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

learn more… | top users | synonyms

3
votes
0answers
92 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
4
votes
2answers
262 views

Optimization problem using Reproducing Kernel Hilbert Spaces

I am encountering a problem concerning Reproducing Kernel Hilbert Spaces (RKHS) in the context of machine learning using Support Vector Machines (SVMs). With refernce to this paper [Olivier Chapelle, ...
0
votes
0answers
15 views

weak local minima sufficient conditions

I have the functional $F(u)=\int^b_a f(x,u(x),u'(x)) dx$ with $u \in C^1([a,b])$ such that $u(a)=\alpha,u(b)=\beta$ and I'm trying to prove the following fact if $\delta F(u_0,v)=0$ and $\exists ...
1
vote
0answers
25 views

Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
0
votes
0answers
270 views

Geodesics of a cone

To find Geodesics on a cone I used the cylindial coordinates $x=rcos\theta$ $y=rsin\theta$ $z=z$ Is this parameterization correct.How can I know how to parameterize? Then arc length $ds^2=r^2 ...
1
vote
1answer
46 views

minimal surface of revolution integrating Euler Lagrange result

In finding minimal surface of revolution after applying to euler lagrange equation $d u \over dx \sqrt{1+(u')^2}$$=0$. Then $ u \over \sqrt{1+(u')^2}$$=constant$. Then solving $\int{ c \over ...
2
votes
1answer
60 views

Proof of Euler Lagrange equation in calculus of variation

I am learning calculus of variations from the article page 9 There it says $J[u]=\int_a^bL(x,u,u')\, dx$ and $u'$ is represented by $p$. $$\langle\nabla J[u],v\rangle={dJ[u+tv] \over dt} \text{ ...
2
votes
2answers
67 views

Doubt on calculus of variation

In wiki http://en.wikipedia.org/wiki/Calculus_of_variations#Example, the first example of calculus of variation is the minimize distance between 2 points. In my understanding, value of functional $J$ ...
0
votes
1answer
18 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
1
vote
1answer
61 views

Variational Methods, why KL divergence is the difference between true distribution and approximating distribution.

Likelihood = $L(\textbf{w}) = P(V\mid \textbf{w})$. $$\ln P(V\mid \textbf{w}) = \ln \sum_H P(H,V\mid \textbf{w})$$ $$= \ln \sum_H Q(H\mid V)\frac{P(H,V\mid \textbf{w})}{Q(H\mid V)}$$ $$\geq ...
0
votes
0answers
127 views

Integration of the Euler-Lagrange equation with explicit dependence on x

In the variational calculus, if the Lagrangian $L[y(x),y'(x)]$ is not an explicit function of $x$, the Euler-Lagrange equation takes on the following form: $$\frac{d}{dx}\left(L-y'\frac{\partial ...
0
votes
1answer
75 views

Integrating the Euler-Lagrange equation

Let us have a Lagrangian $L(y,y') = f(y)\sqrt{1+y'^2}$, where $y=y(x)$. The corresponding Euler-Lagrange equation is $$\frac{f'}{\sqrt{1+y'^2}} - f\frac{y''}{(1+y'^2)^{3/2}}=0$$ This expression should ...
0
votes
1answer
75 views

Minimizing cost function (Eikonal)

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

Equivalent norm in Sobolev space

Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional $$ I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2. $$ I'm asking if it is equivalent to the norm $$ ...
1
vote
0answers
28 views

Minimization Problem for Winding number

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

Quadratic Minimization

Consider a functional $I\colon H \to R$ on $H$ Banach space, sufficiently regular. Is in generally true that $$ \inf_{\rho \in H}{I^2(\rho)}=\Big(\inf_{\rho \in H}{I(\rho)} \Big)^2 \quad ? $$ If ...
1
vote
1answer
46 views

Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
0
votes
0answers
32 views

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 ...
1
vote
0answers
60 views

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 ...
3
votes
1answer
37 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 ...
0
votes
0answers
18 views

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 = ...
1
vote
0answers
16 views

Minimization Problem and Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
0
votes
1answer
67 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 ...
4
votes
1answer
77 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 ...
0
votes
0answers
48 views

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 ...
2
votes
0answers
49 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 $$ \iiint_V (u_x^2+u_y^2+u_z^2) \, dx \, dy \, dz $$ ...
3
votes
0answers
48 views

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 ...
1
vote
0answers
22 views

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 ...
4
votes
1answer
97 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 ...
1
vote
0answers
28 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 $x \in H$ ...
0
votes
0answers
30 views

Help with Euler Equations!

I'm having real difficulty and can't find this anywhere! Does anyone know how to solve it? Thanks!
1
vote
0answers
32 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 ...
1
vote
0answers
29 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 ...
1
vote
0answers
35 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 ...
1
vote
0answers
29 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 ...
0
votes
0answers
28 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 ...
0
votes
1answer
41 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 ...
1
vote
1answer
22 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.
1
vote
3answers
70 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 ...
0
votes
1answer
27 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 ...
1
vote
2answers
97 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
votes
2answers
55 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 ...
0
votes
1answer
176 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 ...
1
vote
0answers
37 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 ...
0
votes
0answers
82 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 ...
0
votes
0answers
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 ...
0
votes
1answer
41 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)\\ ...
1
vote
0answers
52 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 ...
0
votes
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
votes
1answer
48 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\; ...