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

learn more… | top users | synonyms

0
votes
1answer
31 views

Finding the unique weak solution of Non-linear boundary problem

We are given the equation \begin{cases} -\Delta u=0&x\in \Omega\\ \partial_\nu u+\beta(u)=0&x\in\partial\Omega \end{cases} where $\Omega$ is bounded bounded smooth boundary and $0<a\leq ...
3
votes
0answers
20 views

Calculus of Variations with discontinuous Lagrangian

Consider the classical problem of extremizing a functional of the form $$S[x] = \int_a^b L\left(t,x,\dot{x}\right)\ dt.$$ In almost all cases of consideration, the integrand $L$ is considered to be a ...
2
votes
1answer
52 views

Solve Integral equation with convolution

I have to solve the following integral equation \begin{align*} \int_{-\infty}^\infty e^{-y^2} \log \left( \int_{-\infty}^\infty e^{-(y-x-t)^2} f(t) dt\right) dy=-cx^2 \end{align*} where $c$ is some ...
2
votes
1answer
80 views

Functional Derivative (Gateaux variation) of functional with convolution

I have the following functional \begin{align*} F[f]=\int f(x) \log(g(x)) dx$ \end{align*} where $g(x)$ is given by convolution $g(x)=y(x) * f(x)=\int y(\tau) f(x-\tau) d\tau$, so \begin{align*} ...
0
votes
0answers
46 views

Reference request: calculus of variations

I am searching for a good book to self-study calculus of variations. It should be fairly complete; build up gradually from the very basics; offer detailed explanations; have some emphasis on ...
0
votes
0answers
20 views

Calculus of Variations-First and Second Order Deviations

I'm new to Calculus of Variations and the Method of Least Action (L=T-V) What I'm unsure about is how first and second order deviations are used in finding the least action. I know it's used to find ...
7
votes
3answers
267 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 ...
3
votes
1answer
86 views

Euler–Lagrange equation

Does this PDE $\nabla\cdot( \frac{ \nabla u}{u} )+a\, \Delta u+b\,u=0$ (*) have a variational structure? Here $a$ and $b$ are constants. In other words, the question I am asking is: Does there ...
4
votes
1answer
307 views

Derivation of Euler-Lagrange equation

Here is a simple (probably trivial) step in the derivation of the Euler-Lagrange equation. If we denote $Y(x) = y(x) + \epsilon \eta(x) $, I want to know why is $\dfrac{\partial ...
2
votes
2answers
63 views

Calculus of Variations. Lagrangian Hamiltonian Mechanics Mathpages.

Over at http://www.mathpages.com/home/kmath523/kmath523.htm is an article about Lagrangian and Hamiltonian Mechanics with a derivation of the Euler-Lagrange equations of motion. Mid-way through is ...
4
votes
1answer
34 views

Independence of function and its derivative in calculus of variations

It's common to see in calculus of variation that the integrand $f$ of functional $F[y]=\int f(y,y',x)dx$ is a function of $y,y'$ and $x$. Why do we regard the derivative $y'$ as an independent ...
2
votes
1answer
35 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
3
votes
0answers
111 views

A conjectural inequality of the form $\int_a^b g(B'_1(t)) dt \le \int_a^b g(B'_2(t)) dt $ with convex increasing $g$

Assume $ g:[0,\infty)\to [0,\infty) $ be strictly convex and increasing monotonic function and $B_1:[0,\infty)\to [0,\infty) $ be convex and increasing monotonic function and $B_2:[0,\infty)\to ...
0
votes
0answers
15 views

Multivariable calculus of variations (second variation)

So I understand that, if the multivariable Euler-Lagrange equations are satisfied, then the first variation is zero. But that doesn't mean we really are dealing with a minimum of the action or a ...
0
votes
0answers
16 views

Fixed vs. variable functions in functional variation

Given the functional $$F[y,y'] = \int L[y(x),y'(x),y_0(x),y_0'(x)]\ dx\ ,$$ I want to find the function $y(x)$ that extremizes $F$ for known functions $y_0(x)$ and $y_0'(x)$. Since both $y_0(x)$ and ...
1
vote
0answers
27 views

Existence of measure given the margin is a step function

Suppose $Q:[0,1]\to [0,1]$ is given by a nondecreasing step function $$Q(x)=A, if \phantom{0}0\leq x < x^*$$ $$\phantom{0000} = B, if\phantom{0} x^*\leq x\leq 1 $$ s.t. $$A,B\in[0,1] ...
1
vote
1answer
67 views

Minimization of Variational - Total Variation (TV) Deblurring

Under the Linear Blurring Model - $ f = H \ast u $. I'm trying to calculate the Euler Lagrange of with respect to $ u $ of the functional: $$ E \left( u \right) = {\left\| f - H \ast u ...
0
votes
1answer
39 views

How to find the curve extremizing a given functional?

Given a functional $$I(y)=\int_1^2 {\frac {\sqrt{1+(y'(x))^2}}{x}}dx ,$$ with $y(1)=0$ and $y(2)=1$. How to find the curve extremizing this functional?
0
votes
1answer
22 views

Smooth and Lipschitz domains

We know that an open ball $B_{r}\subseteq R^{n}$ is a smooth domain. It follows that this is a Lipschitz domain. How can I show explicitly the function $\varphi_{x}\in C^{0,1}(R^{n-1})$ that is ...
0
votes
1answer
19 views

Calculation for volume [closed]

How do I find the volume of grain in a round bin when the grain is sloped? The cylinder diameter measures 17.9 feet The top grain height measures 13.25 feet The grain at the bottom of the linear ...
4
votes
2answers
65 views

Proving the shortest function connecting two points is a straight line WITHOUT assuming the Euler-Lagrange equation

For learning purposes, I'm trying to prove that the shortest function passing through the two points $(x_1, y_1)$, $(x_2, y_2)$ is a straight line, without using the Euler-Lagrange equation. My ...
0
votes
0answers
24 views

Derivative with constraint

Consider function $F=F(q(t),p(t))$ with constraint $p(t)=q'(t)$, where $'$ denotes time derivative. Let $\displaystyle G=F'=\frac{\partial F}{\partial q}q'+\frac{\partial F}{\partial p}p'$, I want to ...
0
votes
0answers
71 views

Solving differential equation by weak formulation and minimizing a functional

I want to give a weak formulation of the boundary value problem \begin{align*} -(c(x)(u'(x)-1))' & = 0 \textrm{ on } \Omega = (-1,1) \\ u(-1) = u(1) & = 0 \end{align*} where $c(x)$ is ...
2
votes
1answer
61 views

What does this sentence regarding the Riemannian metric mean?

I am slowly working through a text on ordinary differential equations and I don't understand what this particular exercise is even asking of me. The exercise says to determine the geodesics in ...
2
votes
2answers
314 views

It is physically intuitive that the catenary is unique?

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
37 views

Solving for f(t) in presence of f'(t)

Here's the situation: I have a function $$e(t) = \frac{a~d(t)}{b + d(t)}$$ with first derivative $$e'(t) = \frac{a~b~d'(t)}{[b+d(t)]^2}$$ where $a$ and $b$ are constants. For a given constant $K$ I ...
2
votes
0answers
28 views

Doubt in the derivation of the field Euler-Lagrange equations

I'm looking at a derivation of the Euler-Lagrange equations in a field setting, and one step in the proof is continually eluding me. Let $\phi(\vec x,t)$ be a field and $\mathscr ...
5
votes
1answer
393 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 ...
0
votes
0answers
24 views

Minimum of functional

Find for which $w:\mathbb{R}^2\rightarrow \mathbb{R}$ attains following functional a minimum $$ F[w] = \int_0^1 \frac1{p(x)} \left(\int_0^1 w(x,y)f(x,y)\,dy\right)^2 \,dx + \int_0^1 ...
0
votes
0answers
25 views

Ritz Approximation

I have the following problem: $1/r * d/dr(r*d \theta/dr) - N_1(\theta-\theta_a)^2 -N_2(\theta^4-\theta_a~^4)=0$ where $N_1, N_2,$ and $\theta_a$ are constants Subject to $r=1$ : $\theta=1$ $r=L$ ...
3
votes
3answers
410 views

Procedure for Gâteaux Derivative with functionals

Not after an answer, just the method/procedure as I'm stumped... We have the functionals: $$ T[y] = \int_2^3 \left( 3\left| \frac{dy}{dx}\right|^2 - 8y \right)dx $$ $$ S[y] = \cosh(T[y]) $$ Now, to ...
0
votes
0answers
22 views

Euler Lagrange of a Curve

Let $C(s) = (x (s), y(s))$ be a closed curve inside a plane where $s$ is the parametric arc length parameter. What is Euler Lagrange equation for the following functional $$-\int_0^L \nabla C ds$$ ...
0
votes
0answers
26 views

Euler Lagrange Equations for the following minimization problem

Say I have a curve C in the plane Using Euler-Lagrange I am trying to solve the following minimization problem: $$\int \int (E - R) \nabla C dx dy$$
0
votes
0answers
22 views

second variation of a modified Dirichlet energy

Given: $E(h)=\int (\frac{1}{2}h_x^2-F(h))dx$ $f(h)=F'(h)$ and $f'(h) $is integrable $\int \phi dx=0$ (may not need this) Trying to show: $E(h+\epsilon\phi)=E(h)+\epsilon \int h_x \phi _x \, ...
0
votes
0answers
15 views

Euler Lagrange of the following equation.

I've worked the Euler -Lagrange for the following equation $$E(u(x,y),v(x,y)) = \int \int (I_x u + I_y v + I_t) + \alpha ((u_x^2 + u_y^2) + (v_x^2 + v_y^2))dx dy$$ and got the following equations: $$0 ...
0
votes
0answers
29 views

Using Euler-Lagrange to find the first variational curve of

I have been working on this optimization problem for days but I cannot figure out the right way to finish it off. I am reading from Optimization Theory by Pierre, and this is problem 3.3. Note that ...
1
vote
1answer
269 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points $A(3,9)$ and $B(-2,4)$ lie on the parabola $y=x^2.$ The line $y=x+6$ joins $A$ ...
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 ...
3
votes
1answer
77 views

Relation between Fourier components of a positive function

Here's a problem that has recently come up in my physics research: Let f be a function on [0, 2 $\pi$], which yields positive real numbers. Let the integral of $\int_0^{2\pi}f(x)= 1$. (Just for the ...
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, ...
0
votes
1answer
36 views

I want to find Euler-Lagrange equation for the given functional.

I want to find Euler-Lagrange equation for the following: $$J(u) = \int \left( \frac{\psi(x) u + \dot{u}}{\psi(x)u - \dot{u}} \right)dx, \text{where} \ \psi(x) \ \text{is an explicit function of} \ ...
1
vote
1answer
50 views

How do first integrals help you solve differential equations?

I am reading about Euler-Lagrange equations and this particular section is a little unclear. Consider the differential equation $$\begin{bmatrix} \dot{x}\\ \dot{y} \end{bmatrix} = \begin{bmatrix} ...
-1
votes
0answers
30 views

how to numerically solve a problem in calculus of variations?

I am asked to solve a problem in calculus of variations via numerical methods as below in 2 states: I have no idea what to do? can any one help me urgently? or offer me some tutorials, books, or ...
1
vote
0answers
20 views

Functional derivative of a repeated integral

For a given function $f$, the functional derivative of the functional $\mathcal{F}[\rho]=\int f(x,\rho(x))\,dx$ is well-known to be $\frac{\delta}{\delta \rho(x)}\mathcal{F}[\rho]=\frac{\partial ...
1
vote
0answers
21 views

Determine whether it's min or max of functional.

so I have such functional: $$\phi(y)=\int\limits_0^1 (y^2+2y'^2+y''^2)dx, \ \ y(0)=y(1)=0, \ y'(0)=1, \ y'(1) = -\sinh1.$$ By using Euler-Lagrange formula, I get $$y^{IV} - 2y'' + y = 0$$ After ...
0
votes
0answers
20 views

Write an equation for variations

Write an equation for variations in relation to the parameter: $$ \frac{dx}{dt}=x+ \mu y^2$$ $$\frac{dy}{dt}=x+y$$ with initial condition x(0)=1, y(0)=0 in the point $ \mu=0 $
1
vote
2answers
39 views

Find the first-variational curve which corresponds to the functional $\int_{-1}^1 t^2 \dot{x}^2 dt$ when $x(-1) = -1$ and $x(1) = 1$.

Find the first-variational curve which corresponds to the functional $$\int_{-1}^1 t^2 \dot{x}^2 dt$$ when $x(-1) = -1$ and $x(1) = 1$. Here is what I did: \begin{align} \delta J(x)(h) &= ...
1
vote
1answer
26 views

Does $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ contain corner points?

I want to know if $x^*(t) =(\frac{2 - e + e^2}{2 - 2e^2})e^t + (\frac{e - 3e^2}{2 - 2e^2})e^{-t} + \frac{1}{2}te^{-t}$ can contain corner points. This $x^*(t)$ is the solution to the differential ...
0
votes
0answers
28 views

Rayleigh-Ritz method to solve the P.D.E.

How we take an approximate solution from boundary conditions to find the solution of a partial differential equation by Rayleigh-Ritz method?
3
votes
2answers
37 views

Solution verification for finding an extremal under constraints

Find the extremal of $\int_0^1 \left[\dot{x}^2 + 2x\dot{x} + 2x\right] dt$ with $x(0)=0$, and $x(1)=\frac12$ subject to the constraint $\int_0^1 12tx dt=24$ Could anyone verify the anwer to this? I ...