Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

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Deduce Euler-Lagrange equation

I'm trying to get the Euler-Lagrange equation: $$\frac{d}{dx}\Big(\frac{\partial\mathfrak{L}}{\partial \dot f}\Big)-\frac{\partial \mathfrak{L}}{\partial f}=0$$ But I can't see this: Let $p$ and $q$ ...
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1answer
37 views

Euler-Lagrange equation and unknown coefficients

I want to show that the nonlinear functional $$ J(u) = \int_0^1 (u'(x))^2 + b(x)u^2(x) + f(x) u(x) \,\textrm{d}x $$ attains its minimum in exactly one point of the Sobolev space $W_0^{1,2}(0,1)$. ...
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34 views

Characterizing Bounded Symmetric Bilinear Functions on Hilbert Spaces

Context: I am reading about Sobolev spaces and the Poisson equation from Eberhard Zeidler's Applied Functional Analysis book/article, and a key tool seems to be what Zeidler calls the "Main theorem ...
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2answers
70 views

Solving Euler-Lagrange Equation with delta function

I am trying to understand a physical system and have arrived at the following equation: $$\mathcal{S} = \int_{z = -\infty}^{z = \infty} dz \left\lbrace f_\rho[\rho] + \dfrac{m}{2} \mid \dfrac{\...
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1answer
34 views

How to compute a geodesic (homework question)

I am interested to compute the a geodesic in the Euclidean space $\mathbb{R}^1$. I am not familiar with differential geometry but I know that a geodesic is the solution of the following optimization ...
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59 views

Norm in finding local extrema for functional

In "The Calculus of Variations" by Bruce van Brunt, he says: Let $J:C^2[x_0,x_1]\to\mathbb{R}$ be a functional of the form $$J(y)=\int_{x_0}^{x_1}f(x,y,y^\prime)dx,$$ where $f$ is a function ...
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24 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
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2answers
32 views

Chain rule for variational derivatives and differentiation of an integral?

Assume that I have the following functional: $$ F[u_1,u_2,...u_N,\nabla u_1,...,\nabla u_N,...;t]=\int_{\Omega} f( u_1(x,t),u_2(x,t),...,u_N(x,t),\nabla u_1 , \nabla u_2,... ) dV $$ where $\Omega \...
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16 views

Using the Fourier Series in Variational Optimization Problems

Say I have a functional $L(f)$ which takes as input the function $f:\mathbb{R}\to\mathbb{R}$, and I want to find the function that optimizes $L$. Unfortunately, there's no way to define a functional ...
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1answer
22 views

How to calculate the variation of a matrix?

Suppose we have two diagonal matrices $$ A_{\mu \nu}=\left(\begin{array}{cccc} \rho(t) & 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0 \end{array}\...
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2answers
80 views

Geodesics in Poincare Disk

I would like to find the geodesics in the Poincare disk. I know that the metric is $$\frac{dx^2+dy^2}{(1-x^2-y^2)^2}$$ so $$s=\int \frac{\sqrt{1+y'^2}}{1-x^2-y^2}\, dx$$ Then I try to find y(x) using ...
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19 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
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1answer
25 views

Chain rule in partial derivatives

I've come across the following expression in my textbook about the chain rule in partial differentiation that I don't quite follow To be more specific, it's the diferentiation of (6.9) right at the ...
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46 views

Ordinary differential equations as variational problems

Considering an ordinary differential equation of first order in the implicit form $$ F(q(t),\dot q (t))=\alpha,\,\,\, q(0)=q_0 $$ with $\alpha,\, q_0$ constants, what is the relation of the solution ...
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35 views

For a convex function, why does lower semicontinuity imply weak lower semicontinuity?

I have come across the statement that, for a convex function, all notions of lower semicontinuity are equivalent. That is: weak lower, sequential lower, and weak sequential lower semicontinuity are ...
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1answer
25 views

A tighter family of Markov-like inequalities

I believe there should exist tighter-than-Markov Inequalities that use the same information as the markov inequality (just expectation). Consider the proof of the markov inequality in the link below: ...
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1answer
22 views

Calculus of variation for geodesic

I need to minimize $$J[v]=\int\sqrt{P(x)+R(x)(v')^2}dx$$ By Euler equation, I get $$\frac{d}{dx}\frac{Rv'}{\sqrt{P+Rv^{'2}}}=0$$ Then I need to solve a complex ODE, but I don't know how to deal ...
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1answer
36 views

calculus of variations or optimize over function form

I have a question about optimizing the following quantity over function form . Given unknown function $f(\theta)$ such that $f(\theta)\geqslant 0$ and $\int f(\theta)d\theta\leq \infty$. And ...
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25 views

Minimal surface with radially symmetrical function

The following image is from the book "Regularity Theory for Mean Curvature Flow", by Ecker. I consider the plateau problem, whose goal is to solve minimal surface given fixed boundary values. In ...
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40 views

How to obtain the closed form solution to this nonlinear system of ODEs

I have the following simple but nonlinear ODE system problem below: $$\left\{ \begin{align} &f'(t)=a\cdot (1-f(t)-g(t))^{13/10}\tag{1}\\ &g'(t)=b \cdot f(t) \tag{2}\\ &f(0)=0,\quad g(0)=...
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1answer
49 views

test function and boundary condition

For example, if we consider the Dirichlet energy $\int\frac 12 |\nabla u|^2$ and the solution space as follows: $$X=\{u\in W^{1,2}(\Omega) \text{ | } u = 0 \text{ on } \partial\Omega \}$$ , then the ...
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19 views

Does this mean a new boundary condition for ODE and how to handle it?

In order to to calculate desired light path in continuous medium with gradient refraction index, for schematic see the Figure below. $O:(0,0)$ is the disk center of light source $\odot{O}$ with ...
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66 views

Extreme of functional - Calculus of variations: Euler-Langange equation

Find all extremes of $$I[y] = y(0)^2 \int_{0}^{1} y(y')^2 dx,$$ with the given initial condition $y(1)=0$. I was thinking of writing $y(0)^2 = -\int_{0}^{1}2yy' dx$ and then what (differentiating ...
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2answers
199 views

Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
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1answer
41 views

Find the function that minimizes $\int_{0}^{1}e^{-(y'-x)}+(1+y)y'dx$

Suppose among all the continuously differentiable functions $y(x), x\in \mathbb{R}$, with $y(0)=0$ and $y(1)=\frac{1}{2},$ the function $y_0(x)$ minimizes the functional, $$\int_{0}^{1}e^{-(y'-x)...
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2answers
138 views

Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by \begin{equation} f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } \...
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13 views

Formulation of boundary constrained minimal surface

Using standard notation of classical surface theory how is the standard Plateau problem formulated as an iso-perimetric one minimizing area for given boundary length $$ \int \sqrt{ E \, du^2 + 2 F ...
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2answers
132 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
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1answer
50 views

Shortest path to the apex of a cone

This is something I thought about today but have no idea how to approach. We are given a right circular cone with lateral length L and angle at the base $\alpha$. A curve along the surface of the ...
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2answers
55 views

Which advanced mathematics book do you recommend for a college starter?

I'm supposed to start my preparation for the college admission math exam so I'm looking for a good explanatory/textbook/problem book which covers most of or all of precalculus topics with a hint of ...
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1answer
27 views

Any critical point $u_0\in M$ of $I|_M$ satisfies $I'(u_0)=\mu\gamma'(u_0)$

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}u_r^2-\xi u^2+\ln(1+u^2)\right\}r\,dr,$$ $\xi\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in C^1[0,R]:u(0)=0=...
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Calculus of variation: Maximize fuctional

"Among the curves of given length l, at the upper half-plane, passing through the points (-a, 0) and (a, 0) , find the one that encloses the largest surface area together with the space [-a,a] ." I ...
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0answers
19 views

Palais-Smale condition for a functional

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}ru_r^2-\gamma ru^2+r\ln(1+u^2)\right\}dr,$$ $\gamma\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in C^1[0,R]:...
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8 views

Variation of a d'alambertian operator

Let $M$ be a pseudo Riemannian manyfold, $H$ be function of a scalar curvature $R$. Assume that variation of the metric tensor and it's first derivatives is zero on the border $\partial M$. Which ...
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49 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} \mathbb{...
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1answer
140 views

Obtaining an estimate for the Lagrangian $L=H^*$ from the Hamiltonian $H$

This is from C. Evans' PDE book, page 130. The convex function $H:\mathbb{R}^n\to\mathbb{R}$ is $C^2$ and satisfies $$ H\big(\frac{p_1+p_2}{2}\big) \leq \frac{1}{2}H(p_1) + \frac{1}{2}H(p_2) - \frac{\...
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2answers
48 views

How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = y'\frac{d}{dx}\left(\frac{\...
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2answers
53 views

Does Constant factor rule in integration hold for functionals?

The constant factor rule in integration states the following relation is valid $$\int a f(x)dx=a \int f(x) dx$$ for all constants $a$(or $a$ that are constant functions of x, that is $\dfrac{da}{dx}=...
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2answers
98 views

Why does $\mathrm{tr}(\mathrm{ln}g_{\mu\nu})$ vary as $g^{\mu\nu}\cdot\delta g_{\mu\nu}$ under $\delta g_{\mu\nu}$?

For a pseudo-Riemannian manifold, under the variation $g_{\mu\nu}\mapsto g_{\mu\nu}+\delta g_{\mu\nu}$, the determinant $g=\mathrm{det}g_{\mu\nu}$ varies as $$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}$$ ...
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1answer
44 views

Euler-Lagrange equation of energy of length function on Riemann manifold

$(M,g)$ is a Riemann manifold. $\gamma:[0,1]\rightarrow M$ is a curve.The length of $~\gamma $ is $$ L(\gamma)=\int^1_0 ||\dot\gamma (t)||_g ~dt $$ The energy is $$ E(\gamma)=\frac{1}{2}\int^1_0 ||\...
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1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln u^2-nu^2](4\pi\tau)^{-n/...
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1answer
36 views

Variation with constraint condition

For example, let $$ I[u]=\int_M |\nabla u|^2+u dV $$ It's not hard to compute the variation of $I[u]$. If $I[u]$ reach the minimum at $u_0$, I can get that $$ i(t)=I[u_0+tv] \\ i'(0)=0 \\ i'(0)=\...
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0answers
52 views

Maximization of the Expectation of a function

Recently I was thinking in general on how to maximize the expectation of a function (not necessarily a utility function, but apparently this is a common case). To give an idea of the kind of problem, ...
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41 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
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1answer
22 views

Movable end points Calculus of Variation.

Given problem is $J[y] =\int_{0}^{x_1}y'^2dx$ with $y(0)=0$ and $y(x_1)=-x_1-1$. After solving Euler Lagrange equation I got $y=Ax+B$ . And using first boundry conditon I got $y=Ax$ We have ...
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1answer
167 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
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1answer
34 views

Finding minimal area of a cylindrically symmetric surface

I am told that I have a cylindrically symmetric surface that is bounded between two circles $r=a$ at $z=\pm b$. I'm meant to use the Euler-Lagrange equation, so I'm trying to a functional for the ...
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1answer
65 views

a continuous path between two sobolev functions without increasing energy

This question has been post on MO a week ago. I move it here to get more luck. Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that $T[u_1]=T[...
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3answers
43 views

Find the double integral by changing to polar coordinates [duplicate]

So I have the following double integral $$\int_{-2}^{0} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} dydx$$ If I integrate with respect to y first I get: $\frac{1}{2}(y\sqrt{x^2+y^...
2
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0answers
42 views

Functional integration and Feynman path integrals in wolfram alpha

Is it possible to do Feynman path integrals in wolfram alpha? Say for a free quantum mechanical particle. The reason I am interested in this is because I would like to see how it arrives at the ...