Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on an infinite dimensional spaces.

learn more… | top users | synonyms

0
votes
0answers
40 views

How to obtain the closed form solution to this nonlinear system of ODEs

I have the following simple but nonlinear ODE system problem below: $$\left\{ \begin{align} &f'(t)=a\cdot (1-f(t)-g(t))^{13/10}\tag{1}\\ &g'(t)=b \cdot f(t) \tag{2}\\ &f(0)=0,\quad ...
1
vote
1answer
49 views

test function and boundary condition

For example, if we consider the Dirichlet energy $\int\frac 12 |\nabla u|^2$ and the solution space as follows: $$X=\{u\in W^{1,2}(\Omega) \text{ | } u = 0 \text{ on } \partial\Omega \}$$ , then the ...
2
votes
0answers
19 views

Does this mean a new boundary condition for ODE and how to handle it?

In order to to calculate desired light path in continuous medium with gradient refraction index, for schematic see the Figure below. $O:(0,0)$ is the disk center of light source $\odot{O}$ with ...
1
vote
0answers
66 views

Extreme of functional - Calculus of variations: Euler-Langange equation

Find all extremes of $$I[y] = y(0)^2 \int_{0}^{1} y(y')^2 dx,$$ with the given initial condition $y(1)=0$. I was thinking of writing $y(0)^2 = -\int_{0}^{1}2yy' dx$ and then what (differentiating ...
5
votes
2answers
189 views

Book on applied mathematics/analysis

My Applied Mathematics course covers these subjects: -Calculus of Variations -Laplace Transform -Fourier Analysis -Special Functions -Integral Equations And as an introduction to the subject it has ...
2
votes
1answer
40 views

Find the function that minimizes $\int_{0}^{1}e^{-(y'-x)}+(1+y)y'dx$

Suppose among all the continuously differentiable functions $y(x), x\in \mathbb{R}$, with $y(0)=0$ and $y(1)=\frac{1}{2},$ the function $y_0(x)$ minimizes the functional, ...
1
vote
2answers
136 views

Calculus of Variations Problem involving mixed constraints

Motivation Let $X$ be $\mathcal{N}\Big(-\frac{\sigma^2}{2},\sigma^2\Big)$ random variable, i.e. probability density function $f(x)$ is given by \begin{equation} f(x)=\frac{1}{ \sqrt{2\pi\sigma^2} } ...
0
votes
0answers
13 views

Formulation of boundary constrained minimal surface

Using standard notation of classical surface theory how is the standard Plateau problem formulated as an iso-perimetric one minimizing area for given boundary length $$ \int \sqrt{ E \, du^2 + 2 F ...
1
vote
2answers
124 views

Calculus of variation with inequality constraints

I want to find the function $y$ which maximizes the functional $J[y] = \int_0^1 g(x) y(x) dx$ subject to $0 \leq y(x) \leq 1$ for all $x\in [0,1]$ and $\int_0^1 y(x) dx = k$ where $g$ is a strictly ...
1
vote
1answer
45 views

Shortest path to the apex of a cone

This is something I thought about today but have no idea how to approach. We are given a right circular cone with lateral length L and angle at the base $\alpha$. A curve along the surface of the ...
1
vote
2answers
54 views

Which advanced mathematics book do you recommend for a college starter?

I'm supposed to start my preparation for the college admission math exam so I'm looking for a good explanatory/textbook/problem book which covers most of or all of precalculus topics with a hint of ...
1
vote
1answer
27 views

Any critical point $u_0\in M$ of $I|_M$ satisfies $I'(u_0)=\mu\gamma'(u_0)$

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}u_r^2-\xi u^2+\ln(1+u^2)\right\}r\,dr,$$ $\xi\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in ...
0
votes
0answers
20 views

Calculus of variation: Maximize fuctional

"Among the curves of given length l, at the upper half-plane, passing through the points (-a, 0) and (a, 0) , find the one that encloses the largest surface area together with the space [-a,a] ." I ...
1
vote
0answers
19 views

Palais-Smale condition for a functional

Consider $I:H\rightarrow\mathbb{R}$ defined by $$I(u)=\int_0^R\left\{\dfrac{1}{2}ru_r^2-\gamma ru^2+r\ln(1+u^2)\right\}dr,$$ $\gamma\in(0,1)$, where $H$ is the completion of $$X=\left\{u\in ...
0
votes
0answers
8 views

Variation of a d'alambertian operator

Let $M$ be a pseudo Riemannian manyfold, $H$ be function of a scalar curvature $R$. Assume that variation of the metric tensor and it's first derivatives is zero on the border $\partial M$. Which ...
2
votes
0answers
48 views

Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf. Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} ...
4
votes
1answer
139 views

Obtaining an estimate for the Lagrangian $L=H^*$ from the Hamiltonian $H$

This is from C. Evans' PDE book, page 130. The convex function $H:\mathbb{R}^n\to\mathbb{R}$ is $C^2$ and satisfies $$ H\big(\frac{p_1+p_2}{2}\big) \leq \frac{1}{2}H(p_1) + \frac{1}{2}H(p_2) - ...
2
votes
2answers
48 views

How to verify this identity?

From Weinstock, "Calculus of Variations", p.24: We have the readily verifiable identity \begin{align}\frac{d}{dx}\left(y'\frac{\partial f}{\partial y'}-f\right) = ...
1
vote
2answers
52 views

Does Constant factor rule in integration hold for functionals?

The constant factor rule in integration states the following relation is valid $$\int a f(x)dx=a \int f(x) dx$$ for all constants $a$(or $a$ that are constant functions of x, that is ...
2
votes
2answers
90 views

Why does $\mathrm{tr}(\mathrm{ln}g_{\mu\nu})$ vary as $g^{\mu\nu}\cdot\delta g_{\mu\nu}$ under $\delta g_{\mu\nu}$?

For a pseudo-Riemannian manifold, under the variation $g_{\mu\nu}\mapsto g_{\mu\nu}+\delta g_{\mu\nu}$, the determinant $g=\mathrm{det}g_{\mu\nu}$ varies as $$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}$$ ...
0
votes
1answer
43 views

Euler-Lagrange equation of energy of length function on Riemann manifold

$(M,g)$ is a Riemann manifold. $\gamma:[0,1]\rightarrow M$ is a curve.The length of $~\gamma $ is $$ L(\gamma)=\int^1_0 ||\dot\gamma (t)||_g ~dt $$ The energy is $$ E(\gamma)=\frac{1}{2}\int^1_0 ...
0
votes
1answer
46 views

Variation under constraint

I always can't compute right.$u=u(x),R=R(x)$ and $\tau$ is constant, and $M$ is compact manifold.If $u$ is the minimizer of $$ \inf\{\int_M [\tau(4|\nabla u|^2+Ru^2)-u^2\ln ...
1
vote
1answer
36 views

Variation with constraint condition

For example, let $$ I[u]=\int_M |\nabla u|^2+u dV $$ It's not hard to compute the variation of $I[u]$. If $I[u]$ reach the minimum at $u_0$, I can get that $$ i(t)=I[u_0+tv] \\ i'(0)=0 \\ ...
1
vote
0answers
50 views

Maximization of the Expectation of a function

Recently I was thinking in general on how to maximize the expectation of a function (not necessarily a utility function, but apparently this is a common case). To give an idea of the kind of problem, ...
1
vote
0answers
38 views

Everything about Legendre transform

The Legendre transform, or transformation, seems to have many properties which are useful in different fields. For example: It switches between Lagrangian and Hamiltonian formalism in mechanics / ...
2
votes
1answer
22 views

Movable end points Calculus of Variation.

Given problem is $J[y] =\int_{0}^{x_1}y'^2dx$ with $y(0)=0$ and $y(x_1)=-x_1-1$. After solving Euler Lagrange equation I got $y=Ax+B$ . And using first boundry conditon I got $y=Ax$ We have ...
10
votes
1answer
163 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
0
votes
1answer
34 views

Finding minimal area of a cylindrically symmetric surface

I am told that I have a cylindrically symmetric surface that is bounded between two circles $r=a$ at $z=\pm b$. I'm meant to use the Euler-Lagrange equation, so I'm trying to a functional for the ...
6
votes
1answer
65 views

a continuous path between two sobolev functions without increasing energy

This question has been post on MO a week ago. I move it here to get more luck. Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Let $u_1$, $u_2\in H^{1}(\Omega)$ such that ...
2
votes
3answers
39 views

Find the double integral by changing to polar coordinates [duplicate]

So I have the following double integral $$\int_{-2}^{0} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} dydx$$ If I integrate with respect to y first I get: ...
2
votes
0answers
40 views

Functional integration and Feynman path integrals in wolfram alpha

Is it possible to do Feynman path integrals in wolfram alpha? Say for a free quantum mechanical particle. The reason I am interested in this is because I would like to see how it arrives at the ...
2
votes
1answer
80 views

A function $u$ that grows from $0$ to $1$ with small integral of $|u'|^2+(1-u)^2$

I am trying to prove that there exists a function $u$ satisfies the following conditions: Here $\eta>0$ small is a given constant. $u$ is defined on $[0,T]$ where $T$ might depends on $\eta$. ...
0
votes
0answers
16 views

Variational Inequality of a reduced cost functional

Given a reduced cost functional $g(u) = \frac{1}{2}\|CSu - y_d\|^2_H + \frac{\lambda}{2}\|u\|^2_U$, where $C$ is an observation, linear map: $\mathrm{Dom}(C) \to H$ ($H$ is a Hilbert space), and $S: ...
0
votes
0answers
23 views

Brachistochrone problem with extra derivative condition

I was given for HW in classical mechanics the brachistochrone problem but they also mention their that besides the given end points they demand that at the final point: $y'(x_f)=0$. Now, since I ...
1
vote
1answer
110 views

Alternative to Arnold's mathematical methods

I have difficulties understanding Arnold's book of mathematical methods of classical mechanics. Yet I should get some familiarity with the subjects found at chapters 3,4,7,8 before next semester to ...
1
vote
0answers
51 views

Maximum over Probabilistic Distribution Functions Space

Suppose $P$ is the set of functions where $p\in P: R^{+2}\to R^+$ and $p(t,s)$ is differentiable in $t$. $\forall t, p(t,\cdot)$ is a probability distribution on the positive axis $s\in [0,\infty)$, ...
1
vote
1answer
44 views

Difficulty in solving calculus of variations problem.

I am solving a problem on calculus of variation in which $F(x,y,y')$ is given as $F(x,y,y')=e^yy'^2$ After solving Euler equation I got this $2y'' +2y'-y'^2=0$. I don't know how to proceed further. ...
1
vote
0answers
45 views

textbook on calculus of variation which focuses on the following topics

I need a textbook (or set of online lecture notes) on calculus of variation which focuses on the following topics "Variation of a functional, Euler-Lagrange equation, Necessary and sufficient ...
1
vote
1answer
29 views

Calculate $\Delta J$ for a functional

$J(y)=\int_{0}^{1} (x^2-y^2+(y')^2)dx$ $y(x)=x, h(x)=x^2$ I need to calculate $\Delta J$ and I am given this from the answer key: $\Delta J = J(y + \epsilon h)-J(y) = J(x + \epsilon x^2) - J(x)$ ...
1
vote
1answer
37 views

When do minimizers exist?

I'm trying to solve a problem set for my functional analysis course and I'm stuck at the following problem: Decide if the following problem has a minimizer Let $g\in C^0([0,1])$. Minimize ...
1
vote
0answers
31 views

Is this correct use of Lagrange multipliers?

Given the standard isoperimetric problem: Minimize the functional $$ A[y]=\int_a^bF(y,y',x)dx $$ subject to the functional-constraint $$ B[y]=\int_a^bG(y,y',x)dx = c=\text{constant}$$ (where $ ...
2
votes
2answers
26 views

Continuous representative for functions in $W^{1,2}(\mathbb{R})$

I want to prove that $K(x,y) = \frac{1}{2}e^{-|x-y|}$ is a reproducing kernel for $W^{1,2}(\mathbb{R})$ and as a hint I have given that for $f\in W^{1,2}(\mathbb{R})$ I should use its continuous ...
1
vote
1answer
89 views

Feynman problem on action

It is very weird for me that a newbie can ask a new (may be silly, sorry...) question but must have 50 reputation to comment. When I see a good question like this but have no answer what I have to ...
1
vote
0answers
32 views

Weak derivative of $\int_0^x g(t)\,dt$ is $g$ on $(0,1)$

I'm working on a functional-analysis problem set, and the question is: Let $g\in L^1(0,1)$, and define $f(x) = \int_0^x g(t)\,dt$. Show that $f\in W^{1,1}(0,1)$ and that the weak derivative of $f$ ...
1
vote
1answer
81 views

Minimization of Expected Value

I'd like to know how I can minimize, with respect to $\hat{y}(x)$, $$ \DeclareMathOperator{\Tr}{Tr} \mathbb{E}_{p(x,y)}[(\hat{y}(x)-y)^2 + (\hat{y}(x)-y)\Tr(\nabla^2_x\hat{y}(x)) + ...
0
votes
1answer
30 views

Total variation of a vector valued measure

If I have understood correctly, a vector valued measure $\mu$ is simply a vector of measures, that is $\mu=(\mu_1,\dots,\mu_n)$, where $\mu_i$ is a possibly signed measure on the measure space ...
0
votes
1answer
52 views

Examining $\inf \int_0^1(x')^{2}(t)\mathrm dt$

I need to examine $$\inf \int_0^1(x')^{2}(t)\mathrm dt$$ with such conditions: \begin{cases} x(0)=0, x(1)=1 & \\ \int_0^1x(t)dt=0 & \end{cases} I started with writing Euler-Lagrange ...
0
votes
0answers
88 views

Maximization of a complex function with real outputs

Let $a$ and $b$ be functions from $\mathbb{C}$ to $\mathbb{R}$. Suppose that for some real $\alpha$ and $\beta$, there exist complex $z$ and $w$ such that $a(z) = \alpha$ and $b(w) = \beta$. Given ...
0
votes
1answer
40 views

How would you solve this surface integral?

Suppose you had the surface integral $\iint \limits_{A} = x^{3}(1-x^{4}-y^{4})dx \ dy$ where $A$ is the region defined by $x \geq 0, \; y \geq 0, \; x^{4}+y^{4} \leq 1$. How would you solve this ...
2
votes
1answer
19 views

What does it mean “Distance between $k$-planes induced by the identification plane-projection matrix”?

I'm reading some parts of Functions of bounded variation and free discontinuity problems by Ambrosio, Fusco, Pallara. At the very beginning of page 82 there's written "Let $G_k$ be the complete ...