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

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1answer
67 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
66 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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0answers
27 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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0answers
42 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
127 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
52 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
34 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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0answers
26 views

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
378 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
44 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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0answers
24 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
35 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 ...
2
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1answer
37 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 ...
2
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1answer
117 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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0answers
26 views

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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0answers
27 views

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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0answers
38 views

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
58 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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0answers
32 views

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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1answer
52 views

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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1answer
48 views

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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0answers
29 views

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
78 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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0answers
62 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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0answers
33 views

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
45 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.
3
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1answer
96 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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0answers
39 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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0answers
35 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} ...
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1answer
26 views

The shortest line problem extremum extinction

Let's consider a functional $J(y)=\int_{0}^{1}{ \sqrt{1+\frac{dy}{dx} ^{2}} dx}$, $y(x_{1})=y_{1}, y(x_{2})=y_{2}$. Using Euler-Lagrange equations we can get that the set of functions which attains ...
1
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1answer
53 views

Integration of partial derivative $\frac{dL}{dq}$ with respect to $t$ where $q$ is implicitly a function of $t$

Is $\int_{t1}^{t2} \frac{\partial L}{\partial q}\delta{q} dt$ equal to $\left[\frac{\partial L}{\partial \dot{q}}\delta{q}\right]_{t1}^{t2} $ if $q$ implicitly depends on $t$ ? If not I ...
4
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1answer
107 views

Is there any situation in which a geodesic maximize the path length between two points?

Some people (even in here) claim that geodesics are, in general, stationary curves. Locally speaking, geodesics always minimize arc length (see Manfredo, for example). But I can't visualize a surface ...
6
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0answers
174 views

Euler-Lagrange Equation and “Eigen Value ”

The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable): $$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$ ...
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19 views

Variational calculus on integrals and derivatives

I am studying mechanics but am a novice in variational calculus. While reading a book on Lagrangian mechanics, I blocked when the author states that by calculating the variation of the following ...
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17 views

Stable geodesics

Consider a function defined on some space of smooth curves in a manifold (think of the "action functional"). I understand what a "critical point" of such a function is, but what is understood by a ...
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1answer
79 views

Solving a functional problem with break points

I'm totally out with solving functional problems when we need to find broken extremals, can someone show me how to solve the following problem: Can the following problem have break (corner) ...
2
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1answer
67 views

Find maximum $\int^{1}_{0}\{f(x)\}^3dx$

I would appreciate if somebody could help me with the following problem: Question: Find maximum $\int^{1}_{0}\{f(x)\}^3dx$ when (1). $f(x) : \text{conti-and} \int^{1}_{0}f(x)dx=0$ (2). $-1\leq ...
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0answers
65 views

Understanding derivation in Calculus of Variations book

I'm reading about Calculus of Variations and about the general variation of a functional. I bumped into few obstacles in my book I can't get over with. I have scanned the pages where I have my ...
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1answer
34 views

A little confusion about calculating variance?

given $a = 10% $, $b = 50%$, $c = 25%$ and $d = 15% $ calculate the variance? ( this is a made up question to make thinks clear for me) what i have done is 1st eliminate percentages? then i ...
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1answer
31 views

Harmonic solutions

Assume that $\Omega\subset R^2$ is an open bounded set with a smooth boundary, $g:\partial\Omega\to R$ is a continuous map and $\{b_i \ | \ i=1,2,\ldots,d\}$ is a finite subset of $\Omega$. ...
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0answers
13 views

Minimal distance in expectation problem

Two random variables $\tilde{X},\tilde{Y}$ have the joint distribution $F:X\times Y\to [0,1] $ The given function is $g^*=(g_1^*,g_2^*):X\times Y\to R^2$ The control function is ...
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0answers
73 views

Second variation of the domain functionals.

I am reading a paper which deals with the second variation of the domain functionals and applications . The following problem is solved . The eigen value $\lambda(t)$ which is characterised by the ...
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116 views

solving euler-lagrange equation in constrained functional optimization

The problem to solve is the minimization of a functional of two functions, $F(y,z) = \int_a^b f(y,z)dx$ , subject to a constraint $g(y,z,y',z') = 0$. The augmented functional is then $L(y,z,y',z') = ...
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1answer
32 views

Finding function that maximizes ratio of area to length

I'm new to variational analysis, so I need someone to check, if I'm going in the right direction. Let's say I need to find a curve $y(x)$ with $y(0) = 1$ and $y(1) = 0$ that maximizes ratio of area ...
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1answer
61 views

Is this functional well defined?

I came across the following formulation of the problem. Minimize the functional $L[u]$ given by $L[u] = \int^b_a \sqrt{(1+(u'(x))^2}$ over $U = \{u\in C([a,b])\cap C^1((a,b)):u(a)=\alpha, u(b) = ...
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0answers
21 views

Checking Weierstrass's condition in a functional problem

My problem is to check for the Weierstrass's (necessary) condition in the following functional problem in my Calculus of Variations course: $$\min J[y]=\int_0^1xy'^3\;dx,\;\;\;\;y(0)=0,\;y(1)=1$$ ...
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0answers
34 views

Determine whether the extremals of a functional are weak or strong?

Is there any thumb rule for determining whether the extremals of a functionals (which are determined using Euler Lagrange equations) are maximum or minimum, weak or strong. I read about Weierstrass ...
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24 views

Numerical optimization in function space

I'm new to calculus of variations. I'm curious about how to apply simple numerical optimization techniques in function space. Consider the classical problem: finding the shortest path between two ...
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27 views

Prove a formula after change of variable?

If I have a change of variable $(x,u)\to (X,U)$ given by $$X=x+\epsilon u,U=u-\epsilon u.$$ How to prove the formula $$\frac{\partial U(X,0)}{\partial \epsilon}=\phi(X,u(X))-u'(X)\xi(X,u(X)),$$ where ...
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1answer
44 views

The minimizing problem over a sequence of shrinking balls

Let $B(0,r)\subset \mathbb R^3$ be a ball centered at $0$ with radius $r$. Define $$ \mathcal A_r:=\{u\in H_0^1(B(0,r)),\,\,\|u\|_{L^{q+1}}=1\}$$ where $1<q<5$. Hence we know that each ...