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

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52
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
936 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 ...
15
votes
3answers
533 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 ...
14
votes
1answer
246 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
8answers
4k 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 ...
13
votes
1answer
343 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
1answer
329 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
1answer
283 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 ...
10
votes
3answers
515 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 ...
10
votes
4answers
312 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 ...
9
votes
2answers
332 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 ...
9
votes
1answer
146 views

Equivalence of variational inequalities

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 ...
8
votes
3answers
489 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 ...
8
votes
1answer
994 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
307 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
1answer
275 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
0answers
390 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 ...
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 ...
7
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 + ...
7
votes
3answers
317 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 ...
7
votes
1answer
186 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 ...
6
votes
3answers
157 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
313 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
1k 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}) &= ...
6
votes
2answers
106 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
246 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
1answer
176 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
61 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
0answers
250 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. ...
5
votes
1answer
638 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 ...
5
votes
1answer
262 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
565 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 ...
5
votes
1answer
56 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
1answer
90 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
136 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
350 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 $\Omega$, ...
4
votes
1answer
175 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 ...
4
votes
5answers
253 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 ...
4
votes
2answers
732 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 ...
4
votes
3answers
324 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 ...
4
votes
1answer
980 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 ...
4
votes
1answer
186 views

Assumption that $\delta q'$ is small in the derivation of Euler-Lagrange equations.

I have never completely understood the justification of this step in the derivation of the E-L equation: $\delta L = L(q + \delta q, q' + \delta q', x) - L(q, q', x) = \partial_q L \delta q + ...
4
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 ...
4
votes
1answer
186 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
2answers
34 views

How to take partial derivatives of functions whose inputs depend on the same variable?

I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me: The Euler–Lagrange equation, then, is given by ...
4
votes
2answers
159 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
4
votes
1answer
445 views

Polar coordinates, line integrals, and the Beltrami Identity

Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
4
votes
2answers
229 views

fundamental lemma for variational calculus

Is it possible to use the fundamental lemma of calculus of variations in some way in the following case: $F(x,y)$ is a locally integrable function on $\mathbb{R}^n \times \mathbb{R}^n$. We know that ...
4
votes
1answer
189 views

Divergence Theorem/ partial integration of higher order

I'm in the middle of a proof and i'm trying to understand a step of the proof which does give me a hard time. The proof is about minimal surfaces and at the moment I'm trying to understand why the ...
4
votes
1answer
217 views

Existence of a Lagrange multiplier (Euler Lagrange equations + holonomic constraints )

Let $I=[a,b]\subset \mathbb{R}, G:\mathbb{R}^n\to \mathbb{R}^k$ smooth, $0<k<n, M=G^{-1}(0)$. Assume that $DG(x)$ has full rank for all $x\in M$. Fix $p_1,p_2\in M$ and assume $u\in ...