Questions on the calculus of variations, a subfield of calculus that deals with the optimization of functionals.

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derivative wrt to a function

Suppose $\phi(x+V\Delta t)$-$aV{^2}\Delta t$ is a function to be maximized w.r.t the function V which is a function of (x), $a$ and $\Delta t$ being scalar constants. Assuming $\phi()$ is ...
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3answers
55 views

Proof or counterexample : Supremum and infimum

If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum. How to Prove This.
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22 views

The Initial and boundary conditions of a 2nd order nonlinear ODE

The problem is derived from: Original gradient index optics problem See the Figure above. $O:(0,0)$ is the disk center of light source $\odot{O}$ with radius $3$. Then the profile light rays of ...
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0answers
15 views

Optimize monotonic function in calculus of variations

I'm interested in the variational problem $$\min_{y} \int_a^b F(x,y(x),y'(x))dx \qquad \text{subject to} \quad -y'(x)\leq 0 \quad \forall x \tag{1}$$ i.e. $y(x)$ has to be monotonic. I ...
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24 views

Finding a function of minimal arclength via Euler Lagrange theorem, stuck on solving differential equation

I want to minimize the arclength of a function $u(x) \geq 0$ for $x\in [-1,1]$ that is contrained by $u(-1)=0=u(1)$ and $\int_{-1}^{1} u(x) \, dx = A$, where $0<A<\pi/2.$ I have reduced the ...
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0answers
25 views

What is the difference between $L$ and $\mathcal{L}$? How does one find the Lagrangian

I'm following a course of Lagrangian and Hamiltonian mechanics, but I'm getting somewhat confused. Could someone explain the difference between $L$ and $\mathcal{L}$? I'm calling both "the Lagrangian ...
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1answer
32 views

How to find direction of velocity V2 to reach an object travelling at velocity V1, such that it takes least time?

If an object A is currently at point P1 moving with constant velocity V1, and there is another object, object B which currently at point P2 which can move with velocity v2, then what should be the ...
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53 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and ...
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0answers
30 views

Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
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1answer
38 views

Difficult Problem on Calculus of variation

My problem is: consider the functional $J(y)=y^2(1)+\int_0^1 y'^2(x) dx$ and $y(0)=1$ where $y\in C^2([0,1])$. If y extremizes J then find $y(x)$ . Any Hint will be appreciated.
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86 views

$H^1$ convergence of eigenfunctions of Schrödinger operators

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
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0answers
35 views

Convergence in the distributional sense (mean field games dynamics)

I am trying to go through the papers by Gueant, Lions and Lasry on Mean field games. One of their examples is the Mexican wave (which happens in football stadia). Straight to the point the Lagrangian ...
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0answers
32 views

Partial Differentiation with respect to a functional

Suppose that there are two independent variables $x, y \in \mathbb{R}$. Define two functionals $f_i : \left(\mathbb{R} \times\mathbb{R}\right) \rightarrow \mathbb{R}, i=1,2$ \begin{align} ...
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1answer
24 views

The shortest line problem extremum extinction

Let's consider a functional $J(y)=\int_{0}^{1}{ \sqrt{1+\frac{dy}{dx} ^{2}} dx}$, $y(x_{1})=y_{1}, y(x_{2})=y_{2}$. Using Euler-Lagrange equations we can get that the set of functions which attains ...
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1answer
48 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
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1answer
79 views

Is there any situation in which a geodesic maximize the path length between two points?

Some people (even in here) claim that geodesics are, in general, stationary curves. Locally speaking, geodesics always minimize arc length (see Manfredo, for example). But I can't visualize a surface ...
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165 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
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0answers
14 views

Variational calculus on integrals and derivatives

I am studying mechanics but am a novice in variational calculus. While reading a book on Lagrangian mechanics, I blocked when the author states that by calculating the variation of the following ...
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0answers
14 views

Stable geodesics

Consider a function defined on some space of smooth curves in a manifold (think of the "action functional"). I understand what a "critical point" of such a function is, but what is understood by a ...
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1answer
43 views

Solving a functional problem with break points

I'm totally out with solving functional problems when we need to find broken extremals, can someone show me how to solve the following problem: Can the following problem have break (corner) ...
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1answer
66 views

Find maximum $\int^{1}_{0}\{f(x)\}^3dx$

I would appreciate if somebody could help me with the following problem: Question: Find maximum $\int^{1}_{0}\{f(x)\}^3dx$ when (1). $f(x) : \text{conti-and} \int^{1}_{0}f(x)dx=0$ (2). $-1\leq ...
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0answers
47 views

Understanding derivation in Calculus of Variations book

I'm reading about Calculus of Variations and about the general variation of a functional. I bumped into few obstacles in my book I can't get over with. I have scanned the pages where I have my ...
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1answer
33 views

A little confusion about calculating variance?

given $a = 10% $, $b = 50%$, $c = 25%$ and $d = 15% $ calculate the variance? ( this is a made up question to make thinks clear for me) what i have done is 1st eliminate percentages? then i ...
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1answer
26 views

Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
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0answers
11 views

Minimal distance in expectation problem

Two random variables $\tilde{X},\tilde{Y}$ have the joint distribution $F:X\times Y\to [0,1] $ The given function is $g^*=(g_1^*,g_2^*):X\times Y\to R^2$ The control function is ...
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68 views

Second variation of the domain functionals.

I am reading a paper which deals with the second variation of the domain functionals and applications . The following problem is solved . The eigen value $\lambda(t)$ which is characterised by the ...
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65 views

solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = ...
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1answer
24 views

Finding function that maximizes ratio of area to length

I'm new to variational analysis, so I need someone to check, if I'm going in the right direction. Let's say I need to find a curve $y(x)$ with $y(0) = 1$ and $y(1) = 0$ that maximizes ratio of area ...
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1answer
57 views

Is this functional well defined?

I came across the following formulation of the problem. Minimize the functional $L[u]$ given by $L[u] = \int^b_a \sqrt{(1+(u'(x))^2}$ over $U = \{u\in C([a,b])\cap C^1((a,b)):u(a)=\alpha, u(b) = ...
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17 views

Checking Weierstrass's condition in a functional problem

My problem is to check for the Weierstrass's (necessary) condition in the following functional problem in my Calculus of Variations course: $$\min J[y]=\int_0^1xy'^3\;dx,\;\;\;\;y(0)=0,\;y(1)=1$$ ...
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0answers
20 views

Determine whether the extremals of a functional are weak or strong?

Is there any thumb rule for determining whether the extremals of a functionals (which are determined using Euler Lagrange equations) are maximum or minimum, weak or strong. I read about Weierstrass ...
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0answers
17 views

Numerical optimization in function space

I'm new to calculus of variations. I'm curious about how to apply simple numerical optimization techniques in function space. Consider the classical problem: finding the shortest path between two ...
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27 views

Prove a formula after change of variable?

If I have a change of variable $(x,u)\to (X,U)$ given by $$X=x+\epsilon u,U=u-\epsilon u.$$ How to prove the formula $$\frac{\partial U(X,0)}{\partial \epsilon}=\phi(X,u(X))-u'(X)\xi(X,u(X)),$$ where ...
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1answer
44 views

The minimizing problem over a sequence of shrinking balls

Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each ...
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0answers
40 views

First variation — A differentiation problem.

My question: This is just differentiation and I did it. I got $$T'(\epsilon)[L(T(\epsilon),x(T(\epsilon);\epsilon),\dot x(T(\epsilon);\epsilon))]+\int_0^{T(\epsilon)}L_x(t,x(t,\epsilon),\dot ...
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0answers
18 views

How to determine convexity of functional

I am a rookie to functional and calculus of variation. I want to know whether there exists any sufficient condition for convexity of a functional besides the definition. Actually for function, one ...
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1answer
33 views

Find the extremals of a functional of the form $\int^{x_1}_{x_0}F(y',z')dx$

I was working on Problem 3 in Ch. 2 of Gelfand & Fomin's Calculus of Variations, which reads: Find the extremals of a functional of the form $$\int^{x_1}_{x_0}F(y',z')dx$$ given that ...
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1answer
27 views

Calculation of the second variation of the functional $I(y)=\int_{-1}^1 [x^2(y')^2+x(y')^3]\,dx$

My question: I don't understand the last equation about second variation. According to definition, shouldn't it be $\int_{-1}^1 [2x^2+6xy'] (\eta)^2$? Can anyone help me with this? Where am I ...
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19 views

Calculus of variation: question about Lagrange multiplier?

In the example of Newton's problem with volume constraint. When using the method of Lagrange multipliers,the new Lagrangian is $L(x,h,h')=\frac{x}{1+h'^2}+\lambda xh $. I don't understand why we ...
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1answer
36 views

Differentiation under the integral sign problem

I have to take the derivative of the following function with respect to $\varepsilon$: $$\phi(\varepsilon)=\int_{a}^{b+\varepsilon C}F(x, y(x)+\varepsilon\eta(x), y'(x)+\varepsilon\eta'(x))\;dx$$ My ...
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1answer
23 views

How to determine a function whose minima falls on a specified curve?

I have a family of curves given by $g(x,y)=C_0 yx^{-n}$. How can I determine the function $f(x,y)$ for the family of curves that satisfies the condition that the local minima $\frac{\partial ...
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Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
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0answers
31 views

Mountain pass theorem

Let $I$ be a real functional over a Hilbert space $H$, satisfying all the conditions in the Mountain pass (M-P) theorem. My question is, can the assumption in the M-P theorem that $I[v]\leq 0$ for a ...
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How to minimize $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $

I'm new in optimizations and i am trying to understand how to obtain $ v $ that minimizes $ \int_{\Omega} \frac1{2 \theta} (u(x) - v(x))^2 + \lambda|\rho(v(x))| dx $ where $\rho(x)$ - continuous ...
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2answers
29 views

Understanding part of a theorem of Calculus of Variations

I have trouble understanding the following statement (From Gelfland's Calculus of Variations book): If $\phi[h]$ is a linear functional and if ...
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1answer
54 views

Why Dirichlet's energy uses a **squared** norm?

$E = \int_{\Omega}\left \| \nabla u(x)\right \|^2 dx$ So, Dirichlet's energy measures the integral of the squared norm of the gradient. Why squared norm? What would we get if we use just a norm? It's ...
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Problem about deformation theorem

I'm reading Evans PDE, on chapter 8.5 the proof of deformation theorem about the calculus of variation. On page 504 Evans wrote on the top: "we verify that the map $u\to dist(u,A)+dist(u,B)$ is ...
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10answers
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Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.

Contest: Give an example of a continuous function $f$ that satisfies three conditions: $f(x) \geq 0$ on the interval $0\leq x\leq 1$; $f(0)=0$ and $f(1)=0$; the area bounded by the graph of $f$ and ...
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0answers
35 views

Check whether the extremal has weak minima or weak maxima

The functional $$\int_0^1(y'^2 + x^3)dx,$$ given $y(1)=1,$ achieves its weak maximum on all its extremals weak minimum on all its extremals weak maximum on some, but not on all of its extremals weak ...
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1answer
41 views

What does the term “perturb” mean?

I've been studying Calculus of Variations and I came a cross with the term "perturb" in my study material, but the term was not defined. The sentence where I read it from was: "Rigid extremals are ...