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

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Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
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2answers
47 views

If $\sum_{k=0}^{n}\binom nk=2^n$ then how is $2(\binom n0+\binom n2+\binom n4+…)=2^n$ [duplicate]

$$\sum_{k=0}^{n}\binom nk=2^n$$ then how is $2(\binom n0+\binom n2+\binom n4+...)=2^n$ ?? I don't think it could be because half of the members of the sum are chosen, that seems a bit intuitively ...
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0answers
14 views

How should the Calculus of Variations deal with $\delta(t-t_0)$ variations?

I'm familiar with using the Calculus of variations to find the condition for which first order variations of a functional wrt a function are zero: We start with a functional $J[x]= ...
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14 views

Gradient of the Fourier transform of a function

I am wondering if there is a simple way to express the first variation of the Fourier transform of a function as a function of said function. In other words, if $g:x\mapsto F(f)(x)$, where $F(f)$ is ...
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1answer
19 views

Lower semicontinuous energy functional on compact space of Lipschitz functions

Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open subset containing $0$ and let $L>0$ be some positive constant. Consider the space $A_{0}=\{f \in C^{\infty}(\overline{\Omega}) \mid f \text{ ...
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1answer
41 views

A maximization problem parametrized by a function

Let $f$ be a smooth positive monotonically increasing real function which is defined and finite in $[0,1]$, and define the following two quantities (see the figure below): $F=\int_{x=0}^1{f(x)dx}$ = ...
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16 views

Getting the minimum of a mixed functional

I have a functional $T$ defined on the attached picture. The functional always gives non-negative values. So it has a non-negative infinum I'm trying to figure out whether this infinum is ...
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weak lower semi-continuity of high oreder functionals [on hold]

Let Ω be a bounded open set in Rn. if we have the functional F(u)=(1/p)∫Ω|D2u|p dx defined in W(2,p) where 1
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34 views

Calculus Questions on Fibonacci and Length of Curve

Hi All, I have an issue trying to do part (i) and (ii). How do you go about doing it? As for the fibonacci sequence, I keep getting really big numbers, i can't seem to get the number of digits that ...
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35 views

Lagrange multipliers in Calculus of Variations

I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things ...
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2answers
34 views

Meaning of an Extremum of a Functional

Consider the following minimisation problem: $$\int_0^3\left(0.5\dot{x}^2-x\right)\,\mathrm{dt}$$ Subject to $x_0=0$ and $\dot{x}=0$. Using the Euler lagrange equation one can get: ...
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Direct method in the calculus of variations

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$. $$ \mathcal{F(u)} = \int_{\Omega}\frac{1}{2}|Du|^2dx $$ $$ u \in \mathcal{A}: = \{v \in W_{0}^{1,2}: \int_{\Omega}v^2dx = 1\} $$ Does this ...
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0answers
25 views

Variational optimization problem with several constraints

I am looking for solutions, approaches or hints to solve this variational optimization problem: Let $f:\mathbb{R}\rightarrow [0,\infty)$ be such that $\int f(x)\,dx=1$ and $\int x\,f(x)\,dx=0$ and ...
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2answers
42 views

Minimize a non-convex function subject to linear dynamics constraint

I want to solve the following problem: $$\min\limits_{\bf u} \frac{\bf c^T {\bf x} (T_f)}{\| \bf c\|\|{\bf x} (T_f)\|}$$ subject to $$\dot{\bf x} (t) = A {\bf x}(t) + B {\bf u}(t)$$ $$x(0) = x_0$$ ...
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8 views

How to optimize this types of problems?

Given that $min [ t_{f} - t_{0} ]$ such that $x(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $y(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $z(x'(t_{0}),y'(t_{0}),z'(t_{0}),t_{0}) = 0$ $x(t_{f}) = ...
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0answers
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Calculus of Variations: What if the functional is an integral with boundaries at infinity?

I am trying to grasp the basics of Calculus of Variations. The problem seems to be concentrated on functionals of the form : $$ F[y] = \int_{a}^{b} G(y,y(x),y'(x))dx$$ where $y(x)$ is assumed to be ...
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1answer
32 views

The Euler-Poisson equation

$$\int_{0}^\pi (x''^2+4x^2) dt$$ $$ x(0)=x'(0)=0; x(\pi)=0;x'(\pi)=sinh(\pi)$$ This is The Euler-Poisson equation, i found: $$\frac {\partial f}{\partial x}-\frac {d}{dt} \frac{\partial f}{\partial ...
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4answers
100 views

Use of $L^2$ norm in calculus of variations

I am trying to make an introduction to the calculus of variations. This field has many connections with functional analysis, in which I do not have an experience. I recently learned about function ...
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1answer
42 views

Dominated convergence theorem and fundamental lemma

this is a proof of the fundamental lemma of calculus of variation. Some preparations: Let $g(x):=e^{\frac{-1}{1-||x||}} \chi_{||x||<1},$ with characteristic function $\chi,$ then $$c:= ...
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Lagrange Multipliers with Calculus of Variations

We wish to extremize $$S = \int \mathcal{L}(\mathbf{y}, \mathbf{y}', t) dt $$ subject to the constraint $$g(\mathbf{y}, t) = 0 \;.$$ We move away from the solution by $$y_i(t) = y_{i,0}(t) + \alpha ...
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15 views

How is the functional differentiation derived?

I am trying to understand how the functional derivative is derived but I consistently fail to find a convincing resource explaining it. I want to understand how it is derived from the regular ...
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1answer
42 views

Why is the Lagrangian a function on the tangent bundle?

I understand that empirically the state of a dynamical system (at a given instant in time) is determined by specifying it's position and velocity, but I'm slightly unsure as to why the Lagrangian is ...
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1answer
42 views

variational problem with constraints

Let me bring to your attention the following problem. Suppose we have the functional $$ F = \int\limits_{a}^{b} f(y(x))\cdot\frac{dy}{dx} dx .$$ It is easy to see that that the Euler-Lagrange ...
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14 views

Energy functional for a differential equation

Is there a variational formulation for the following differential equation: $\frac{\partial}{\partial x}(D(u,x)\frac{\partial u}{\partial x})=0 $ $x$ varies over $[0,1]$, $D$ is bounded, is ...
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19 views

Fixed Length Catenary

Doing a fixed length catenary problem, why is it that adding the constraint $L=\int_A^B ds$ gives us more solutions. A little background: the catenary problem involves minimizing the integral ...
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3answers
118 views

Prove $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for smooth $g$ with $g(0)=g(1)=0$ [closed]

This came up in an optimization problem. How do you prove that $\int_0^1 ((g'(x))^2-1)^2dx \geq 1$ for any $g$ which is twice continuously differentiable on $[0,1]$ and such that $g(0)=g(1)=0$?
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1answer
39 views

Derivative of an Infinitesimal?

I am currently studying calculus of variations (for my classical mechanics course). I have, on multiple occasions, seen the derivative of an infinitesimal quantity defined like below $$\frac{d}{dt} ...
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1answer
30 views

Why am I getting two different answers to this simple calculus of variations problem?

A worker is disposing of radioactive material of mass $\mu$ and needs to minimize her exposure. Being near the radioactive material exposures her to radiation at a rate of $\frac {dE_n}{dt}=c\mu$, ...
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How is this simplication of the characteristic equation derived?

The characteristic equation in the calculus of variations (at least that's what my book calls it) is $$\frac {\partial F}{\partial y} - \frac{d}{dx}\frac{\partial F}{\partial y'}=0$$ where ...
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4answers
333 views

How can $y$ and $y'$ be independent in variational calculus?

In variational calculus, functionals are written as \begin{eqnarray} F = \int f(x,y,y') dx \end{eqnarray} Where $F$ depends upon choice of $y,y'$. But for smooth regular functions specifying the $y$ ...
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1answer
37 views

Directional derivative of $f=f(\nabla \cdot\mathbf{u})$

How do you evaluate the directional derivative of $$f=f(\nabla \cdot\mathbf{u})\tag{1}$$ I've tried this but I'm not sure that my answer is correct, here is my attempt: The definition of the ...
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22 views

What can we say about variational energies?

Suppose that $U \subset \mathbb{R}^d$ is open and let $V_{ij}^{kl}(r)$ $(1 \leq i,j,k,l \leq d)$ be functions on $V$ to $\mathbb{R}$ which are as smooth as the coming problem may require. For the ...
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2answers
34 views

Fastest approach path to a moving object

Say you have a point $A$ which's coordinates at the time $t$ are given by $(0, tv_0)$ for some constant $v_0$. You have another point $B$ with coordinates given by the function $x$ with ...
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1answer
33 views

Shortest path in the plane under derivative constraint

A colleague posed a toy problem to me today that degenerates to finding the curve y(x) of shortest length than connects two points in the plane (WLOG: y(0) = 0, y(a) = b), such that y'(0) = 0. This ...
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1answer
61 views

Why does the Hamilton Jacobi Bellman Equation imply Pontryagin's Minimum Principle

I'm having difficulty understanding the proof that allows us to go from the Hamilton-Jacobi-Bellman equation to to the Pontryagin Min(Max) Principle. Lets consider $x(t)$ and $u(t)$ as real valued ...
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Calculus of Variations for discrete functionals

Question: How does one determine the optimal function f that either maximize or minimizes: $$\int_{x_1}^{x_2} L\left(x, f, D_{h,x}[f ]\right) dx$$ Whereas: $$ D_{h,x}[f] = \frac{f(x + h) - ...
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maximising expected value with a variance constraint

suppose I have a portfolio, say the assets X_1, X_2 and Z where Z is risk free. Also these are all independent. Then I want to maximize $aE(X_1) + bE(X_2) + cE(X_2)$ subject to: (1) $a + b + c = 1$, ...
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derivative wrt to a function

Suppose $\phi(x+V\Delta t)$-$aV{^2}\Delta t$ is a function to be maximized w.r.t the function V which is a function of (x), $a$ and $\Delta t$ being scalar constants. Assuming $\phi()$ is ...
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3answers
54 views

Proof or counterexample : Supremum and infimum

If $($An$)_{n \in N}$ are sets such that each $A_n$ has a supremum and $∩_{n \in N}$$A_n$ $\neq$ $\emptyset$ , then $∩_{n \in N}$$A_n$ has a supremum. How to Prove This.
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The Initial and boundary conditions of a 2nd order nonlinear ODE

The problem is derived from: Original gradient index optics problem See the Figure above. $O:(0,0)$ is the disk center of light source $\odot{O}$ with radius $3$. Then the profile light rays of ...
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Optimize monotonic function in calculus of variations

I'm interested in the variational problem $$\min_{y} \int_a^b F(x,y(x),y'(x))dx \qquad \text{subject to} \quad -y'(x)\leq 0 \quad \forall x \tag{1}$$ i.e. $y(x)$ has to be monotonic. I ...
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Finding a function of minimal arclength via Euler Lagrange theorem, stuck on solving differential equation

I want to minimize the arclength of a function $u(x) \geq 0$ for $x\in [-1,1]$ that is contrained by $u(-1)=0=u(1)$ and $\int_{-1}^{1} u(x) \, dx = A$, where $0<A<\pi/2.$ I have reduced the ...
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What is the difference between $L$ and $\mathcal{L}$? How does one find the Lagrangian

I'm following a course of Lagrangian and Hamiltonian mechanics, but I'm getting somewhat confused. Could someone explain the difference between $L$ and $\mathcal{L}$? I'm calling both "the Lagrangian ...
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1answer
26 views

How to find direction of velocity V2 to reach an object travelling at velocity V1, such that it takes least time?

If an object A is currently at point P1 moving with constant velocity V1, and there is another object, object B which currently at point P2 which can move with velocity v2, then what should be the ...
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51 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and ...
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Hardy inequality punctured space

given the minimization problem: $inf \ \frac{\int_{\Omega} |\nabla u|^p }{ \int_{\Omega} \frac{|u|^p}{|x|^p} } ,\ \ p>1$ infimum taken on all smooth functions with compact support in the ...
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1answer
36 views

Difficult Problem on Calculus of variation

My problem is: consider the functional $J(y)=y^2(1)+\int_0^1 y'^2(x) dx$ and $y(0)=1$ where $y\in C^2([0,1])$. If y extremizes J then find $y(x)$ . Any Hint will be appreciated.
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1answer
86 views

$H^1$ convergence of eigenfunctions of Schrödinger operators

Consider the Schrödinger-Operator with Potential $V\in L^\infty(\Omega)$ with Dirichlet boundary conditions $$ H^D=-\Delta + V $$ and let $u_{i,n}\in H_0^1(\Omega)$ be the first, nonnegative ...
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35 views

Convergence in the distributional sense (mean field games dynamics)

I am trying to go through the papers by Gueant, Lions and Lasry on Mean field games. One of their examples is the Mexican wave (which happens in football stadia). Straight to the point the Lagrangian ...
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30 views

Partial Differentiation with respect to a functional

Suppose that there are two independent variables $x, y \in \mathbb{R}$. Define two functionals $f_i : \left(\mathbb{R} \times\mathbb{R}\right) \rightarrow \mathbb{R}, i=1,2$ \begin{align} ...