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

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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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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
252 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
78 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
31 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
160 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} = ...
3
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1answer
488 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
110 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
47 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
95 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
33 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
131 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 $J$ ...
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1answer
226 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 ...
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41 views

Regularity questions in constrained variational problem

Consider the problem of minimizing $$ I(u) = \int_a^b F(t,u(t),u'(t)) d t $$ over, say, $W^{1,\infty}(]a,b[)$. Then regularity theory tells us that if $F$ and $F_{\dot q}$ are $C^k$, and in addition ...
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0answers
26 views

Maximum value for $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt$?

This is what I have done thus far: $F(y)=\int_0^1[y'\sin(\pi y)-(y-t)^2]dt=-\frac{1}{\pi}\int_0^1[(\cos(\pi y))\frac{d}{dt}]dt-\int_0^1(y-t)^2dt$ (as $-\frac{1}{\pi}[(\cos(\pi ...
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1answer
43 views

Measure of surface quadrature

In the article you can find at http://www.cc.gatech.edu/~turk/my_papers/schange.pdf, precisely at page 2 of the .pdf, there is a functional E which is said to be a measure of the aggregate squared ...
3
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1answer
153 views

Dido's problem with Euler equations

I'm considering Dido's problem: Consider 2 differentiable arcs $C$ and $C_0$ in $\mathbb{R}^2$ from the point $P$ to $Q$ and back. We keep $C_0,P,Q$ fixed, and want to choose the arc $C$ such that ...
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1answer
76 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
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1answer
35 views

A calculus of variations equation

Let $L$ be as smooth as needed a function of the arguments $(q_1,\dots,q_n,\dot q_1,\dots,\dot q_n,t)$, where the dot denotes the derivetive with respect to $t$. Let $\delta$ denote the variation of ...
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1answer
33 views

Variation equations and integral invariants

I am reading Whittaker's Analytical Dynamics. This is chapter 10 Hamiltonian Systems. Paragraph 112 is Variation Equations. Let ${dx_r\over dt}=X_r(x_1,\dots,x_n,t),\quad (r=1,\dots,n)$ be a system ...
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0answers
60 views

The equvalence of the virtual work and the Hamiltonian equations

I am reading Whittaker's Analytical Dynamics. This is chapter 10 *Hamiltonian Systems&. Paragraph 109 is Hamiltonian Systems & Their integral invariants. Whittaker starts with the Lagrangian ...
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1answer
66 views

Deriving Hamiltons principle for conservative holonomial systems

I am reading Whittaker's Analytical Dynamics. This is chapter 9 Principle of least action and least curvature. The paragraph is 99 Hamilton’s principle for conservative holonomial systems. Let us ...
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2answers
54 views

Existence of solution in Hölder spaces

Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space. My question ...
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1answer
56 views

Show that $ M$ is constant on $[a,b]$ (variational calculus)

Let $F:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ be $C^2$ on $[a,b]$ and $u$ be a solution for the Euler-lagrange equations for the functional given by $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
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1answer
120 views

Determine the minimum and maximum values of an integral subject to end conditions

The question: determine the minimum and maximum values of the integral $$\int_0^1 yy'dx$$ subject to the conditions $y(0)=0$ and $y(1)=1$. There is no explicit y dependence, so our Euler-Lagrange ...
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1answer
85 views

Find the Minimum value of a Functional Constrained to end-point Conditions

The question: Find the minimum value of $\int_0^1 y'^2 dx$ subject to the conditions $y(0)=y(1)=0$ and $\int_0^1y^2dx=1$. In another question, I proved that, if we have an integral of the form ...
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2answers
75 views

Euler-Lagrange problem solution

Hi, Can anyone solve this question? I have no clue.
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1answer
571 views

Euler lagrange equation solving

Find the Euler-Lagrange equation for the functional $$I(y) = \int_0^1(py\,'\,^2-qy^2)\mathrm dx$$ subject to the constraint $$\int_0^1ry^2 = 1.$$ Answer: $\frac{d}{dx}(py') + (q-\lambda r)y = 0$. ...
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101 views

Find the Euler-Lagrange equation of $F=p(x)y'^2-q(x)y^2+2f(x)y$

The question statement: obtain the Euler-Lagrange equations associated with extremizing $\int_a^b F\, dx$. Our F is $$F=p(x)y'^2 -q(x)y^2 +2f(x)y$$ I haven't used the EL equations for some time, ...
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1answer
68 views

Piecewise $C^1$ function is element of $W^{1,\infty}$

Hey I'm confused about the following (apparantly) fact: Let $u:[a,b]\to\mathbb{R}$ a piecewise $C^1$ function, i.e. there exists $a=t_1<t_2<\cdots < t_n = b$ such that $u|_{[t_i,t_{i+1}]} ...
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1answer
29 views

On the controllability function (minimising a functional)

Consider a system of ODEs $$\dot{x}(t)=f(x(t))+g(x(t))u(t),$$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ and $g:\mathbb{R}^n\to\mathbb{R}^{n\times m}$ are smooth. Let $L:\mathbb{R}^n\times ...
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0answers
55 views

Optimal parametric curve via calculus of variations

Assume we have $N+2$ points in $\mathbb R^2$, $(x_0,y_0),(x_1,y_1)\dots(x_N,y_N),(x_{N+1},y_{N+1}),$ where $(x_0,y_0)=(x_{N+1},y_{N+1}).$ We would like to find a pair of functions \begin{cases} x = ...
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1answer
349 views

Why Euler-Lagrange equation does not depend on the second derivative of the function?

Why Euler-Lagrange equation does not depend on the second derivative of the function? I.e. why it's $L[q, \frac{dq}{dt}]$ but not $L[q, \frac{dq}{dt}, \frac{d^2q}{dt^2}]$, neither not $L[q, ...
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1answer
75 views

Calculus of Variation: small variation in functions

I am reading a mathematical physics book, and I am trying to follow along. In the section about functionals ( $J[y] = \int_{x_1}^{x_2} f(x,y,y',\ldots,y^n)$ ), they let $y(x) \rightarrow y(x) + ...
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2answers
259 views

fundamental lemma for variational calculus

Is it possible to use the fundamental lemma of calculus of variations in some way in the following case: $F(x,y)$ is a locally integrable function on $\mathbb{R}^n \times \mathbb{R}^n$. We know that ...
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1answer
47 views

how to solve this optimization problem with functions?

Suppose $\theta\in[0,1]$ and $s(\cdot)$ is strictly increasing in $\theta$ with $0\leq(0)<s(1)\leq1$ and $s(0)+s(1)>1$ How could I solve the program, $\max_{s(\cdot)\in ...
3
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1answer
63 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \smallint_a^b f(x,y,y')dx$$ & find it's Euler-Lagrange equations $$\tfrac{\partial f}{\partial y} - \tfrac{d}{dx}\tfrac{\partial f}{\partial ...
2
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1answer
58 views

Is a geodesic the least curved path?

It is clear that, in $\mathbb{R}^n$, straight lines are the lines with minimum possible curvature. That is, given the Frenet-Serret ($n$-dimensional equivalent) matrix, and taking its squared norm, ...
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64 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
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2answers
345 views

Moment of inertia about center of mass of a curve that is the arc of a circle.

Let $(x(s),y(s))$ be a smooth 2-d plane curve which is an arc of a circle of a certain radius $r$. Assume it is represented by an inelastic string $S$ of finite length, lying in a 2-d plane. Let there ...
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0answers
106 views

A maximization problem in Sobolev space

For $k>0$, let $f_k$ be a sequence of positive functions in $H_N^1(0,1)$, where $H_N^1(0,1):=\{u\in H^1(0,1)|u^{'}(0)=0=u^{'}(1)\}$, $H^1(0,1)$ is the usual Sobolev space consisting of $L^2(0,1)$ ...
2
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2answers
63 views

How does one evaluate this function of several variables?

In deriving the Euler-Lagrange equation, one step involves evaluating this: $$\frac{\partial f(y(x)+\alpha\eta(x), y'(x)+\alpha\eta'(x), x)}{\partial \alpha}$$ (this is from pg. 220 of 'Classical ...
2
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1answer
40 views

Lagrangian's arguments only up to the first derivative

A question related to a previous question I've asked. I am wondering why in QFT the arguments of the Lagrangian only go up to the first derivative? I remember hearing someone mention that it has to ...
0
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0answers
115 views

Linear programming with countably “infinite variables” and “finite constraints”!

Is it possible to do a linear programming with countably "infinite variables" and "finite constraints"? If not, what do you purpose? (Example Link): Maximum and minimum of an integral under integral ...
0
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1answer
140 views

first variation of function defined by an integral

Let $f$ be a function defined by $f(x) = \int_0^x \sin \phi(t) dt$. What is the first variation $\delta f(x)$ and how it is calculated?