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

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Find the infimum value of a functional.

Consider the functional $$F(u)=\int_{0}^{1}x^{\alpha}|u'(x)|^pdx,\ \ \ u\in W^{1,p}(0,1)$$ where $\alpha\ge 0$ and $1<p<\infty$. Given $a<b$, find the value of $$\inf\{F(u): u\in ...
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40 views

General Brachistochrone problem

Let $\gamma$ is a continuous curves that connect $(0,0), (x,y), x>0,y<0$ in $\mathbb{R}^2$ and $u : [0,L(\gamma)]\rightarrow\mathbb{R}\times(-\infty,0]$ is the arc length parameterrization of ...
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2answers
240 views

Intuition behind variational principle

Hofer-Zehnder, in section 1.5, proves that every Hamiltonian field on a strictly convex compact regular energy surface carries a periodic orbit. I have understood the proof. What I am wondering about ...
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1answer
102 views

How to take a partial derivative of an integral containing a fourier series?

(Chapter 2, p. 68, Problem 24) From Goldstein Classical Mechanics. Problem: The one-dimensional harmonic oscillator has the Lagrangian $L=m\dot{x}^2/2-kx^2/2.$ Suppose you did not know the ...
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1answer
34 views

Why does the Lagrange equation have to be zero?

I know it's a pretty basic question, but I still don't get it since starting Lagranian mechanics this year. I tried to read Stone and Goldbart's "Mathematics for Physics" and they said: Suppose ...
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1answer
150 views

Functional optimization: maximize a double integral where the functional appears twice

Please help me solve the following optimization problem. Suppose that you have to choose a function $U: [0,1]\mapsto [0,1],$ which must be nondecreasing ($U'\geq 0$) to maximize the following ...
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64 views

Short introduction script to calculus of variations

Does anybody know of a short lecture script/website/small book that gives an introduction to the calculus of variations? It should be suitable for total beginners, with steps explained in detail. At ...
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1answer
52 views

Stuck with a problem of calculus of variation in the proof that a minimizing curve is a geodesic

I'm reading the proof of the proposition that states that every minimizing curve is a geodesics when it is given an unit speed parametrization. In the proof appears the following quantity : $$ ...
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87 views

Functional derivative or chain rule?

Just a quick question... I have two functions – $V(a,b,c)$ and $F(a,b,c)$ – and I wish to calculate the derivative of one with respect to another ($\frac{\partial V}{\partial F}$). Am I right in ...
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1answer
36 views

Linearization of divergence of a vector field?

Let's $X$ is a fixed smooth vector field on semi-Riemannian manifold $(M,g)$. For a symmetric 2-tensor field $s$, and for sufficiently small values of $t$, $\tilde{g}=g+ts$ is a semi_Riemannian metric ...
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1answer
29 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
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1answer
37 views

Variational calculus - inequality

Let $u \in L^1_{loc}(\Omega)$, and suppose that $\int _{\Omega} u(x)\eta (x)\; dx \geq 0$, $\forall \eta \in C ^{\infty}_{0} (\Omega)$ . Then $u(x) \geq 0$ , a. e. $x \in\Omega$
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19 views

Characterize $|\nabla f|$ as minimal function which satisfies an upper gradient inequality

Let $f \in C^1( \mathbb R^n, \mathbb R) .$ Then one by chain rule has $$ (*)\qquad |f(g(1))-f(g(0))| \leq \int_0^1 |\nabla f|(g_t)|g'(t) |\ dt, \quad \forall g \in C^1([0,1],\mathbb R^n). $$ I have ...
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2answers
23 views

Proof for special concave form of functions

If $f(x)$ is a non decreasing concave function between $0$ and $1$ for $x\ge 0$, then for $a>1$ I am confident that $af(x)>f(ax)$, but I am not quite sure how to prove it. Any help would be ...
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22 views

Sufficient conditions for an optimization problem

Given an optimization problem \begin{equation} \max{F(x)} \text{ subjected to }T(x)=u \end{equation} Where $F:\mathbb{R}^n \rightarrow \mathbb{R}$ is concave function and $T:\mathbb{R}^n ...
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105 views

Calculation of a Frechet derivative

Say I have an infinite sequence $X=(x_i)$, $i=1,2,3,\ldots$ such that it's in $\ell^2$ space, i.e. $\sum_{i=1}^\infty|x_i|^2<\infty$. Now, this function that takes this infinite sequence to a real ...
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67 views

How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
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15 views

Variational function versus variational solution

I want to minimize the functional $F[f(x)]$ and I'm going to try this in two different ways: First I am going to numerically minimize the functional $F[f(x)]$, leading to the "true solution" $f(x)$. ...
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1answer
33 views

Fastest path with limited acceleration

An object on point $A$ with the initial velocity of $\dot{\bf{x}} (t)$ have a maximal acceleration of $a$. What is the fastest path for the object to get to point $B$? I thought this should be all ...
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1answer
38 views

Partial derivative of an integral transform

I have a Lagrangian of the form $\mathcal{L}(x,f)=[s(x)-\int_a^b A(x,x')f(x')dx']g(x)$, where $a,b$ are constants, and $g(x),s(x)$ and the kernel $A(x,x')$ are given . I am interested in computing ...
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Are triangles the strongest shape?

They are according to the buzz on the Internet (and most stable too), despite competition from circles. Mythbasters even proclaim that "triangles are the strongest shape because any added force is ...
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1answer
37 views

Does Euler-lagrange Equation hold with discontinuous integrand?

Suppose I have a function $Heaviside(x-1) \sqrt{1+f'(x)^2}$ or some other discontinuous function, how do I find the minima with E-L Eqn or in other way?
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53 views

Why would integrating acceleration give the following solution?

Suppose I have a mass with equation of motion described by: $x^{''}(t) = F(t) - 1$, $0<t<T$, all initial conditions equal to zero $F(t)$ is some unknown force My text claims that the equation ...
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30 views

Analytic version of Hilbert's XIX problem

The famous Hilbert's nineteenth problem, initially stated in the $C^\omega$ category, was reduced by Bernstein and Petrowsky to the analogous statement in the $C^\infty$ category (and, after ...
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1answer
270 views

Poincaré hyperbolic geodesics in half-plane and disc models

The objective of this post is to state that 1) the Poincaré hyperbolic metric results in a solution of complete geodesic circles in both half-plane and disk models. 2) the choice of one or other ...
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44 views

Minimizing sequences and topology (direct method)

To show the importance of the choice of the topology for the direct method we have been assigned the following exercise which I've not been able to solve due to my lack of understanding on how strong ...
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1answer
24 views

How to calculate route variations/permutations

If I have 5 trucks and 10 deliveries to make per truck, that's 50 deliveries total, but how many different route variations could there be? You could give each truck the list of deliveries and they ...
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2answers
81 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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41 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 ...
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27 views

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
45 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
43 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
52 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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1answer
41 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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40 views

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
32 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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28 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
81 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
32 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
49 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
30 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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123 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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108 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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27 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
33 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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59 views

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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54 views

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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40 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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51 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
43 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 $C^1$ ...