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

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3
votes
1answer
323 views

Does a maximum entropy probability distribution with KL-divergence constraint not exist?

In my earlier question I asked about a technical aspect of solving a system of equations arising from looking for an entropy-maximizing distribution $p(x)$ continuous on $\mathbb{R}$ and constrained ...
2
votes
1answer
521 views

Troublesome functional derivative: second term of Euler-Lagrange equation

I am attempting to calculate the functional derivative of a functional $$E[\rho] = \int G(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\mathbf{r})d\mathbf{r},$$ where ...
15
votes
1answer
422 views

Gradient flow of a surface

I found the following definition in a book (S. Osher, R. Fedkiw, "Level Set Methods and Dynamic Implicit Surfaces", p. 140): [the context is reconstruction of surfaces from unorganized point sets] ...
5
votes
3answers
346 views

Treacherous Euler-Lagrange equation

If I have an Euler-Lagrange equation: $(y')^2 = 2 (1-\cos(y))$ where $y$ is a function of $x$ subjected to boundary conditions $y(x) \to 0$ as $x \to -\infty$ and $y(x) \to 2\pi$ as $x \to ...
1
vote
1answer
281 views

Wikipedia Article — Legendre Transform

I was reading the wiki article on Legendre Transform. I would be grateful if someone could explain the section at http://en.wikipedia.org/wiki/Legendre_transformation#Examples ie how they arrived at ...
1
vote
2answers
273 views

Question on the catenary

The catenary minimizes the potential energy of a cable and has equation $y - y_0 = A \cosh (\frac{x-x_0}{A})$. It is physically intuitive that the catenary is unique, but is there a mathematical ...
3
votes
1answer
152 views

Help deriving geodesic of $S^2$ by considering small deviations

For $s(t)$ the geodesic confined to the surface of a (3D) sphere, how does one get $\|\dot{s}\|^2 s + \ddot{s} = 0$ by setting $\frac{d}{d\delta} \left( \int \|\frac{d}{dt} \frac{s(t)+\delta ...
6
votes
2answers
254 views

Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem

A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume". At that time for me to write ...
1
vote
1answer
167 views

The uniqueness of the brachistochrone

How does one show the uniqueness of the solution to the brachistochrone problem? Doesn't the fact that the solution is of the form $x=a-c(2t+\sin2t)$ and $y=c(1+\cos2t)$ naturally guarantee ...
4
votes
2answers
833 views

Geodesics of a Sphere in Cartesian Coordinates

I want to minimize $I = \int |\dot{x}|^2 dt$ subject to the constraint $|x|^2=1$ (sphere) which gives an Euler equation of $\lambda x - \ddot{x} = 0$. I have to show that the Euler equation is ...
-2
votes
2answers
184 views

Calculus of variations question from Darcogona

I asked the question in the next forum, the 4th post, hopefully someone can help me with this here or there: https://nrich.maths.org/discus/messages/7601/151442.html?1310911861 Thanks in advance.
6
votes
1answer
736 views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

I have a question about Euler-lagrange equation which you can check this file. http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf, specifically in page 6,equation 8 , not equation 9... There ...
2
votes
2answers
92 views

An extremal property of the distance function

Let $\Omega \subseteq \mathbb{R}^N$ be open and bounded, let $\mathcal{I}:C(\overline{\Omega}) \ni u\mapsto \int_\Omega u(x)\ \text{d} x \in \mathbb{R}$ and set: $$\phi(x):=\text{dist} (x,\partial ...
4
votes
1answer
197 views

Why weak formulations in numerical mathematics?

Regard the Poisson equation on the domain $\Omega = [-1,1]^n$ with $f \in H^{-1}$ $- \triangle u = f$ with homogenous Neumann boundary conditions. From standard regularity theory we know $u \in ...
4
votes
0answers
222 views

Calculus of Variations: Contains an integral of my goal function

OK, using the calculus of variations, I want to find a function $f$ that maximizes: $$J = \int_0^n L(x,f(x)) \text{d}x$$. But $L$ has multiple integrals in it (for example, $\displaystyle \int_0^n y ...
16
votes
3answers
550 views

What's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$?

The question is as in the title: what's the minimum of $\int_0^1 f(x)^2 \: dx$, subject to $\int_0^1 f(x) \: dx = 0, \int_0^1 x f(x) \: dx = 1$? (Assume suitable smoothness conditions.) A problem in ...
7
votes
5answers
2k views

Introductory text for calculus of variations

I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up ...
5
votes
2answers
375 views

minimizing the norm of a curl over a domain

According to my computations: The function which minimizes $$\int_\Omega \|\operatorname{curl} f\|^2\,dx$$ should satisfy $$\operatorname{curl}(\operatorname{curl}f) = 0$$ everywhere on $\Omega$, ...
3
votes
1answer
227 views

Is it possible to combine the Euler-Lagrange equations with the method of Lagrange multipliers?

In particular, say we seek a sufficiently smooth function $ u : [a,b] \to \mathbb{R} $ such that the solution $x$ to the differential equation with given initial conditions $$ G(x, x', \dots, ...
2
votes
2answers
1k views

Partial derivative with respect to a function

Let $p$ be a function of the form $\mathbb{R}^2 \to \mathbb{R}$. How do i find the derivatives of the following expressions with respect to p(x,y) : a. $\int_\mathbb{R^2} p$ b. $ \dfrac{\partial ...
1
vote
1answer
266 views

Maximizing a function by finding derivative

I want to find the value of $\vec{p}$, $p_s$, $p_t$ each of which is a function of the form $f:\mathbb{R}^2 \to \mathbb{R}$ that maximize the following function : $$\begin{align} \int_\mathbb{R^2} ...
1
vote
0answers
90 views

Variability of a curve with Vapnik-Chervonenkis dimension 4

Say I have 10,000 data in 2-D and I want to fit a curve to them. There are many functional forms this curve could take -- polynomial, B-spline, trigonometric, and so on. I've decided that I only want ...
3
votes
1answer
314 views

Functional derivative of $\int \left( \frac{df^2 }{d^2 x} \right)^2 dx$

According to page 7 of the PDF document $$ \frac{\delta}{\delta f} \int \left( \frac{df^2 }{d^2 x} \right)^2 dx = \int \frac{df^4}{d^4 x} dx $$ I would like help proving this statement. Although ...
3
votes
2answers
456 views

Divergence Theorem, Laplacian, Energy Minimization

I am trying to understand a proof for critical points of certain energy functions being harmonic functions. It goes as follows: For a function $u(x_1,..,x_n)$, a functional E(u) is defined as $E(u) ...
8
votes
2answers
1k views

Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For an image denoising problem, the author has a functional $E$ defined $$E(u) = \iint_\Omega F \;\mathrm d\Omega$$ which he wants to minimize. $F$ is defined as $$F = \|\nabla u \|^2 = u_x^2 + ...
1
vote
1answer
173 views

Euler-Lagrange expression uniformly non-negative

If you form the Euler-Lagrange equation for some calculus of variations problem in x, f(x) and f'(x), and the resulting expression is always non-negative over the domain of x (because the expression ...
4
votes
5answers
258 views

Function that maximizes a function

Let's say we have a real, continuous, positive function f(x) for which we define the quantity: $$\pi(f,a) = \frac{\int_0^a f(x) dx}{\int_0^a \sqrt{1+\left(\frac{df(x)}{dx} \right)^2 ...
8
votes
1answer
1k views

Calculus of Variations and Lagrange Multipliers

A general problem for the Calculus of Variations asks us to minimize the value of a functional $A[f]$, where $f$ is usually a differentiable function defined on $\mathbb{R}^n$. What if, however, the ...
9
votes
3answers
509 views

Supremum length of space curves contained in the open unit ball having always less than unity curvature

I am in the process of proving that if a space curve (in R^3) has infinite length and the curvature tends towards 0 as the natural parameter s tends to infinity, the curve must be unbounded - i.e. not ...
3
votes
1answer
429 views

Quadratic minimization in a Hilbert space

If $A$ is a positive definite matrix, then the solution to the minimization problem $(1/2)x^TAx - b^Tx$ is given by $A^{-1}b$. I'm interested in the generalization of this to a Hilbert space. What ...
12
votes
8answers
5k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...