on what variations would the minimum action problem be considered, what would ...
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Functional Calculus in QFT

Does anybody know some good sources with detailed derivations of the main results we need to compute generating functionals in QFT (and functional calculus used in the subject in general). I find that ...
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Search for maximum and minimum functionality

How can I prove that the trajectory I found after applying the principle of least action corresponds to the minimum of action, and not to the maximum?
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Christoffel symbols for diagonal metric with the Lagrangian [closed]

I have been trying to find an explicit formula for the Christoffel symbols of an arbritrary diagonal metric $g_{\mu\nu}$. I do not wish to expand the symbols with the formula which contains the metric ...
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How can the Euler-Lagrangian equation be applied to the Schroedinger Lagrangian of several fields?

In a Lagrangian like the one that follows: $$\mathcal{L} = {i} \Psi^*\dot{\Psi} - \frac{1}{2m} \nabla{\Psi} ^* \nabla \Psi.\tag{1}$$ How can I apply the Euler-Lagrangian equation, shown in $(2)$, to ...
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Action in Lagrangian Mechanics [duplicate]

I editted this question since it was closed because it is a duplicate. However, answers in the referenced question didn't solve my question, so I am writing it again. Lagrangian mechanics is built ...
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Metric variation of a four-vector

I am trying to calculate the metric variation $\frac{\delta T^{\alpha\beta}}{\delta g^{\mu\nu}}$. Here, $T^{\alpha\beta}$ is the stress-energy tensor of a perfect fluid, given by $T^{\alpha\beta} = (\...
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Deriving the equation of motion of an action involving colour structure

I would like to know the explicit equation of motion (up to quartic order) from the following action \begin{equation} S=-\frac{1}{2} \int \mathrm{d}^{d} x \partial_{\mu} \phi_{a} \partial^{\mu} \phi^{...
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Mass Distribution to turn Hanging Chain into Parabola

I've learned recently about how a uniform chain hanging between two points will form a catenary curve (of the form $a \cdot \cosh (\frac{x}{a})$), and I reflected on the fact that this is only because ...
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Deriving the conformal current of the Polyakov action

In string theory, the Polyakov string action is given by \begin{align} S = -\frac{T}{2}\int d^2\sigma\:\sqrt{-\gamma}\gamma^{ab}\eta_{\mu\nu}\frac{\partial X^{\mu}}{\partial \sigma^a}\frac{\partial X^{...
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Physical interpretation of a hypermomentum tensor

In Einstein-Cartan gravity, the metric and connection are independent objects. Therefore, a matter action coupling to gravity may imply the existence of $$ \frac{1}{\sqrt{g}}\frac{\delta S_{matter}}{\...
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Lagrangian depending on running integral

In the theory of calculus of variations, the Lagrangian generally depends on the unknown function and its first derivative. This assumption leads to the Euler-Lagrange equations. However, I found that ...
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Mathematical background required for Lagrangian Field Theory? [closed]

I want to start teaching myself Lagrangian Field Theory. I can do multivariable calc, tensor calc, Lagrangian mechanics, and some calculus of variations. Are there other math fields I should study ...
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