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

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Isoperimetric Euler Equation When Objective Does Not Depend on Derivitive

I'm solving the optimization problem (simplification of my actual problem) $$ \max \int_{0}^{1} b(t) \left(t - \frac{b(t)}{2}\right)dt \\ \text{st} \\ \int_0^1 \frac{b^2(t)}{2}dt = B $$ Without the ...
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Brachistochrone Problem to find out the path by which a bead travels in least time

The question is to find the shape of the curve down whcih a bead sliding from rest and accelerated by gravity will slip(without friction) from one point to another in the least time. So I proceeded ...
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34 views

how to use variational principle to find the best value for parameter $\lambda$?

I need to minimize the following integral by varying parameter $\lambda$: $$\int_0^\infty(f(x)-g(x,\lambda))^2dx$$ The functions $f(x)$ and $g(x,\lambda)$ are known and they satisfy ...
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Double standards on recognizing expression as functional

The Calculus of Variations starts with a definition of functional Such an expression, the argument of which is a function, is called a functional. Particularly, they say that $J = ...
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Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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63 views

Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
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62 views

Shortest distance between two points calculus of variation

Ok this problem might be trivial but when solving it using calculus of variations it's not so stupid. Suppose we have a fixed boundary condition $f(a) = f(b) = 0$ and we want to find the shortest ...
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First variation of convolution of two nonlinear functions, how to reexpress $\left[x \delta x * x^2 \right]$?

A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ...
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33 views

Calculus of Variation - Question with Integral constraint

I'm stuck on this question, Let $V$ be an open ball in $\mathbb R^3$ in $ { x,y,z \text{ st }x^2 +y^2 +z^2<1}$ I need to minimise the integral $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
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Maximize an integral

I have the following integral to maximize but I don't know what to do with this $f(b,y)$ in the first integrand: $$ F=\intop_{a}^{b}[f(b,y)+\intop_{a}^{b}f(x,y)dx]dy $$ a and b are constants. Do I ...
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Finding extremals over y in two ways

We can find the extremal of $$\int_0^1(\frac{1}{2}y'^2+yy'+y'+y)dx$$ amongst all y with $y(0)=1$ by imposing the natural boundary condition $\frac{\partial F}{\partial x}=y'+y+1=0$ at $x=1$.Solving ...
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75 views

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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26 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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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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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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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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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
31 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
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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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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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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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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. ...
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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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96 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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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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62 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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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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36 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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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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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 ...
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1answer
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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\; ...
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1answer
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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
31 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?
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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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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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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: ...
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344 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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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 ...
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1answer
170 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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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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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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41 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
42 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 ...
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2answers
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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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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 ...
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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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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 ...