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

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1answer
9 views

Palais–Smale compactness condition

Can someone explain the essence of Palais–Smale compactness condition used in the Mountain Pass Theorem, in particular its weak formulation?
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0answers
4 views

$\Gamma$-convergence (Gamma-convergence) and PDEs?

My question is about the applying calculus of variations to solving Partial Differential Equations. In particular, what is the idea behind using $\Gamma$-convergence to find weak solutions of PDEs? ...
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0answers
18 views

Question about the following proof

Below is a proof of the theorem in Du's paper "Centrodial Voronoi Tessellations:Applications and Algorithms": I have the following questions about the proof. (1) To understand the proof better, I ...
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0answers
30 views

Finding an Extermal of Hard Examples?

Who Can show me the calculation for solving extermal for $$\int_0^1 (x^2+ \dot {x}^2+2xe^t) dt \quad \text{ when }\quad x(0)=0,\;x(1)=free.$$ My TA say a short answer and I Couldn't reach to ...
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1answer
40 views

Why using liminf instead of limsup?

In Chapter 8: Calculus of variations of Evan's Partial Differential Equations, Evan writes as follows: I am wondering about the last paragraph where he says that knowing $I[u] \leq ...
-3
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1answer
49 views

finding extermal on old exam questions? [on hold]

I ran into a question that wants to find Extermal of following function: $$\int_0^2 \frac{ \dot{x}^2}{x^3} dt \quad \text{ with }\quad x(0)=1,\;x(2)=4$$ who can help me how we can solve this old ...
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1answer
24 views

Why is this inequality true?

In Evan's Partial Differential Equations, he writes Then, he continues to write: But I do not understand how he gets $I[w] \geq \delta ||Dw||^q_{L^q(U)} - \gamma$. I tried to write it out and I ...
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0answers
14 views

About the definition of functional derivative and the $L^2$ inner product

There is something I do not understand well about the definition of the functional derivative. In the wikipedia page http://en.wikipedia.org/wiki/Functional_derivative it says: 1) This definition ...
-1
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0answers
11 views

Boundary conditions and Lagrange Constraints in Calculus of Variations

I am trying to learn about Calculus of Variations for some time now. In many problems, there are some boundary conditions defined, for example when we want to maximize a functional ...
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0answers
8 views

calculus of variations with free endpoint

I have a Lagrangian $L(x,\dot x)$ and want to solve $$\arg\min_{\gamma(t)} \int_0^\infty L(\gamma, \dot \gamma)\,dt$$ subject to holding only one of the endpoints fixed: $\gamma(0) = \gamma_0$. Now ...
-1
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0answers
19 views

the extremals of the functional with boundary condition

This question is about the extremals of the functional J using method of variation. But I know how to calculate the extremals, the exact question is slightly different and I have no idea what title is ...
1
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1answer
18 views

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
52 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
15 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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0answers
17 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 ...
3
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1answer
23 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{ ...
1
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1answer
45 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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0answers
18 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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0answers
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 ...
4
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0answers
38 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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0answers
51 views

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 ...
0
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0answers
29 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
43 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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0answers
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
15 views

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
36 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 ...
7
votes
4answers
102 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 ...
1
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1answer
46 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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0answers
23 views

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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0answers
16 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 ...
1
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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 ...
1
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1answer
43 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
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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0answers
20 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
120 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
40 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} ...
1
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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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0answers
22 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 ...
13
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4answers
341 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 $y$ ...
1
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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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0answers
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 ...
2
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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 ...
2
votes
1answer
62 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
19 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
24 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
32 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
55 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
21 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 ...