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

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56
votes
1answer
2k views

What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
28
votes
5answers
1k views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
18
votes
2answers
216 views

Time-optimal control to the origin for two first order ODES - Trying to take control as we speak!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
17
votes
3answers
567 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 ...
15
votes
1answer
441 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] ...
14
votes
1answer
259 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
13
votes
1answer
445 views

Hilbert's 19th problem: Why do we care?

Hilbert's 19th problem asks: Are the solutions of regular problems in the calculus of variations always necessarily analytic? This was proven to be true (through the work of Sergei Bernstein, ...
12
votes
8answers
6k 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 ...
12
votes
3answers
127 views

Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.

Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in ...
12
votes
1answer
362 views

Arnol'd's trivium problem #68

I came across this blog that says that its French version has answers to most of Arnol'd's trivium problems, and I figured I'd try my hand at some of the ones they don't have. Number 68 raised my ...
11
votes
3answers
580 views

Calculus of variations: find $y(a/2)$ if $y(x)$ maximizes the volume of rotation

A curve $y(x)$ of length $2a$ is drawn between the points (0,0) and (a,0) in such a way that the solid obtained by rotating the curve about the $x$-axis has the largest possible volume. Find ...
11
votes
1answer
440 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
11
votes
4answers
434 views

When is the moment of inertia of a smooth plane curve is maximum?

Given a smooth plane curve $(x(s),y(s))$, parameterized in arc length $s$, of fixed finite length $L$, its moment of inertia about its center of mass (axis perpendicular to the plane) is given as $$MI ...
10
votes
2answers
372 views

Can there be a cubical bubble?

Although not physically perfect, a reasonable mathematical model for a bubble's shape is that it minimizes surface area subject to fixed volume. A single floating bubble is usually a sphere, but ...
10
votes
1answer
335 views

Checking my understanding of $T^*M$ as a symplectic manifold and the links between the classical description of Lagrangians vs this invariant way.

I am working through a book titled "An introduction to mechanics and symmetry" by Marsden and Ratiu. I have written up a brief summary trying to solidify my understanding of the general principles. ...
9
votes
3answers
514 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 ...
9
votes
2answers
2k views

Conceptual difference between strong and weak formulations

What are the conceptual differences in presenting a problem in strong or weak form? For example for a 2D Poisson problem the strong form is: \begin{split}- \nabla^2 u(\pmb{x}) &= ...
9
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 + ...
9
votes
1answer
158 views

Subdifferential boundary conditions: Testing pointwise or with $L^2$ functions

Let $\phi \colon \mathbb R^n \to \mathbb R$ be convex, proper and lower semi-continuous (lsc). Let $M$ be a measurable subset of $\mathbb R^n$. We can define a functional $\Phi \colon L^2(M) \to ...
9
votes
0answers
415 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
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 ...
8
votes
2answers
415 views

Finding Euler-Lagrange equations

Maybe you can help here. There is kind of a lengthy setup to understand what the question is asking. There is a paper I'm reading, and in one section of it I can't make heads or tails of the result. ...
8
votes
3answers
370 views

calculus of variations question

I would like to find a continuous function $y : [0,4] \to \mathbb{R}$ that minimizes the following functional $$I (y) := \displaystyle\int_{0}^4\sqrt{y\left(1+(y^{\prime})^2\right)} dx$$ subject to ...
8
votes
1answer
327 views

Minimizing Lagrangian with two functions

I read this problem where I have to minimize a functional $E[L]$ using calculus of variations, but I'm not sure what is the procedure to follow. The functional is the expected loss: $$E[L] = ...
8
votes
1answer
82 views

Shallow tent like soap film

A soap film circle in $x-y$ plane with center at origin can be carefully pricked with a blunt soapy pin at center and drawn out a little bit on $z$-axis forming a surface of revolution somewhat like a ...
7
votes
5answers
3k 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 ...
7
votes
1answer
785 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 ...
7
votes
1answer
292 views

existence of a minimizer for functional

My problem is the following: Show that the mapping $u \rightarrow ||\nabla u||^2 + (fu,u)$ has a minimum $u$ in $M:=\{ w \in H^1(\Omega): ||w||=1\}$ . The function $f$ is in $L^\infty$. I dont see ...
7
votes
3answers
268 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 ...
6
votes
3answers
186 views

Beach Path math question

Anyone who has walked on the beach knows that walking speed is dependent upon how far away from the ocean one walks. If you walk on the wet sand you can walk much more quickly than if you walked on ...
6
votes
1answer
322 views

Prove that $\int_0^1[f''(x)]^2dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$ such that $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1[f''(x)]^2dx\ge4.$ Find all $f$ for equality to occur.
6
votes
2answers
111 views

How to solve $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$?

How to find $\max_{f}\int_{0}^{\infty}f\left(x\right)dx$ subject to $\int_{0}^{\infty}xf\left(x\right)dx=x_{0}$, where $f$ is a function and $x_{0}$ is a constant?
6
votes
2answers
746 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
6
votes
1answer
676 views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
6
votes
1answer
178 views

Is there a fundamental misunderstanding here or have I made an algebraic slip?

Is there a fundamental misunderstanding here or have I made an algebraic slip? I have a Riemannian metric of the form $ds^2={du^2+dv^2\over 1-u^2-v^2}$ on an open disc and I want to prove that radial ...
6
votes
0answers
130 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
6
votes
1answer
177 views

Noether's theorem

I am now reading the book Calculus of Variations written by Jost and I have a problem in the proof of Noether's theorem: Theorem 1.5.1. Let $F\in C^2([a, b]\times \mathbb R^d \times \mathbb R^d, ...
5
votes
2answers
109 views

Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$ \tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2 $$ Called the "energy" of $f$? I am ...
5
votes
3answers
358 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 ...
5
votes
1answer
336 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
5
votes
1answer
2k views

Constrained variational problems intuition

Problem: minimise $F(x,y,y')$ over $x$, constrained by $G(x,y,y')=0$. $$J_1(x,y,y')=\large \int_{x_0}^{x_1}F(x,y,y')+ \lambda (x) G(x,y,y')dx$$ I understand the Euler-Lagrange equation and Lagrange ...
5
votes
1answer
2k views

Simple simple Euler Lagrange Equation

Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do .. One of the questions from my ...
5
votes
1answer
59 views

Show that $ M$ is constant on $[a,b]$ (variational calculus)

Let $F:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ be $C^2$ on $[a,b]$ and $u$ be a solution for the Euler-lagrange equations for the functional given by $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
5
votes
2answers
164 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
5
votes
1answer
394 views

Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
5
votes
1answer
95 views

Proving that this function must be even

Let $u:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function that is not identically equal to zero. Suppose further that $u$ is an odd function (ie. $u(\mathbf{x})=-u(-\mathbf{x})$). Let ...
5
votes
1answer
145 views

Smoothness of a non-local functional

While studying a nonlinear PDE arising from quantum mechanics, I met a statement that I cannot prove easily. Let us write $E=W^{1,2}(\mathbb{R}^3)$ for the usual Sobolev space, and define the ...
5
votes
2answers
382 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 $\Omega$, ...
5
votes
0answers
538 views

Maximum and minimum of an integral under integral constraints.

Find the maximum and minimum of the following integral in terms of $f(x),a,C$: \begin{align}I=\int_{0}^{a} \frac{x}{f(x)}p(x)dx \end{align} s.t.: 1) $\int_{0}^{a} p(x)dx=1$ 2) $\int_{0}^{a} ...
4
votes
1answer
206 views

Why can't we construct a counter-example to the Fundamental Lemma of the Calculus of Variations?

"The fundamental lemma of the calculus of variations states that if the definite integral of the product of a continuous function $f(x)$ and $h(x)$ is zero, for all continuous functions $h(x)$ that ...