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

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Calculus of variations of multiple integrals and multiple constrains

Let $q(x_1,\dots, x_6)$ be a arbitrary multivariate density that we choose beforehand. What I want is to calculate a new multivariate density $p(x_1, \dots, x_6)$, which is obtained by minimizing the ...
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2answers
24 views

What does first and second approximations mean in this context?

In the Feynman lectures on physics, Feynman in talking about the principle of least action, discusses how we should be able to find the true path $x(t)$ which has the least action, and the way to do ...
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21 views

Calculus of Variations transformation

In the Calculus of Variations book by Gelfand and Fomin it says to consider the transformation $$x^{*} = \Phi(x,y,y')$$ $$y^{*} = \Psi(x,y,y').$$ Here it seems that $y'$ is the derivative of $y$ with ...
2
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Why does minimizing $H[f] =\sum^{N}_{i=1}(y_i-f(x_i))^2+\lambda \| Pf \|^2 $ leads to solution of the form $ f(x) =\sum^N_{i=1}c_iG(x; x_i)+p(x)$?

I was reading the following paper of dimensionality reduction (1) and also one on theory of networks for approximations and learning (2) and was trying to understand how the regularization problem ...
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3answers
31 views

Derivative with respect to $y'$ in the Euler-Lagrange differential equation

I am having trouble understanding the $ \dfrac{\partial L}{\partial y'} $ part in Euler-Lagrange Equation. For example, if $ L = y^2(z) $, what is the symbolic expression for $ ...
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1answer
25 views

Clarifying A Calculus of Variations Problem

Let F[y] be a functional defined like so: F[y] = $\int y(x)^2 + (y'(x))^2 dx$. I'm trying to find the function y which maximizes the value of F, and because the Euler Lagrange equation specifies a ...
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1answer
26 views

The Euler-Lagrange equation yields a constant function?

My functional is $J[f] = \int_{-\infty}^{\infty} f(x) \log f(x)\,dx$. I want to maximize it using the calculus of variations. In order to use the Euler-Lagrange equation, I define $L(t, y, y')$ such ...
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19 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
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+50

Jacobi field geodesic and calculus of variations.

How can we show that the second order variation to a geodesic is given by the Jacobi differential equation? In essence, \begin{equation} \frac{D^2}{dt^2}J(t)+R(J(t),\dot \gamma (t))\dot \gamma ...
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geodesic of Stiefel manifold

Define a metric on Stiefel manifold $V_{n,p}$ as $$\left<\Delta_1,\Delta_2\right>=\text{tr}\Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2$$ $\forall \Delta_1,\Delta_2\in T_YV_{n,p}$ how to ...
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1answer
17 views

Double integral of a product in calculus of variations

Let's say I have an integral of the form $$ V(u) = \iint\limits_{[0,T]^2}f(x,y)u(x)u(y)\mathrm dx\mathrm dy $$ which I would like to optimize over smooth functions $u$. For the variation I get $$ ...
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0answers
10 views

strong minima and maxima condition in calculus of variation

I am going through the topic CALCULUS OF VARIATION. There are not many examples on the topic strong/weak maxima minima. Can anybody provide the link of the source or book name where this topic is ...
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Current value Hamiltonian questions [closed]

Would really appreciate it if somebody could solve this question, we will have similar ones on the next math exam and I would love to know how to answer it. Minimize $$ \int_0^2 (x^2 + ...
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24 views

What type of self-adjoint operator does $\hat{P}$ has to be for Green's function to result in a radial exponetial $e^{-\| x-t \|^2}$

I was reading the following paper on hyper basis function (HBF) (similar to radial basis function RBF network) and was trying to understand when is it the case that the network has radial basis ...
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1answer
33 views

Defining a partial derivative with respect to an antisymmetric tensor/matrix

I'm looking at some nonlinear electrodynamics, and have been following a textbook which contains a primer on some of the stuff I'm interested in following up. However, I seem to have fallen at the ...
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1answer
17 views

Derivation of Euler Equation

In the following notes here I don't understand the very last line of proof of theorem 6.1 . We now use the fact that $\frac{\partial}{\partial a}S[x_a(t)]$ must be zero for any function ...
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1answer
36 views

Dirichlet energy and Fourier transform

Is there a direct relationship between the Dirichlet energy of a function: $$E(f)=\int_{\Omega}\lvert\nabla f(\mathbf{x})\rvert^2\mathrm{d}V$$ and its Fourier transform ...
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2answers
23 views

Bivariate optimal density

Consider any feasible $p:[0,1]^2\to [0,1]$ that allows discontinuities and the problem $$\min_{p(.)} \int_0^1\int_0^1 p(x,y)^2 dF(x) dG(y)$$ s.t. $$\int_0^1 p(x,y)dG(y)=k\phantom{0} for \phantom{0} ...
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1answer
36 views

What is the significance of the integral of the Hessian determinant?

The integral of a function over some region measures the total value of the function in that region: $$T(u)=\int u\thinspace\mathrm{d}V$$ The integral of the squared norm of the gradient of the ...
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1answer
59 views

Arc length function of a helix/spiral is convex?

Given the arc-length of a parametric curve, $\int_a^b\|\gamma'(t)\|$ if the parametric curve was non-convex, can the arc length be a convex function?If the parametric curve was convex, will the arc ...
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18 views

Non-linear hyperbolic systems

This question is about a naive approach to non-linear hyperbolic systems, thinking in the context of elasticity. To set up the problem suppose $\Omega\subset \mathbb{R}^n$ is open and bounded. ...
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1answer
26 views

Euler equation formula

When I am using Euler equation for Fourier transform integrals of type $$\int_{-\infty}^{\infty} dx f(x) exp[ikx] $$ I am getting following integrals: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
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Fundamental Lemma of the Calculus of Variations with higher derivatives

The fundamental lemma of the calculus of variations is often presented as: If $M(x) \in C[a,b]$ such that $\int_{a}^{b}{M(x)\eta(x)} = 0 ~~\forall\eta\in C^1[a,b],\eta(a)=\eta(b)=0$, then $M(x)=0$ for ...
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Differentiate functional with delta function when calculating Euler-Lagrange equation

The paper "active contours without edges" by Chan and Vese http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=902291, My goal is to understand how to derive the corresponding euler-lagrange ...
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1answer
25 views

The meanings of some symbols in “Calculus of variations”

Could someone tell me the meanings of the "C" and its superscript "1" and subscript "0" in the equation which I have marked. Thank you very much!!!
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36 views

What curved ramp transports a ball from (1,1) to (0,0) most quickly, under the acceleration of gravity, with no friction or air resistance?

An infinitisemally small ball is placed at the top of a ramp which has a height of 1m and ends 1m away horizontally. What is the optimal curve of the ramp to minimize time taken for the ball to reach ...
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1answer
29 views

calculus of the measure of a $C^1 $ hypersurface

I have to prove that: $$\lim_{r \to 0} \frac {\mathcal H ^{n-1}(M \cap B(x,r))}{\omega_{n-1} r^{n-1}}=1, $$ where $\mathcal H ^{n-1}$ is the ($n-1$)-dimensional Hausdorff measure, $M$ is a $C^1$ ...
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77 views

A variational approach to symplectomorphisms

Let $(\Sigma,\omega)$ be a compact symplectic surface, and let $\mathcal{D}$ be the group of diffeomorphisms of $\Sigma$. Is there some known variational approach to determine if an element ...
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1answer
25 views

Absolute Continuity defined by Necas

I read a definition of the absolute continuity in Necas' book "Direct Methods in the Theory of Elliptic Equations": Let $\Omega$ be a domain in $\mathbb{R}^n$ , $P$ a line verifying ...
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1answer
37 views

Elementary Examples of Functionals

I'm working on a research project that's a little over my head, so forgive some simple questions. Is a composite function $f(g(x))$ a functional? What are a handful of other simple examples of ...
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1answer
18 views

Regularity of limit measure and prove that $|\mu_h|\stackrel{*}{\rightharpoonup}|\mu|$

I have some questions. First of all, let $\mu_h$ a sequence of Radon measures and suppose that $\mu_h$ weakly-converge to another measure $\mu$. Now, this limit measure $\mu$ is still Borel? Is it ...
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1answer
48 views

How to Invert the Euler Lagrange Equations?

Suppose I have a functional L. For example $L = y+3y'$. Where y is itself a function of real variable x It's easy for me to evaluate the Functional Derivative of L via the Euler Lagrange Equations: ...
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2answers
29 views

Minimum curve for the distance between two points at the plane

The problem is to determine the curve y=y(x) in the plane, the lenght of which is given by the functional: \begin{equation} I(y)=\int_{x_1}^{x_2}\sqrt{1+(y')^2}dx=\int_{x_1}^{x_2}F(x,y,y')dx ...
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1answer
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Calculus of Variation: Euler-Lagrange Equation in 1D

I am currently trying to get into calculus of variation for a course of Image processing. In the lecture we learned that a smooth function u[a,b]->R that minimises: $$\int_a^b F(x,u,u') dx $$ ...
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Prove that a sequence of measures weak-star converges to another measure

We have a set of locally finite perimeter and a sequence of sets $\{E_h\}_h$ with $C^1$ boundary such that $$E_h\to E \text{ and } \mu_{E_h}\stackrel{*}{\rightharpoonup} \mu_E,$$ where $\mu_{E_h}$ and ...
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0answers
44 views

The Dirichlet problem for the Laplace equation: classical solutions versus weak solution

Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in ...
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How to use find the Lagrange Multipliers in multidimensional Calculus of Variations

Suppose I wish to minimise the integral $$I = \int_{s_0}^{s_1}\int_{t_0}^{t_1}F\, dt ds$$ Where $F$ is a function of the six variables $x(s,t)$, $y(s,t)$, and their four partial derivatives, ie $$F ...
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1answer
45 views

Euler-Lagrange equation with constraints outside the integral

So I've been studying Euler-Lagrange equations, and on an assignment I have the problem to find them for $J(y)=\int_a^bF(x,y,y')dx-By(b)+Ay(a)$ Where $y(a)$ and $y(b)$ are free, $A$ and $B$ are ...
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1answer
74 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
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Definition of the Second Variational Derivative In terms of The first

I know that for functional $F$ the first variational derivative at $f$ with increment $h$ is defined as \begin{align*} \delta F[f,h]= \lim_{\alpha \to 0 } \frac{F[f+\alpha h]-F[f] ]}{\alpha }. ...
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2answers
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dirichlet principle: why $u-g\in W_0^{1,2}(\Omega)$?

Let $D\subset\mathbb R^n$ be open and bounded. Consider $\Delta u=f$ in $\Omega$ and $u=g$ on $\partial\Omega$. Let $g\in W^{1,2}(D)$ and $f\in L^\infty(D)$. Then the minimizer of $$ I(u)=\int_\Omega ...
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Calculus of variations - unilateral constraints [duplicate]

Question about Evans states, chapter 8.4.2! We have $I[w] := \int_U \frac{1}{2}|Dw|^2 - fw\, dx$, among all functions $w$ belonging to the set $$\mathcal{A} : = \{w \in H_0^1(U) : w \geq h \, ...
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1answer
54 views

The functional $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ is weakly lower semicontinuous

I am studying calculus of variation, and I need to prove that $I[w] = \int_U \frac{1}{2} |Dw|^2 - fw \, dx$ with $f \in L^2(U)$ is weakly lower semicontinuous on $H_0^1(U)$. In classes, I only ...
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1answer
15 views

Euler equation for the functional has the form: $f_y-f_xy'-\frac{fy''}{1+y'^2}=0$

I want to show that the Euler equation for the functional $J(y)= \int_a^b f(x,y) \sqrt{1+y'^2}dx$ has the form: $$f_y-f_xy'-\frac{fy''}{1+y'^2}=0$$ $$L(x,y,y')= f(x,y) \sqrt{1+y'^2} dx$$ ...
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1answer
37 views

optical flow Euler-Langrange equation

I have a problem understanding how optical flow functional is plugged into Euler-Lagrange equation. The functional is: $\iint[(I_xu+I_yv+I_t)^2+\alpha^2(||\nabla u||^2 +||\nabla v ||^2)]dxdy$ ...
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25 views

Geodesic equation for surface of sphere

One of the standard problems of calculus of variations is showing that geodesics on the surface of the sphere are great circles. But I don't understand the equation. The equation for great circle ...
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1answer
33 views

Dirichlet energy

From PDE Evans, 2nd edition: Chapter 8, Exercise 17: Let $u,\hat{u} \in H_0^1(U)$ both be positive minimizers of the Dirichlet energy $$I[w] := \int_U |Dw|^2 \, dx,$$ subject to the constraint ...
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2answers
245 views

Elliptic regularization of the heat equation

This is from PDE Evans, 2nd edition: Chapter 8, Exercise 3: The elliptic regularization of the heat equation is the PDE $$ u_t - \Delta u -\epsilon u_{tt}=0 \quad \text{in }U_T, \tag{$*$}$$ where ...
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0answers
47 views

Derivation of Von Karman Equations

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...
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1answer
99 views

Euler-Lagrange equation [duplicate]

This is PDE Evans, 2nd edition: Chapter 8, Exercise 2: Find $L=L(p,z,x)$ so that the PDE $$-\Delta u + D\phi \cdot Du = f \quad \text{in }U$$ is the Euler-Lagrange equation corresponding to the ...