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

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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 ...
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51 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} \mathbb{...
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140 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) - \frac{\...
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2answers
49 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) = y'\frac{d}{dx}\left(\frac{\...
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2answers
54 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 $\dfrac{da}{dx}=...
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2answers
98 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 $$\delta g=gg^{\mu\nu}\delta g_{\mu\nu}$$ ...
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44 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 ||\...
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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 u^2-nu^2](4\pi\tau)^{-n/...
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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 \\ i'(0)=\...
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55 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, ...
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43 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 / ...
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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 ...
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167 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 ...
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1answer
37 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 ...
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66 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 $T[u_1]=T[...
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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: $\frac{1}{2}(y\sqrt{x^2+y^...
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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 ...
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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\...
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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: U\...
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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 didn'...
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1answer
124 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 ...
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55 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 $s\in [0,\infty)$, i....
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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. ...
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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 ...
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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: $\Delta J = J(y + \epsilon h)-J(y) = J(x + \epsilon x^2) - J(x)$ ...
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1answer
38 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 $\|...
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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 $ y(x)\...
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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 ...
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1answer
92 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 ...
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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 $f$ ...
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1answer
85 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)) + ||\nabla_x\hat{y}...
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1answer
40 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 $(X,\...
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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 ...
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89 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 ...
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41 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 ...
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1answer
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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 ...
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Can't understand where the second term in this derivative comes from

Given an integral $$S=\int_{\theta_1}^{\theta_2}q(p,t,\theta)f(\theta)\:d\theta $$ My understanding is that I can partially differentiate this w.r.t., say, t, in the following way: $$\frac{\partial ...
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A homogeneous double integral equation

I've happened to stumble upon an interesting double integral equation: $$ 0=\int_0^\ell\int_0^\ell f(s,t)\mu(t)\mu^\prime(s)\,ds\,dt $$ Here $f$ and $\mu$ are at least continuous (if you want higher ...
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Minimizing distance between two curves. Can the Calculus of Variations be used?

Given two curves, one might want to find the minimum distance between two points. It is fairly straightforward to find minimums of the function $$(x_1-x_2)^2+(y_1-y_2)^2$$ which corresponds to the ...
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1answer
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Conceptual problems when minimizing a simple functional

I have a problem with what seems a very simple functional maximization. Let's define: $$ J[z]=\int \left( u(z)-\frac{\dot z^2}{2} \right) dt $$ Where $u(z)=-z^2+5$. The problem is to find $$ \arg\...
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variational calculus with probabilistic boundaries

I'm interested to find the solution to the following variational problem: $$ J[y]=\int_{T=0}^{\infty}\int_{t=0}^{T}L(t,y(t),y'(t))p(T)dtdT $$ where $p(T)$ is a probability distribution function ...
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Find the variation of this integral functional

Find the variation of $A(Y) = 2\pi\int \limits _0 ^1 |(Y(x)|\sqrt{1+Y'(x)^2} \Bbb dx$ with respect to $Y$. I have no idea how to solve this problem.
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$\text{min }\frac{1}{2} \int (f')^2$ in $C^1([0,1])$ given both D and N boundary conditions.

Does there exist a minimizer in $C^1([0,1])$ (or $H^1([0,1])$) for $$\frac{1}{2}\int (f')^2 dx, \text{ given the boundary conditions: } f(0)=0, f(1) = a, f'(1) = b?$$ When $a=b$, we have the ...
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Sturm-Liouville Variational Problem

I'm entirely clueless with this problem. No formal training in variational methods. Show that for function $\phi\left ( x \right )$ with $$\phi\left ( a \right )=\phi\left ( b \right )=0$$ and ...
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43 views

Differentiation under an integral with respect to a function

Consider the functional $F$ defined via the integral $$ F(\mu)=\int_0^\ell\int_0^\ell f(s,t)\mu(s)\mu(t)\,ds\,dt. $$ How would I differentiate this with respect to $\mu$? I realize that this has ...
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Are these critical points minima to the variational problem?

Let $\Omega\equiv (0, 1)\times(0, 1)\subset\mathbb{R}^2$ and consider the variational integral \begin{equation*} I[u]\equiv\int_{\Omega}\frac{1}{2}|Du|^2\ \mathrm{d}x-\frac{5\pi^2}{2}|u|^2\ \mathrm{d}...
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2answers
42 views

Variation Problem with Euler-Lagrange Differential Equation

I'm just trying to understand that type of equations, and I can't solve this, kind of a simple minimization problem. Maybe someone can help me ? Here is my equation $$\eqalign{ & A(y(x)) = \...
2
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1answer
71 views

Energy functional and Euler Lagrange equation

We know that for potential energy functional, its derivative is called the Euler Lagrange equation and physically, it means that at the given point there is a force balance. Now if the energy ...
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38 views

Time it takes a particle of $2x^2+3y^2+4z^2=9$ to reach the xy plane using derivatives.

Consider in $\mathbb{R}^3$ the surface $2x^2+3y^2+4z^2=9$. Suppose a particle leaves the point $(1, 1, 1)$ located in the surface along the normal at that point to the $xy$ plane at a speed of one ...
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9 views

Rewriting the reparametrization of a function as a sum

Say I have some function $$\mu:[0,\ell]\to\mathbb{R}^+,$$ and a bijective function $$\phi_\varepsilon:[0,\ell]\to[0,\ell].$$ Let $s$ represent the variable in $[0,\ell]$ and say that $\varepsilon\in(...