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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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Variation of the Palatini action with respect to the triad and spin conection

I am trying to understand how could we get these two equations of motion by varying the action with respect to the triad and spin connection. Where F is the curvature in terms of the spin connection ...
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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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Integration by parts, Weinberg Cosmology p.526 [on hold]

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

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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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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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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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 ...
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Functional derivative commutes with total derivative

I have a question about a rule from the calculus of variations. Assume we consider the space of differentiable functions on $C^1(\mathbb{R})$ (or for the sake of simplicity the smooth functions $C^{\...
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Necessity and sufficiency of Euler-Lagrange equations in making an integral stationary

Suppose we want to make an integral $S$ of the form $$S = \int_{x_1}^{x_2} f\left[y_1(x), \dots, y_n(x), y'_1(x), \dots, y'_n(x), x\right]dx$$ stationary with the constraint $y_1\left(x_1\right) = \...
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Einstein Field Equation for a Garfinkle-Horowitz-Strominger theory

When deriving the Einstein field equation from a theory with a dilaton field, we have: GHS's calculation However, in my calculation: My calculation So I found that GHS has missed $R$ and $(▽φ)^2$. ...
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Questions about Euler-Lagrange derivation in Classical Field Theory

I'm new to classical field theory, so I have a few basic questions: From the derivation of the Euler-Lagrange equations, we have the following: \begin{align} \delta S[\phi]&=\int d^4x\delta L(...
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Different definitions of Functional Derivative

In studying QFT and General Relativity, I came across two different definitions of Functional Derivative, and I'd like to know if they are equivalent. Firstly, in Wald's book General Relativity, as ...
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Time-independent Schrödinger equation Lagrangian derivation

Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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Applying the Euler-Lagrange equations to Maxwell's Theory

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\...
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Chern-Simons equation of motion

How do I get the equation of motion of the Chern-Simons Lagrangian below? Is there the product rule at work? Do I have to sum over the indices?
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Euler-Lagrange equations from a complex Lagrangian

I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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How do you actually find the path of least action?

Reading up on Lagrangian mechanics, it's fascinating. Entirely different view, one single rule, a complete alternative to Newton's laws. But how do you actually find the path of least action? Let's ...
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Proving that the Euler-Lagrange Equation has no solution [closed]

I'm trying to show that the Euler - Lagrange equation for the functional $$I(y)=\int_{a}^{b} y\:dx$$ subject to $y(0)=y(1)=0$ has no solutions. The Euler - Lagrange equation states that: $$\frac{d}{...
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Equations of motion from action variation

I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action: $$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
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What's the variation of the Christoffel symbols with respect to the metric?

By the Leibniz rule, I expected it to be $$\delta \Gamma^\sigma_{\mu\nu} = \frac 12 (\delta g)^{\sigma\lambda}(g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu}-g_{\mu\nu,\lambda}) + \frac 12 g^{\sigma\lambda}(\...
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Attaining extrema when a stationarity condition has no solution

I was wondering if someone could shed some light on the following for me: If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
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Understanding the variational / functional derivative (in relation to the Euler-Poincaré equation)

I'm trying to understand the Euler-Poincare equations, which reduce the Euler-Lagrange equations for certain Lagrangians on a Lie group. I'm reading Darryl Holm's "Geometric mechanics and symmetry", ...
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Variational Navier-Stokes: where to find study material “for dummies”?

I have worked with the Navier Stokes equations before but I'm a physicist. I was talking to a mathematician and they use a complete different notation and I am very lost. First of all, I use the ...
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Poincare transformations and “three kinds of infinitesimal variations”

I'm currently reading these$^1$ lec. notes as an introduction to relativistic QFT. In chapter two (pp.57-61) he introduces the concept of field variations along with some formulas for the different ...
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A Naive Question about SUSY Variation

I am following BUSSTEPP Lectures on Supersymmetry to learn supersymmetry. My simple question is the following. My Lagrangian for the Wess-Zumino model in $4D$ is $$\mathcal{L}=-\frac{1}{2}(\...
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What is the physical meaning of the functional π = U-W used in the variational approach to deriving the finite element equations of a system?

In the variational formulation of the finite element equilibrium equations for a system, the functional π = U-W is used, for which we find stationary points in order to derive the governing equation ...
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Time component of momentum four-vector

In Landau-Lifshitz, Classical theory of fields (second chapter), the four-momentum is defined by the equation $$-\frac{\partial S}{\partial x^i}=p_i\tag{9.12},$$ where $S$ is the action integral. The ...
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EoM for scalar field in Brans-Dicke Theory

The action is given by $$ S^{(BD)} = \int d^4 x \sqrt{|g|} \left[ \phi R - \frac{\omega}{\phi} g^{\mu \nu} \, \nabla_\mu \phi \nabla_\nu \phi - V(\phi) \right]$$ I am trying to vary with respect to ...
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Variation of a integration involving derivatives

I'm having problem with calculating the functional derivative of $F$ with respect to $\phi(x)$ while $$F = \int d^{4}x \phi^2 \partial_{\mu}\phi\partial^{\mu}\phi.$$ I want to obtain $\frac{\delta F}...
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Varying a scalar field Lagrangian density

I was varying a scalar field density and I look at this term $${\cal L}~=~-\frac{1}{2}\partial _\mu\phi\partial^\mu\phi.$$ The result that I need to come is $$-\frac{1}{2}\delta(\partial _\mu\phi\...
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Derivatives with Two Indices in Electromagnetic Lagrangian [duplicate]

I was reading about the derivation of Maxwell's equations from an electromagnetic Lagrangian density from Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity. The Lagrangian ...
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Vector calculus in classical fields

The action is defined as: $$S = \int d^2\textbf{x}\,dt \left[\left(\frac{\partial h}{\partial t}\right)^2 + (\nu \,\nabla^2h)^2\right]$$ The equation of motion is asked for, so use Euler-Lagrange: $$\...
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Variation of action for massive point particle (pp)

So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of: $$S_{...
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Commutator generating transformations

Lately I am encountering the commutator of variations of the variables and I'm not quite sure about its physical meaning. Some examples. 1) "The composition of two supersymmetries generates a time ...
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Calculus of Variations help [closed]

I've been studying Chapter 6 in Taylor's Mechanics book. And am working through the odd-numbered problems. I am struggling with 6.13, which reads: In relativity theory, velocities can be ...
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Derivation of EL equations for real scalar field

I am looking to derive Eq. (1.11) from these notes on QFT from Tong’s notes: http://www.damtp.cam.ac.uk/user/tong/qft/one.pdf But Im the second equation, is there not suppose to be a factor of 1/2 ...
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68 views

Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?

His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
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Periodic motion(s) on a torus

I recently came across the Lyusternik-Fet theorem concerning closed geodesics on a compact manifold. For simplicity of description, take the 2-torus, and imagine it represents the configuration space ...
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Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
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Is it possible for the Action $S$ to *not* have a stationary point?

So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary: $$\delta S = \...
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Equation of motion for a massless scalar in 2 dimensions

I'm working through Polchinski's String Theory (Volume 1, Chapter 2, page 34), with $D$ scalar fields: The action is given by $$ S = \frac{1}{2 \pi \alpha'}\int d^{2}z \partial X^{\mu} \overline{\...
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Varying the Einstein-Hilbert action without reference to a chart

In most treatments of General Relativity, when the the Einstein-Hilbert action over some manifold $\mathcal{M}$ (plus Gibbons-Hawking-York term if $\mathcal{M}$ has a boundary), given by $$S=\frac{1}{...
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How to deal with explicit time dependence of the Lagrangian?

Clearly, if the Lagrangian in explicitly time dependent, the Euler-Lagrange equations being satisfied does not extremise the action. I am unclear as to how to deal with systems with an explicitly time-...
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Why can we choose affine parameterization?

In general relativity when deriving the geodesic equation $$\ddot{x}^\mu + \Gamma^\mu_{\alpha\beta}\dot{x}^\alpha\dot{x}^\beta = 0\tag{1}$$ from the action $$S = \int d\tau \sqrt{|g_{\mu\nu} \dot{x}...
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Equations of motion for a Weyl spinor in the context of SUSY

I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114). Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \...
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Polyakov Lagrangian and Lagrange multipliers

I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...