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

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5
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2answers
4k views

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 ...
0
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1answer
36 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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1answer
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 ...
3
votes
0answers
28 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 ...
4
votes
1answer
241 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 ...
2
votes
0answers
40 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 ...
0
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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 ...
1
vote
2answers
78 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 ...
1
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0answers
39 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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0answers
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 ...
0
votes
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 $ ...
0
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1answer
42 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 ...
2
votes
1answer
51 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 ...
0
votes
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 ...
2
votes
0answers
36 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 ...
1
vote
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 $$ ...
0
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0answers
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 ...
3
votes
1answer
78 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 ...
2
votes
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 ...
1
vote
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 ...
2
votes
2answers
29 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} ...
3
votes
1answer
120 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 ...
1
vote
1answer
104 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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0answers
26 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. ...
0
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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: $\int_{-\infty}^{\infty} dx f(x) cos(kx)$ ...
3
votes
0answers
56 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 ...
0
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0answers
53 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 ...
1
vote
1answer
39 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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0answers
49 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 ...
1
vote
1answer
42 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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0answers
83 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 ...
1
vote
1answer
41 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 ...
1
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1answer
60 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 ...
0
votes
1answer
26 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 ...
1
vote
2answers
121 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: ...
0
votes
2answers
50 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 ...
0
votes
1answer
42 views

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: $$\int_a^b F(x,u,u') dx $$ ...
0
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1answer
36 views

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 ...
0
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0answers
60 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 ...
2
votes
0answers
65 views

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 ...
0
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1answer
72 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 ...
2
votes
1answer
146 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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0answers
21 views

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 }. ...
4
votes
2answers
53 views

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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0answers
26 views

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 \, ...
1
vote
1answer
82 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 ...
0
votes
1answer
18 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: $$f_y-f_xy'-\frac{fy''}{1+y'^2}=0$$ $$L(x,y,y')= f(x,y) \sqrt{1+y'^2} dx$$ ...
0
votes
1answer
71 views

optical flow Euler-Langrange equation

I have a problem understanding how optical flow functional is plugged into Euler-Lagrange equation. The functional is: $\iint[(I_xu+I_yv+I_t)^2+\alpha^2(||\nabla u||^2 +||\nabla v ||^2)]dxdy$ ...
2
votes
0answers
217 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 ...
0
votes
1answer
84 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 ...