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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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Lagrangian for Electromagnetic Fields

As an experimental physicist with little knowlegde on (but having interest in) field theory, I am trying to recalculate some field theoretical work. So far, I have recovered Maxwell's equations (in ...
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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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Prove this mathematical equation is useful in classical mechanics [closed]

Prove that this equation is being used in the proofs of classical mechanics. $$ \sum m_i\left(\dot{\vec r_i}\cdot\delta\dot{\vec r_i}\right)=\delta\sum\frac12m_i\left(\dot{\vec r_i}^2\right). $$
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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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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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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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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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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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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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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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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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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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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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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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1answer
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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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1answer
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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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3answers
117 views

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: $$\...