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

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Calculus of variations: the inside function has an integral

It is known that if the functional $$J=\int_a^b L(x,f(x))dx \tag{1}$$ has an extremum, then the Euler equation $\frac{\partial{L}}{\partial{f(x)}}=0$ holds. My question is, for example, what if ...
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1answer
47 views

how to handle a gradient expression

How to prove in a rigorous way that: $$|u|=1 \implies \nabla|u|^2 = 0 \implies (\nabla u)^Tu=0$$ and then $\forall v$ $$\nabla u : \nabla((u.v)\cdot u)= |\nabla u|^2 (u \cdot v)$$
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1answer
49 views

Find the derivatives to transformed variables

Let $\theta \in \mathbb{R}$ and consider the rotational action $X = x \cos\theta - y \sin\theta$ ; $Y = x \sin\theta + y \cos\theta$. Find the transformed derivatives $Y'$ and $Y''$. How do I ...
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1answer
44 views

a variational problem involving $L^p$ norm

Is there any way to prove that there are only finitely many maximizers to the following variational problem: $$\max\left\{||f||_{L^p[0,1]}-\int_0^1(f'(t))^2dt\right\}$$ over all functions $f$ which ...
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1answer
112 views

Euler Lagrange equation derivation and application of the fundamental lemma of the calculus of variations

Say we have: (1) $J(x) = \int_{\textit{to}}^{\textit{tf}} g(x(t),\dot{x}(t),t) dt$. We go through the general derivation and arrive at: (2) $\delta J(x,\delta x) = ...
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5answers
984 views

Can this ant find its way back to the nest?

So the puzzle is like this: An ant is out from its nest searching for food. It travels in a straight line from its nest. After this ant gets 40 ft away from the nest, suddenly a rain starts to ...
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1answer
133 views

Maximum Entropy (The existence of a Calculus of Variations problem)

Take maximum differential entropy as an example: Gaussian achieves the maximum differential entropy when the second order moment is fixed. The calculus of variation form: \begin{equation} ...
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25 views

Optimization problem for integrable functions

For the following optimization problem: find the extremal values of $$ I(x) = \int_a^b F(t,u,x) dt$$ where $x:[a,b]\rightarrow\mathbb{R}$ is a continuous function and $u$ is the primitive of $x$, ...
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0answers
48 views

Euler-Lagrange equation: no fixed endpoints

My aim is to maximize the objective $J(f) = \int_{0}^{\infty}{ F(f(x),x) p(x) dx}$, where $p(x)$ is a fixed probability density. However, the endpoints are not fixed since the class of functions I ...
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0answers
50 views

Lemma of calculus of variation for Green's function!

OK, I know the title is fundamentally wrong! But I guess you know where I'm going with it! Basically, I'm wondering if it's possible to prove that if $\int_{x\in\Omega}G(x,x')h(x')dV=0$ for any ...
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0answers
31 views

Derivative of infimum in variational problem

Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$. In the expression $$\frac{\partial}{\partial \alpha} \inf_{\phi ...
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0answers
23 views

Exchange limit and infimum in variational problem

Let $\{\mathcal{E}_n\}_{n\in \mathbb{N}}$ be a sequence of functionals over the same domain $\mathcal{D}$. What are sufficient conditions on the sequence and possibly the domain such that ...
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1answer
48 views

Prove differentiability of functional.

In $C[0;1]$ space let's consider following functional: $$\phi(f) = \int_{0}^{1}(1+f(t))^{3}dt.$$ Prove differentiability of $\phi$ and find $\mathrm{D}\phi(f)$ for: $f(t)=0$, $f(t)=t$, $f(t)=\cos t$. ...
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1answer
73 views

Euler-Lagrange equation: integral over positive real line of a perturbed functional

My goal is to minimize the functional $I[f] = \int_{0}^{\infty}{L(x,f(x),f'(x)) e^{-x} dx }$ However, the solution of the Euler-Lagrange equation is usually stated as minimizing a functional of the ...
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0answers
53 views

Pontryagin principle: does the abnormal multiplier define a minimum

The Pontryagin principle PM provides the necessary condition for a local minimum of the functional $ J(u)=\int L(x(t),u(t))dt \\$ subject to: $\dot x = f(x(t),u(t)) \ \ \ \ x(t0)=x0, \ \ ...
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0answers
275 views

Using Rayleigh-Ritz Method to approximate solutions to extremum problem.

I know how to use the Rayleigh-Ritz method when given a sturm-Liouville problem. But I am not sure where to start when asked questions like the one above. I know i need to plug the trial functions ...
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66 views

Transformation laws for functional derivative with different sets of variable?

We consider a one-to-one transformation $ x \rightarrow x' $, for scalar functional F(x) and F'(x'), we expect the simple transformation behavior $$ F(x) = F'(x'(x))$$ If we set $ x = {\rho, ...
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1answer
107 views

How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
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2answers
123 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
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1answer
46 views

Euler-Lagrange on restricted set

I am reading a chapter of Evan's book on weak convergence methods for nonlinear PDE's p.49 and it states that the Euler Lagrange equation for the functional \begin{equation} I[w]:=\int_U|Dw|^2 ...
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20 views

Variations Theory

I would like to know like what we have about an increment of a function of an independent variable x which is as follows: What would be correct about functional? I think if we have a functional ...
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1answer
74 views

Variational calculus applied to the strain energy functional in solid mechanics

The question is basically about when to apply the variational operator... Given the general functional representing the strain energy of a solid under a given stress state $\sigma$ and strain state ...
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1answer
152 views

Differential Equations for a Teardrop Shape

My research has led me to a nonlinear system of differential equations which should yield a teardrop shape in the $x-y$ plane. The equations, parameterized by $t$ are ...
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1answer
32 views

Variational Principle for an Elliptic equation

I wish to find the functional whose minimisation yields the follwoing equation on the vector function u $(\lambda + \mu) \nabla (\nabla \cdot u) + \mu \nabla^{2} u = 0$, the Navier equation of linear ...
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45 views

Euler-Lagrange Equation for a functional involving symmetric gradients

I struggle to compute the Euler-Lagrange equation for the following functional $\int_{\Omega} (\nabla^{s} u) D \nabla^{s} u \mathrm{d}\Omega$, where u is a vector valued function u = (u1 (x,y), u2 ...
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1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
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1answer
221 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points A(3,9) and B(-2,4) lie on the parabola y=x^2. The line y=x+6 joins A and B. The ...
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0answers
92 views

Calculus of variations: Lagrange multipliers with functions depend on only one variable

The problem: \begin{align} \min & \iint k(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2 \tag{1}\\ \mathrm{s.t. } & \iint h(x_1,x_2,y_1,y_2) \, dx_1 \, dx_2=l \tag{2}\\ & g(x_1,x_2,y_1,y_2)=0 ...
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1answer
73 views

Is this functional differentiable?

A functional $\Phi$ is differentiable if there exist $F$ and $R$ such that $\Phi(f+h)-\Phi(f)=F(f,h)+R(f,h)$, where $F$ depends linearly on $h$ and $R(f,h) = O(h^2)$. Define a functional $\Phi(f) = ...
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0answers
29 views

Multivariable transversality conditions for infinite time horizon in variational calculus?

I would like to find the optimal paths of $d[t]$ and $k[t]$ that maximize the following function: $$J(x)=\int_0^\infty e^{-pt}(d[t]-d[t]^2-d'[t]^2+k[t]-k[t]^2-k'[t]^2+d[t]k[t])$$ Using Mathematica ...
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1answer
36 views

Finding extremal of function J

Find a curve passing through $\left(0,0\right)$ and $\left(1,1\right)$ that is an extremal for the functional $\displaystyle{{\rm J}\left(x,y,y'\right) = \int\left\{\left[y'(x)\right]^{2} + ...
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0answers
89 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
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23 views

Vectorial derivative

Let's say that I want to perform the functional derivative of a scalar functional $F$, for example $F_{d}=\int ...
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0answers
31 views

Solve an integral equation

How can I solve the following problem for $s(x,T)$, $s(\cdot)$ is continuous and strictly increasing in $x$. $\int_0^T\frac{-x^3+x^2-T(1-T)x}{(s(x,T)-x)^2}dx=0$ s.t. $s(T,T)=T$
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0answers
33 views

Maximize an integral with variable end points

How can I find optimal $s(\cdot)$ in this problem (I have little knowledge about calculus of variation) Thanks. $$ \large{\max_{s(\cdot)} \int_0^{\theta^*} ...
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1answer
251 views

Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = ...
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0answers
77 views

Calculus of variations with constraints (multiple integrals)

I want to find the local extreme of the functional $$\iint f(x_1,x_2,y_1,y_2) dx_1 dx_2, \tag{1}$$ subject to the constraints $$\iint h(x_1,x_2,y_1,y_2)dx_1 dx_2=0, \tag{2}$$ $$g(x_1,x_2,y_1,y_2)=0, ...
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0answers
30 views

Optimal form a differential equation

Suppose I have an objective function: $min(-\int_0^T((C*x_2 + f(x_1))*V) dt$ with the constraints $\dot x_1 = x_2 $ and $ \dot x_2 = -{C\over m} *x_2 -{f(x_1)\over m}-A $ The standard base ...
3
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1answer
153 views

Euler Lagrange equation for harmonic maps

In the paper "The existence of minimal immersions of 2-spheres" by Sacks and Uhlenbeck the authors claim that the Euler Lagrange equation for the modified functional $E_\alpha(s) = \int_M (1 + ...
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1answer
19 views

What's the derivative $\frac{dE[F]}{dF_i}$ of this function $E[F] = |\frac{dF}{dx}|^2$?

The function is $E[F] = |\frac{dF}{dx}|^2$ where we take $\frac{dF}{dx}$ to be the discrete derivative defined by $F_{i+1} - F_i$. Could someone walk through why $\frac{dE[F]}{dF_i} = ...
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1answer
444 views

Calculus of variations: Lagrange multipliers

Given a functional $$J(y)=\int_a^b F(x,y,y')dx, \tag{1}$$ where $y$ is a function of $x$, and a constraint $$\int_a^b K(x,y,y')dx=l, \tag{2}$$ if $y=y(x)$ is an extreme of (1) under the ...
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2answers
109 views

Natural growth conditions and weak solutions for inhomogenous systems.

Let $\Omega\subset \mathbb{R}^n, \ n\geq 2$ be a bounded Lipschitz domain. Consider the following inhomogenous system (in divergence form) subject to zero Dirichlet boundary conditions: ...
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1answer
40 views

Calculation mistake in variation of length functional?

This should be pretty simple to check if you know the basics of variational calculus. I feel like I am making an obvious mistake somewhere like not using chain rule somewhere. Let $g : \mathbb{R}^n ...
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1answer
69 views

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. [duplicate]

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. I do not know how to do it.
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1answer
92 views

Calculus of Variations: Mul-variable-mul-function

My question is: How to find the necessary condition for minimizing/maximizing the functional $$J(f,g)=\int\int_{R}F(x,y,f(x),g(y))dxdy,~~~~~~~~~~(1)$$ where we have two functions $f(x)$ which only ...
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1answer
30 views

Derivation of weak form of Euler Lagrange Equation

In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional: \begin{equation} ...
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32 views

Conserved quantity in a time independent constrained lagrangian system

I've to find a conserved quantity in a constrained lagrangian system which is time independent. So the lagrangian is given by $L(q(t),\dot{q}(t))$. In the a unconstrained system this is clear to me.
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1answer
108 views

Calculus of variations minimum

I have a question that asks: Find the extremal of the functional $$J(x)=\int^{\pi}_02x\sin(t)-\dot x^2 dt$$ with $x(0)=x(\pi)=0$. I found $x(t)=\sin(t)$ It then asks to Show that this ...
2
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1answer
128 views

Finding critical points of functional (Euler equations)

Consider for $T>0$ the functional $$u\mapsto J(u) := \int_0^T (\dot{u}(t)^2-u(t)^2)dt. $$ on the space $W_0^{1,\infty}((0,T),\mathbb{R})$. (a) Depending on $T$, find the critical points of $J$ ...
4
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1answer
225 views

Existence of a Lagrange multiplier (Euler Lagrange equations + holonomic constraints )

Let $I=[a,b]\subset \mathbb{R}, G:\mathbb{R}^n\to \mathbb{R}^k$ smooth, $0<k<n, M=G^{-1}(0)$. Assume that $DG(x)$ has full rank for all $x\in M$. Fix $p_1,p_2\in M$ and assume $u\in ...