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

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Difference between Variation of Calculus problems and Control Theory problems?

Variation of Calculus seems to have problems without the control with variables such as state and time. Then again Control Theory problems seems to have problems with one extra variable that is ...
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25 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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27 views

Help with Euler Equations!

I'm having real difficulty and can't find this anywhere! Does anyone know how to solve it? Thanks!
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25 views

How are Hamilton function and Hamilton-Jacobian-Bellman function related to each other?

I am trying to understand the solution to the problem 2 here that uses the Hamilton conditions for the HJB. I don't understand how the Hamilton conditions can be used with the ...
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0answers
23 views

Euler Lagrange equations

I need to minimise $$\int\limits_\Omega|\nabla H_\epsilon(\phi)|\,dx\,dy$$ with respect to $\phi$. Where $H_\epsilon$ is the regularised Heaviside function, so that it is differentiable. This can be ...
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34 views

Non-convexity of an energy functional

How would I go about showing that the following Mumford Shah functional is not convex? $$E_{MS}(u,C)= \int_{\Omega} |u_{0}(x,y) -u(x,y)|^{2}\ dx\ dy + \mu \int_{\Omega \backslash C}|\nabla ...
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27 views

Non-standard applications of Noether's theorem

The particular "Noether's theorem" that I'm referring to is the one that appears in the calculus of variations: if the Lagrangian in a variational problem is invariant under a one parameter group of ...
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22 views

I need help understanding a proof about hamiltonian formulation of euler-lagrange equation

In Dacorogna's book about calculus of variations there is a proof of a lemma marked 2.8 about the hamiltonian defined as a Legendre transform of a C2 function (see link for the lemma's statement and ...
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1answer
27 views

Using the Euler-Lagrange equation to evaluate an integral

I think this is a very basic question. I'm just not sure how to use $\frac{\mathrm{d}}{\mathrm{d}x}\left(\frac{\partial F}{\partial y'}\right)=\frac{\partial F}{\partial y}$ to find the general ...
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1answer
20 views

minimizing a function involving exponential term

Let $w\ge e$ . I want the following $$ \min_{r\geq0} r(e^r-w) $$ Is there any way to find it. Thanks.
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3answers
54 views

Finding Survival Function given hazard rate

Let X be a random variable defined for 0 < x < 4 with hazard rate $$lambda(t)=1/(4-t)$$ for 0 <= t <= 4. find the survival function, S(x) = P(X>x). Using the formula S(x) e^-integral from ...
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1answer
25 views

Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
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2answers
46 views

Isoperimetric inequality on the sphere via calculus of variations

The isoperimetric inequality on the sphere of radius 1 asserts that for any closed curve on the sphere, $$L^2 \geq A(4\pi - A)$$ where $L$ is the length of the curve and $A$ is the area it encloses. ...
2
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2answers
44 views

Are all critical points of energy geodesics?

Let $\gamma$ be a smooth curve in a Riemannian manifold and consider the arclength functional $L(\gamma) = \int_a^b |\gamma'(t)|\, dt$ and the energy functional $E(\gamma) = \frac{1}{2}\int_a^b ...
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1answer
78 views

DuBois-Reymond Lemma

I know thats the following statement is true. $f,g$ are continuous function $[a,b]$.Suppose $\int\limits_a^bf(t)h(t)+g(t)h'(t) \, dt=0$ for every $h$ belonging to $C_0^{\infty}[a,b]$, then $g$ is ...
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0answers
28 views

Integrating a Functional

Reading the quote the so called "Feynmann path integral", which, as far as I understand, means "integrating" a functional (action) on some infinite-dimentional space of configurations (fields) of ...
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59 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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10 views

How does invariance of $q$ wrt $\lambda$ for a stationary functional, restrict the function?

Suppose I have the following functional: $$S(q) = \int_{b}^{a}L(t, q(t), q'(t))dt$$ and $q(t) = x(t) + \lambda$, where $\lambda$ is a constant independent of t. If $S(q)$ is stationary for a ...
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1answer
35 views

An application of the mountain pass lemma

I am trying to show the existence of classical solution for the following problem using the mountain pass theorem : $$ \left\{ \begin{array}{ccccccc} u^{''} + \lambda u + u³ = 0 (0<t<\pi)\\ ...
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0answers
35 views

Semilinear Poisson Equation Using Direct Method of Calculus of Variations

The following problem comes from: http://people.physics.anu.edu.au/~gvn105/analyticMethPDE.pdf 12.9 Exercises 12.3: Let $\Omega$ be a bounded domain in the plane with smooth boundary. Let $f$ be ...
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0answers
29 views

Case C: Euler's equation in Simmon's textbook

Working through Simmons' Differential Equations with Applications and Historical Notes and we're stuck in Case C, page 360. Case C: If x is missing from the function $f(x,y,y')$, then Euler's ...
2
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1answer
43 views

bilinear continuous, coercive form

Let $k\in \mathbb{R}, k\neq 1$, consider the space $$ V = \{u\in H^1(0,1): u(0) = ku(1)\}$$ Let $$a(u,v) = \int_0^1 (u'v'+ uv)\; dx - \left(\int_0^1 u\; dx\right) \left(\int_0^1 v\; ...
2
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1answer
40 views

Differential of Lagrangian

My professor wrote this $\frac{\partial L}{\partial q}\dot{q}=\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}})$. Due to the fact that I am very very very very bad at Math, could you explain me about ...
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1answer
30 views

Euler Lagrange Theorem doubt

When applying the euler lagrange equation, if we ever obtain $\partial$y/$\partial$y' where y' is dy/dt why do we take it to be zero?
3
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2answers
63 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
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38 views

Existence of a Minimizer $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|} $ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
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0answers
27 views

Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
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0answers
31 views

Image histogram equalization using variational calculus

In an image processing course at Coursera.org, on the section on PDE and calculus of variations, the professor gave the following as the functional to be optimized for image histogram modification: ...
11
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1answer
334 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 ...
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1answer
14 views

about lower semicontinuous functional

Let $X$ a topological space.My book define : A functional $\varphi: X \rightarrow R$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a, + \infty)$ is open in $X$ for any $a \in R.$ (1) And the book ...
3
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1answer
168 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
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1answer
56 views

Question about variational principles involving light rays.

In question 9 (see this link: http://view.samurajdata.se/psview.php?id=28b2e4b5&page=1 ), I've shown the light rays are follow a parabolic paths using the Euler-Lagrange equation and Fermat's ...
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1answer
26 views

Is this differentiation correct?

$J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx = \int_1^2{f(y,y^\prime,x)}$ Need to find $\frac{d}{dx}(\frac{\partial f}{\partial y^\prime})$ $\frac{\partial f}{\partial y^\prime}=x+2y'(x)$ ...
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0answers
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Converse of Noether's (first) theorem

Noether's (first) theorem states that if a Lagrangian $L$ admits a continuous symmetry, then the following quantity are conserved. $$\left(\frac{\partial L}{\partial \dot q}\cdot\dot ...
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1answer
38 views

Elliptic partial differential equations

Consider the following elliptic PDE: $$ \Delta u=f(u), $$ where $f(u)$ is a smooth function. Which references (books, papers,...etc.) about existence of solutions for this PDE do you recommend to have ...
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1answer
41 views

How to take derivative?

Find a curve passing through (0,0) and (1,1) that is an extremal of the function $${\rm J}\left(x,y,y'\right)= \int_{0}^{1}\left[ y'^{\,2}\left(x\right ) + 12\,x\,{\rm y}\left(x\right)\right]\,{\rm ...
2
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2answers
98 views

Function extremal - calculus of variations

Find a curve passing through (1,2) and (2,4) that is an extremal of the function: $J(x,y')=\int_1^2 xy'(x)+(y'(x))^2dx$ I don't know what methods to use at all.
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0answers
56 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 ...
0
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1answer
27 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
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0answers
34 views

Sufficient conditions for the use of the Beltrami identity

For reference, I shall use the notation used in the wikipedia article for the Beltrami identity in the application section (http://en.wikipedia.org/wiki/Beltrami_identity) In the article, the ...
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0answers
88 views

Calculating the maximum of a function

How can one determine $$\max_{f_0,f_1}\frac{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\log\left(\frac{f_1(y)}{f_0(y)}\right)\mbox{d}y}{\int_{\mathbb{R}}f_1(y)^xf_0(y)^{1-x}\mbox{d}y}$$ given ...
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2answers
76 views

Calculus of Variations

In the Calculus of Variations there is a passage from Euler's characteristic equation: $$ \frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0 $$ in ...
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What methods are available for this optimization problem?

I have an intermediate knowledge of the calculus of variations: I can handle constraints in functional or integral forms and extrapolate to multiple variables and functions. If I dig in my notebooks I ...
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32 views

The Volterra derivative of a functional with an argument being a second order derivative.

An exercise demands that we calculate the Volterra derivative of a functional $\Phi(x)(t)$ with respect to argument $x(t)$: $\Phi (x)(t) = x''(t) + \int_a^b \int_a^b K(t,s_1,s_2,x(s_1),x(s_2)) ~dx $, ...
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0answers
20 views

Spectrum of operator with unknown function

I'm working on a variational problem in elasticity which involves a hefty number of Lagrange multipliers. I have calculated the second variation to be [; \int ds ~h \left[ \frac{d^4 }{ds^4 } + ...
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63 views

Stationary action functional

The last part in the derivation of the Euler-Lagrange equations for a stationary action has me confused. It's about the order of differentiation and evaluation, and whichever comes first. I'll ...
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2answers
31 views

Reading Speed for Constant Time to Finish

You open a very long new book on your e-reader and read a few pages. It helpfully informs you that based on your reading speed you have 16 hours of reading left until you are done. You read the rest ...
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1answer
40 views

Counterexample for the Chain rule for the Gateaux-derivative

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state: but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined ...
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1answer
53 views

Calculus of variation: Reduce the order of a differential equation using a 1 parameter lie group adfmitted by it.

We are asked to reduce $y^"+y-y^{-3}=0$ using $X= \sin2x\frac{\partial}{\partial x}+y\cos2x\frac{\partial}{\partial y}$ I know we have to find the first prolongation of X and solve $X^1$F=0 using ...
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86 views

Euler-Lagrange Calculus of Variations Example

I have been working on solving Euler-Lagrange Equation problems in attempts to learn Calculus of Variations, but this one example has me stuck. I am probably making mistakes in my integration. I am ...