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

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Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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1answer
155 views

Who came up with the Euler-Lagrange equation first?

Could someone explain who came up with the specific equation first? http://en.wikipedia.org/wiki/Euler-Lagrange makes it sound like Lagrange got it first, in 1755, then sent it to Euler. but: ...
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26 views

Problem calculating the average power of a vector?

I am calculating the average power of a vector. I would like to compare the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. Thank ...
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45 views

Application of a general “Weierstrass theorem”

http://books.google.at/books?id=9OSrV73a40gC&pg=PA45&lpg=PA45 gives a general Weierstrass theorem. Are there notable applications of this theorem, say in the calculus of variations? (I could ...
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1answer
338 views

Euler-Lagrange Equation example

I have been working on solving Euler-Lagrange Equation problems in differential equations, specifically in Calculus of Variations, but this one example has me stuck. I am probably making mistakes in ...
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55 views

How to prove $\gamma$ is continuous?

In the paper A remark on least energy solutions in $\mathbb{R}^N$, page 2407, it said, if $u_0\in H^1(\mathbb{R}^2)$, set $\gamma(t)=t^{-1/4}u_0(x/t)$. Then $\gamma(t)$ is a continuous path in ...
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35 views

Newton's method for the brachistochrone

Consider the potential $V(x,y)=-y$ and a particle at rest in the beginning of the coordinate system. We are going to examine the brachistochrone - the smooth curve of fastest descent. Assume we are ...
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43 views

Variational Principles: Lagrange Multipliers

I am trying to minimize the functional $$I[\textbf{x}] = \int ||\dot{\textbf{x}}||^2 dt$$ subject to the constraint $\textbf{x}(t) \in \{\textbf{s} \in \mathbb{R}^3 : ||\textbf{s}|| = 1 \}$. The ...
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120 views

Question on Moment of inertia about center of mass of a smooth plane curve.

This question is in continuation to this and motivated to answer this question. If I have an answer to it, then I can prove a special case of this question. Consider two smooth plane curves $C \equiv ...
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52 views

Does calculus of variations have a close connection to Feynman's ''differentiation under the integral sign''?

Most of the calculus I've studied seems separate math problems in to "derivative" or differential applications and integral applications. The one exception seems to be "calculus of variations," which ...
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231 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, ...
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34 views

Functional Extremum

Let a functional $H[\phi]$ of a map $\phi\in\mathbb{R}^{\mathbb{R}^4}$ be given by: $$ H(x^0) = \int_{\mathbb{R}^3} ...
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1answer
242 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 ...
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32 views

Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * ...
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Perturbation of the boundary of a strictly stable minimal surface

Let $\Sigma \subseteq \mathbb{R}^3$ be a minimal surface with boundary $\Gamma$. Now let us assume that $\Sigma$ is strictly stable, that is, $\lambda_1(\Sigma,L) >0$, where $L$ is the stability ...
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412 views

Find the extremal to the functional and discuss whether they provide a max/min

I am having a hard time getting my head around Functionals and Calculus of Variations, My question is: Given a functional and using the Euler-Lagrange equation to find an extremal, how do we show ...
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1answer
68 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
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298 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. ...
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27 views

Why $\widetilde m = argmin_{m \in \mathcal P(X)} E[m]$ implies $\widetilde m (\arg \min \frac{\delta E}{\delta m}[\widetilde m]) =1$?

Consider $E: \mathcal P (X) \rightarrow \mathbb R \cup \{ \infty \}$ a functional (with a convex and dense domain, $E< +\infty$) over $\mathcal P(X)$ the set of probability measures of a metric ...
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1answer
55 views

Variation of the determinant of a Jacobian

I am following a derivation in a Calculus of Variation problem. After introducing a one-parameter family of one-to-one mappings from $R^{2}$ to itself, $$z({x},\epsilon)$$, $x = (x_1,x_2)$, such that ...
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Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$ \mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}. $$ Prove that ...
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1answer
39 views

The curve of shortest length bounding given area

Is formula #7 in this MIT OCW course incorrect? I think it should be $$f(x)=(\sqrt{1-(mx-c)^2}+d)/m$$ Also, presumably this answer to a very similar problem is also wrong. Because this is not the ...
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3answers
373 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 ...
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1answer
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Spherical rearrangement

Let $u\colon\Omega\subset\mathbb{R}^N\to\mathbb{R}$ be a non negative measurable function, and $\Omega$ open and bounded. Consider $u^*$ the spherical rearrangement $$ u^*(x)=\sup\{t\geq0 : \mu\{x: ...
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1answer
42 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{ ...
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64 views

Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
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38 views

Regularity of a Weak Solution

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 ...
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10 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 ...
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1answer
28 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 ...
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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 ...
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24 views

Path followed by a light ray-calculus of variation

Find the path followed by a light ray if the index of refraction (in polar coordinates ) is proportional to $r^{-2}$. In the given answer in the book it says we want to make stationary $\int nds$ or ...
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25 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 ...
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2answers
55 views

Minimizing a Functional with a Path Length Constraint

Say you have some functional of the form $\int_0^{t_f} L(x,\dot{x},y,\dot{y},z,\dot{z}) dt$ that you're trying to minimize. Normally one can solve this using the Euler-Lagrange equations, and when you ...
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2answers
59 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 $J$ ...
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1answer
17 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 ...
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1answer
49 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 ...
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1answer
49 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 ...
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1answer
60 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 ...
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44 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 ...
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2answers
52 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 $$ ...
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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 ...
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Minimization Problem for Winding number

Consider the minimization problem associated to the functional $$ \mathcal{F}(u)=\int_{0}^{2\pi}{\lvert \dot u\rvert\bigg(1+\bigg(\frac{\dot u}{\lvert \dot u\rvert}\cdot m(u)\bigg)^2\bigg),dx} $$ ...
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1answer
44 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 ...
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25 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 ...
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1answer
32 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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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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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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1answer
60 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
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1answer
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 ...